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Thursday, June 30, 2016

Rubik vs. Galois: Preconception vs. Pre-conception

Filed under: General,Geometry — Tags: Bion, Rubik Adoration — m759 @ 1:20 pm

From Psychoanalytic Aesthetics: The British School ,
by Nicola Glover, Chapter 4 —

In his last theoretical book, Attention and Interpretation (1970), Bion has clearly cast off the mathematical and scientific scaffolding of his earlier writings and moved into the aesthetic and mystical domain. He builds upon the central role of aesthetic intuition and the Keats's notion of the 'Language of Achievement', which

… includes language that is both
a prelude to action and itself a kind of action;
the meeting of psycho-analyst and analysand
is itself an example of this language.^29.

Bion distinguishes it from the kind of language which is a substitute for thought and action, a blocking of achievement which is lies [sic ] in the realm of 'preconception' – mindlessness as opposed to mindfulness. The articulation of this language is possible only through love and gratitude; the forces of envy and greed are inimical to it..

This language is expressed only by one who has cast off the 'bondage of memory and desire'. He advised analysts (and this has caused a certain amount of controversy) to free themselves from the tyranny of the past and the future; for Bion believed that in order to make deep contact with the patient's unconscious the analyst must rid himself of all preconceptions about his patient – this superhuman task means abandoning even the desire to cure . The analyst should suspend memories of past experiences with his patient which could act as restricting the evolution of truth. The task of the analyst is to patiently 'wait for a pattern to emerge'. For as T.S. Eliot recognised in Four Quartets , 'only by the form, the pattern / Can words or music reach/ The stillness'.^30. The poet also understood that 'knowledge' (in Bion's sense of it designating a 'preconception' which blocks thought, as opposed to his designation of a 'pre -conception' which awaits its sensory realisation), 'imposes a pattern and falsifies'

For the pattern is new in every moment
And every moment is a new and shocking
Valuation of all we have ever been.^31.

The analyst, by freeing himself from the 'enchainment to past and future', casts off the arbitrary pattern and waits for new aesthetic form to emerge, which will (it is hoped) transform the content of the analytic encounter.

29. Attention and Interpretation (Tavistock, 1970), p. 125

30. Collected Poems (Faber, 1985), p. 194.

31. Ibid., p. 199.

See also the previous posts now tagged Bion.

Preconception as mindlessness is illustrated by Rubik's cube, and
"pre -conception" as mindfulness is illustrated by n×n×n Froebel cubes
for n= 1, 2, 3, 4.

Suitably coordinatized, the Froebel cubes become Galois cubes,
and illustrate a new approach to the mathematics of space .

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Sunday, June 26, 2016

Common Core versus Central Structure

Filed under: General,Geometry — Tags: Apple Tree Children, Bion — m759 @ 2:56 pm

Rubik's Cube Core Assembly — Swarthmore Cube Project, 2008 —

"Children of the Common Core" —

There is also a central structure within Solomon's Cube —

'Children of the Central Structure,' adapted from 'Children of the Damned'

For a more elaborate entertainment along these lines, see the recent film

"Midnight Special" —

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Tuesday, June 14, 2016

Model Kit

Filed under: General,Geometry — Tags: Knight Move — m759 @ 12:14 pm

The title refers to the previous post, which quotes a
remark by a poetry critic in the current New Yorker .

Scholia —

From the post Structure and Sense of June 6, 2016 —

Structure

Sense

From the post Design Cube of July 23, 2015 —

Broken Symmetries in Diamond Space

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Monday, June 6, 2016

Structure and Sense

Filed under: General,Geometry — Tags: Knight Move, Seven Seals — m759 @ 2:01 pm

"… the war of 70-some years ago
has already become something like the Trojan War
had been for the Homeric bards:
a major event in the mythic past
that gives structure and sense to our present reality."

— Justin E. H. Smith, a professor of philosophy at
the University of Paris 7–Denis Diderot,
in the New York Times column "The Stone"
(print edition published Sunday, June 5, 2016)

In memory of a British playwright who reportedly
died at 90 this morning —

Structure

Sense

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Sunday, June 5, 2016

Sunday School: Seven Seals

Filed under: General,Geometry — Tags: Knight Move, Seven Seals — m759 @ 7:00 am

Click image for some background.

Wednesday, May 25, 2016

Framework

Filed under: General,Geometry — Tags: Dirac and Geometry, Priority — m759 @ 12:00 pm

"Studies of spin-½ theories in the framework of projective geometry
have been undertaken before." — Y. Jack Ng and H. van Dam,
February 20, 2009

For one such framework,* see posts from that same date
four years earlier — February 20, 2005.

* A 4×4 array. See the 1977, 1978, and 1986 versions by
Steven H. Cullinane, the 1987 version by R. T. Curtis, and
the 1988 Conway-Sloane version illustrated below —

Cullinane, 1977

Cullinane, 1978

Cullinane, 1986

Curtis, 1987

Update of 10:42 PM ET on Sunday, June 19, 2016 —

The above images are precursors to …

Conway and Sloane, 1988

Update of 10 AM ET Sept. 16, 2016 — The excerpt from the
1977 "Diamond Theory" article was added above.

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Friday, May 13, 2016

Geometry and Kinematics

Filed under: General,Geometry — Tags: One Ring — m759 @ 10:31 pm

"Just as both tragedy and comedy can be written
by using the same letters of the alphabet, the vast
variety of events in this world can be realized by
the same atoms through their different arrangements
and movements. Geometry and kinematics, which
were made possible by the void, proved to be still
more important in some way than pure being."

— Werner Heisenberg in Physics and Philosophy

For more about geometry and kinematics, see (for instance)

"An introduction to line geometry with applications,"
by Helmut Pottmann, Martin Peternell, and Bahram Ravani,
Computer-Aided Design 31 (1999), 3-16.

The concepts of line geometry (null point, null plane, null polarity,
linear complex, Klein quadric, etc.) are also of interest in finite geometry.
Some small finite spaces have as their natural models arrays of cubes .

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Sunday, May 8, 2016

The Three Solomons

Filed under: General,Geometry — Tags: A Sunday in May, Boole vs. Galois, Galois's Space, Sallows, Solomon Marcus, Yale Art Notes — m759 @ 11:00 pm

Earlier posts have dealt with Solomon Marcus and Solomon Golomb,
both of whom died this year — Marcus on Saint Patrick's Day, and
Golomb on Orthodox Easter Sunday. This suggests a review of
Solomon LeWitt, who died on Catholic Easter Sunday, 2007.

A quote from LeWitt indicates the depth of the word "conceptual"
in his approach to "conceptual art."

From Sol LeWitt: A Retrospective , edited by Gary Garrels, Yale University Press, 2000, p. 376:

THE SQUARE AND THE CUBE
by Sol LeWitt

"The best that can be said for either the square or the cube is that they are relatively uninteresting in themselves. Being basic representations of two- and three-dimensional form, they lack the expressive force of other more interesting forms and shapes. They are standard and universally recognized, no initiation being required of the viewer; it is immediately evident that a square is a square and a cube a cube. Released from the necessity of being significant in themselves, they can be better used as grammatical devices from which the work may proceed."

"Reprinted from Lucy R. Lippard et al ., “Homage to the Square,” Art in America 55, No. 4 (July-August 1967): 54. (LeWitt’s contribution was originally untitled.)"

See also the Cullinane models of some small Galois spaces —

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Tuesday, May 3, 2016

Symmetry

Filed under: General,Geometry — Tags: Beadgame Space, Cullinane Cube, Data Cube, Glasperlenspiel, Mythspace, Solomon's Cube — m759 @ 12:00 pm

A note related to the diamond theorem and to the site
Finite Geometry of the Square and Cube —

The last link in the previous post leads to a post of last October whose
final link leads, in turn, to a 2009 post titled Summa Mythologica .

Some may view the above web page as illustrating the
Glasperlenspiel passage quoted here in Summa Mythologica —

“"I suddenly realized that in the language, or at any rate
in the spirit of the Glass Bead Game, everything actually
was all-meaningful, that every symbol and combination of
symbols led not hither and yon, not to single examples,
experiments, and proofs, but into the center, the mystery
and innermost heart of the world, into primal knowledge.
Every transition from major to minor in a sonata, every
transformation of a myth or a religious cult, every classical
or artistic formulation was, I realized in that flashing moment,
if seen with a truly meditative mind, nothing but a direct route
into the interior of the cosmic mystery, where in the alternation
between inhaling and exhaling, between heaven and earth,
between Yin and Yang, holiness is forever being created.”

A less poetic meditation on the above web page* —

"I saw that in the alternation between front and back,
between top and bottom, between left and right,
symmetry is forever being created."

Update of Sept. 5, 2016 — See also a related remark
by Lévi-Strauss in 1955: "…three different readings
become possible: left to right, top to bottom, front
to back."

* For the underlying mathematics, see a June 21, 1983, research note.

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Wednesday, April 27, 2016

Kubrick’s Rube

Filed under: General — m759 @ 9:23 pm

In memory of a culture jammer *—

* "Mr. Lyons … made a living partly by buying,
reconditioning and selling used cars." —
— Ben Ratliff in The New York Times this evening.

See also the previous post and, from Feb. 14 in
this journal, the phrase "more global than local."

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Monday, April 25, 2016

Peirce’s Accounts of the Universe

Filed under: General,Geometry — Tags: Eightfold Cube, Structural Metaphors — m759 @ 8:19 pm

Compare and contrast Peirce's seven systems of metaphysics with
the seven projective points in a post of March 1, 2010 —

Wikipedia article 'Group theory' with Rubik Cube and quote from Nathan Carter-- 'What is symmetry?'

From my commentary on Carter's question —

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Wednesday, April 20, 2016

Symmetric Generation of a Simple Group

Filed under: General,Geometry — Tags: Cullinane Cube, Eightfold Cube, Geometric Theology, Road to Hell, Solomon's Cube, Springer, Symmetric Generation — m759 @ 9:00 pm

The reference in the previous post to the work of Guitart and
The Road to Universal Logic suggests a fiction involving
the symmetric generation of the simple group of order 168.

See The Diamond Archetype and a fictional account of the road to Hell …

'PyrE' in Bester's 'The Stars My Destination'

The cover illustration below has been adapted to
replace the flames of PyrE with the eightfold cube.

For related symmetric generation of a much larger group, see Solomon’s Cube.

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Monday, April 18, 2016

Toy

Filed under: General — Tags: on0418 — m759 @ 9:01 pm

"Finnegan's waterlogged epiphany"

— Phrase by John Lancaster in a Washington Post review of an
autobiography by William Finnegan, a book that today won a Pulitzer Prize.

The review is dated August 7, 2015. This journal on that date —

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Sunday, April 17, 2016

The Thing and I

Filed under: General,Geometry — Tags: Eightfold Cube, The Thing and I — m759 @ 2:01 pm

The New York Times philosophy column yesterday —

The Times's philosophy column "The Stone" is named after the legendary
"philosophers' stone." The column's name, and the title of its essay yesterday
"Is that even a thing?" suggest a review of the eightfold cube as "The object
most closely resembling a 'philosophers' stone' that I know of" (Page 51 of
the current issue of a Norwegian art quarterly, KUNSTforum.as).

The eightfold cube —

Definition of Epiphany

From James Joyce’s Stephen Hero , first published posthumously in 1944. The excerpt below is from a version edited by John J. Slocum and Herbert Cahoon (New York: New Directions Press, 1959).

Three Times:

… By an epiphany he meant a sudden spiritual manifestation, whether in the vulgarity of speech or of gesture or in a memorable phase of the mind itself. He believed that it was for the man of letters to record these epiphanies with extreme care, seeing that they themselves are the most delicate and evanescent of moments. He told Cranly that the clock of the Ballast Office was capable of an epiphany. Cranly questioned the inscrutable dial of the Ballast Office with his no less inscrutable countenance:

— Yes, said Stephen. I will pass it time after time, allude to it, refer to it, catch a glimpse of it. It is only an item in the catalogue of Dublin’s street furniture. Then all at once I see it and I know at once what it is: epiphany.

— What?

— Imagine my glimpses at that clock as the gropings of a spiritual eye which seeks to adjust its vision to an exact focus. The moment the focus is reached the object is epiphanised. It is just in this epiphany that I find the third, the supreme quality of beauty.

— Yes? said Cranly absently.

— No esthetic theory, pursued Stephen relentlessly, is of any value which investigates with the aid of the lantern of tradition. What we symbolise in black the Chinaman may symbolise in yellow: each has his own tradition. Greek beauty laughs at Coptic beauty and the American Indian derides them both. It is almost impossible to reconcile all tradition whereas it is by no means impossible to find the justification of every form of beauty which has ever been adored on the earth by an examination into the mechanism of esthetic apprehension whether it be dressed in red, white, yellow or black. We have no reason for thinking that the Chinaman has a different system of digestion from that which we have though our diets are quite dissimilar. The apprehensive faculty must be scrutinised in action.

— Yes …

— You know what Aquinas says: The three things requisite for beauty are, integrity, a wholeness, symmetry and radiance. Some day I will expand that sentence into a treatise. Consider the performance of your own mind when confronted with any object, hypothetically beautiful. Your mind to apprehend that object divides the entire universe into two parts, the object, and the void which is not the object. To apprehend it you must lift it away from everything else: and then you perceive that it is one integral thing, that is a thing. You recognise its integrity. Isn’t that so?

— And then?

— That is the first quality of beauty: it is declared in a simple sudden synthesis of the faculty which apprehends. What then? Analysis then. The mind considers the object in whole and in part, in relation to itself and to other objects, examines the balance of its parts, contemplates the form of the object, traverses every cranny of the structure. So the mind receives the impression of the symmetry of the object. The mind recognises that the object is in the strict sense of the word, a thing , a definitely constituted entity. You see?

— Let us turn back, said Cranly.

They had reached the corner of Grafton St and as the footpath was overcrowded they turned back northwards. Cranly had an inclination to watch the antics of a drunkard who had been ejected from a bar in Suffolk St but Stephen took his arm summarily and led him away.

— Now for the third quality. For a long time I couldn’t make out what Aquinas meant. He uses a figurative word (a very unusual thing for him) but I have solved it. Claritas is quidditas . After the analysis which discovers the second quality the mind makes the only logically possible synthesis and discovers the third quality. This is the moment which I call epiphany. First we recognise that the object is one integral thing, then we recognise that it is an organised composite structure, a thing in fact: finally, when the relation of the parts is exquisite, when the parts are adjusted to the special point, we recognise that it is that thing which it is. Its soul, its whatness, leaps to us from the vestment of its appearance. The soul of the commonest object, the structure of which is so adjusted, seems to us radiant. The object achieves its epiphany.

Having finished his argument Stephen walked on in silence. He felt Cranly’s hostility and he accused himself of having cheapened the eternal images of beauty. For the first time, too, he felt slightly awkward in his friend’s company and to restore a mood of flippant familiarity he glanced up at the clock of the Ballast Office and smiled:

— It has not epiphanised yet, he said.

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Monday, April 4, 2016

Noon for London

Filed under: General — m759 @ 12:00 pm

(A sequel to today's earlier posts Cube for Berlin and Midnight for Paris.)

See London in this journal.

That search yields …

“The more intellectual, less physical,
the spell of contemplation
the more complex must be the object,
the more close and elaborate
must be the comparison
the mind has to keep making
between the whole and the parts,
the parts and the whole.”

— The Journals and Papers of Gerard Manley Hopkins ,
ed. by Humphry House (London: Oxford University Press, 1959),
as quoted by Philip A. Ballinger in The Poem as Sacrament

(From the post The Inscape of 24, April 24, 2014. The 14 blocks in
the design S(3, 4, 8) of today's previous post are analogous to the 759
blocks in the design S(5, 8, 24).)

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Tuesday, March 15, 2016

15 Projective Points Revisited

Filed under: General,Geometry — Tags: Eightfold Cube, Vigeland — m759 @ 11:59 pm

A March 10, 2016, Facebook post from KUNSTforum.as,
a Norwegian art quarterly —

Click image above for a view of pages 50-51 of a new KUNSTforum
article showing two photos relevant to my own work — those labeled
"after S. H. Cullinane."

(The phrase "den pensjonerte Oxford-professoren Stephen H. Cullinane"
on page 51 is almost completely wrong. I have never been a professor,
I was never at Oxford, and my first name is Steven, not Stephen.)

For some background on the 15 projective points at the lower left of
the above March 10 Facebook post, see "The Smallest Projective Space."

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Thursday, December 17, 2015

Hint of Reality

Filed under: General,Geometry — Tags: Cube Quaternions, Cube School, Eightfold Cube, Lamb's Deathly Hallows, Soul Notes — m759 @ 12:45 pm

From an article* in Proceedings of Bridges 2014 —

As artists, we are particularly interested in the symmetries of real world physical objects.

Three natural questions arise:

1. Which groups can be represented as the group of symmetries of some real-world physical object?

2. Which groups have actually been represented as the group of symmetries of some real-world physical object?

3. Are there any glaring gaps – small, beautiful groups that should have a physical representation in a symmetric object but up until now have not?

The article was cited by Evelyn Lamb in her Scientific American
weblog on May 19, 2014.

The above three questions from the article are relevant to a more
recent (Oct. 24, 2015) remark by Lamb:

"… finite projective planes [in particular, the 7-point Fano plane,
about which Lamb is writing] seem like a triumph of purely
axiomatic thinking over any hint of reality…."

For related hints of reality, see Eightfold Cube in this journal.

* "The Quaternion Group as a Symmetry Group," by Vi Hart and Henry Segerman

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Tuesday, October 20, 2015

Liminal

Filed under: General — m759 @ 3:33 pm

The New York Times has a readable, if not informative,
review of a recent controversial account of history —

"For many, it exists in a kind of liminal state,
floating somewhere between fact and mythology."

— Jonathan Mahler, online Times on Oct. 15, 2015

[See Wikipedia on Liminality.]

Mahler begins his review with a statement by the President
on the night of May 1, 2011.

A more easily checked statement quoted here on that date:

"The positional meaning of a symbol derives from
its relationship to other symbols in a totality, a Gestalt,
whose elements acquire their significance from the
system as a whole."

— Victor Turner, The Forest of Symbols , Ithaca, NY,
Cornell University Press, 1967, p. 51, quoted by
Beth Barrie in "Victor Turner."

A Gestalt from "Verhexung ," the previous post —

Guitart's statement that the above figure is a "Boolean logical cube"
seems, in the words of the Times , to be "floating somewhere
between fact and mythology." Discuss.

(My apologies to those who feel that attempting to make sense
of Guitart makes them feel like Vin Diesel in the Dreamworld.)

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Verhexung

Filed under: General — Tags: Geometric Theology — m759 @ 5:04 am

“Die Philosophie ist ein Kampf gegen die Verhexung
unsres Verstandes durch die Mittel unserer Sprache.”

— Philosophical Investigations (1953), Section 109

An example of Verhexung from the René Guitart article in the previous post —

Monday, October 19, 2015

Symmetric Generation of the Simple Order-168 Group

Filed under: General,Geometry — Tags: Eightfold Cube, Geometric Theology, Springer, Symmetric Generation — m759 @ 8:48 pm

This post continues recent thoughts on the work of René Guitart.
A 2014 article by Guitart gives a great deal of detail on his
approach to symmetric generation of the simple group of order 168 —

“Hexagonal Logic of the Field F₈ as a Boolean Logic
with Three Involutive Modalities,” pp. 191-220 in

The Road to Universal Logic:
Festschrift for 50th Birthday of
Jean-Yves Béziau, Volume I,

Editors: Arnold Koslow, Arthur Buchsbaum,
Birkhäuser Studies in Universal Logic, dated 2015
by publisher but Oct. 11, 2014, by Amazon.com.

See also the eightfold cube in this journal.

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Borromean Generators

Filed under: General,Geometry — Tags: Eightfold Cube, Springer, Triskele — m759 @ 4:10 am

From slides dated June 28, 2008 —

Compare to my own later note, from March 4, 2010 —

It seems that Guitart discovered these "A, B, C" generators first,
though he did not display them in their natural setting,
the eightfold cube.

Some context: The epigraph to my webpage
"A Simple Reflection Group of Order 168" —

"Let G be a finite, primitive subgroup of GL(V) = GL(n,D) ,
where V is an n-dimensional vector space over the
division ring D . Assume that G is generated by 'nice'
transformations. The problem is then to try to determine
(up to GL(V) -conjugacy) all possibilities for G . Of course,
this problem is very vague. But it is a classical one,
going back 150 years, and yet very much alive today."

— William M. Kantor, "Generation of Linear Groups,"
pp. 497-509 in The Geometric Vein: The Coxeter Festschrift ,
published by Springer, 1981

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Saturday, September 5, 2015

For the Machiavelli School*

Filed under: General,Geometry — Tags: Geometric Theology, Operation Blockhead — m759 @ 12:00 pm

See also Weyl + Palermo in this journal —

* The title refers to the previous post, on a current New Yorker cartoon.

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Monday, August 3, 2015

Text and Context*

Filed under: General,Geometry — Tags: Stevens Owl — m759 @ 2:00 am

"The ORCID organization offers an open and
independent registry intended to be the de facto
standard for contributor identification in research
and academic publishing. On 16 October 2012,
ORCID launched its registry services… and
started issuing user identifiers." — Wikipedia

This journal on the above date —

A more recent identifier —

Related material —

See also the recent posts Ein Kampf and Symplectic.

* Continued.

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Monday, July 27, 2015

The Ten

Filed under: General — Tags: Getcha — m759 @ 10:00 pm

The title refers to actress Kristen Wiig. (See Getcha.)

For Wiig, a scene from a 1968 Ingmar Bergman parody…

… and parts of a post from Saturday night, July 18, 2015 —

Some related material from pure mathematics — Design Cube (July 23).

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Friday, June 12, 2015

Core

Filed under: General — m759 @ 12:00 am

Grindhouse Madonna ,

Children at your feet….

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Wednesday, May 13, 2015

Space

Filed under: General,Geometry — Tags: Clooney Omega, Lexicon Omega — m759 @ 2:00 pm

Notes on space for day 13 of May, 2015 —

The 13 symmetry axes of the cube may be viewed as
the 13 points of the Galois projective space PG(2,3).
This space (a plane) may also be viewed as the nine points
of the Galois affine space AG(2,3) plus the four points on
an added "line at infinity."

Related poetic material:

The ninefold square and Apollo, as well as …

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Friday, March 27, 2015

Pursuit of Gestalt*

Filed under: General — Tags: Symmetric Generation — m759 @ 12:00 pm

The art above is by the Copenhagen studio
Hvass & Hannibal. For a photo of the artists,
see a webpage on Beijing Design Week 2011.

Hvass and Hannibal were apparently in Beijing
for the "open workshop," Sept. 17-23, 2011.

Gestalt-related material from this journal that week —

* Title suggested by that of a book by Quine.

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Tuesday, January 27, 2015

Drama Club

Filed under: General — m759 @ 3:27 am

Julianne Moore at the Screen Actors Guild awards
on Sunday evening:

"When I was 17, I decided I wanted to be
an actor. It didn't seem possible because
I'd never met a real actor," Moore said.
"So I want to say to all the kids in the
drama club, you guys are the real actors."

On the main character of the new film "Birdman"—

"Thomson is clearly talented, yet unable to get out of
the shadow of his superhero role. He is filled with
a simmering rage as Robert Downey Jr. appears
on the TV, arguably the highest profile actor alive
courtesy of a role in the Marvel films."

— Grant Pearsall at The Snapper

A midrash for Robert:

See The Stars My Destination and Cube of Ultron.

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Thursday, January 1, 2015

New Year’s Greeting from Franz Kafka

Filed under: General,Geometry — m759 @ 5:01 am

An image that led off the year-end review yesterday in
the weblog of British combinatorialist Peter J. Cameron:

See also this weblog's post final post of 2014,
with a rectangular array illustrating the six faces
of a die, and Cameron's reference yesterday to
a die-related post…

"The things on my blog that seem to be
of continuing value are the expository
series like the one on the symmetric group
(the third post in this series was reblogged
by Gil Kalai last month, which gave it a new
lease of life)…."

A tale from an author of Prague:

The Emperor—so they say—has sent a message, directly from his death bed, to you alone, his pathetic subject, a tiny shadow which has taken refuge at the furthest distance from the imperial sun. He ordered the herald to kneel down beside his bed and whispered the message into his ear. He thought it was so important that he had the herald repeat it back to him. He confirmed the accuracy of the verbal message by nodding his head. And in front of the entire crowd of those who’ve come to witness his death—all the obstructing walls have been broken down and all the great ones of his empire are standing in a circle on the broad and high soaring flights of stairs—in front of all of them he dispatched his herald. The messenger started off at once, a powerful, tireless man. Sticking one arm out and then another, he makes his way through the crowd. If he runs into resistance, he points to his breast where there is a sign of the sun. So he moves forward easily, unlike anyone else. But the crowd is so huge; its dwelling places are infinite. If there were an open field, how he would fly along, and soon you would hear the marvelous pounding of his fist on your door. But instead of that, how futile are all his efforts. He is still forcing his way through the private rooms of the innermost palace. He will never he win his way through. And if he did manage that, nothing would have been achieved. He would have to fight his way down the steps, and, if he managed to do that, nothing would have been achieved. He would have to stride through the courtyards, and after the courtyards the second palace encircling the first, and, then again, stairs and courtyards, and then, once again, a palace, and so on for thousands of years. And if he finally did burst through the outermost door—but that can never, never happen—the royal capital city, the centre of the world, is still there in front of him, piled high and full of sediment. No one pushes his way through here, certainly not with a message from a dead man. But you sit at your window and dream of that message when evening comes.

See also a passage quoted in this weblog on the original
date of Cameron's Prague image, July 26, 2014 —

"The philosopher Graham Harman is invested in
re-thinking the autonomy of objects and is part
of a movement called Object-Oriented-Philosophy
(OOP)." — From “The Action of Things,” a 2011
M.A. thesis at the Center for Curatorial Studies,
Bard College, by Manuela Moscoso

— in the context of a search here for the phrase
"structure of the object." An image from that search:

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Monday, December 29, 2014

Revolution of Forms

Filed under: General — m759 @ 11:49 pm

In memory of Cuban architect
Ricardo Porro, who died
on Christmas Day, 2014:

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Monday, December 15, 2014

Mythic Metaphysics

Filed under: General — Tags: Myth and Memory — m759 @ 9:01 pm

Today’s 8:01 PM post quoted Husserl on
the perception of the cube.

Another approach to perception of the cube,
from Narrative Metaphysics on St. Lucia’s Day —

See also Symplectic Structure and Stevens’s Rock.

From today’s 11:29 AM post —

John Burt Foster Jr. in Nabokov’s Art of Memory and
European Modernism (Princeton U. Press, 1993, p. 224) —

At the time of The Waste Land , in a comment on
Joyce’s Ulysses that influenced many later definitions
of modernism in the English-speaking world, Eliot
announced, “instead of narrative method, we may
now use the mythical method.”¹³

For some illuminating remarks on a mythical approach
to perception of the cube, see Gareth Knight on Schicksalstag 2012.

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Monday, December 1, 2014

Change Arises

Filed under: General,Geometry — Tags: Red Dawn — m759 @ 11:00 am

Flashback to St. Andrew's Day, 2013 —

Saturday, November 30, 2013

Waiting for Ogdoad

Filed under: Uncategorized — Tags: Change Arises — m759 @ 10:30 AM

Continued from October 30 (Devil's Night), 2013.

“In a sense, we would see that change
arises from the structure of the object.”

— Theoretical physicist quoted in a
Simons Foundation article of Sept. 17, 2013

This suggests a review of mathematics and the
"Classic of Change ," the I Ching .

If the object is a cube, change arises from the fact
that the object has six faces…

and is the unit cell for the six -dimensional
hyperspace H over the two-element field —

A different representation of the unit cell of
the hyperspace H (and of the I Ching ) —

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Wednesday, November 26, 2014

Class Act

Filed under: General,Geometry — Tags: Galois-Plane Models, Springer — m759 @ 7:18 am

Update of Nov. 30, 2014 —

For further information on the geometry in
the remarks by Eberhart below, see
pp. 16-17 of A Geometrical Picture Book ,
by Burkard Polster (Springer, 1998). Polster
cites a different article by Lemay.

A search for background to the exercise in the previous post
yields a passage from the late Stephen Eberhart:

The first three primes p = 2, 3, and 5 therefore yield finite projective planes with 7, 13, and 31 points and lines, respectively. But these are just the numbers of symmetry axes of the five regular solids, as described in Plato's Timaeus : The tetrahedron has 4 pairs of face planes and corner points + 3 pairs of opposite edges, totalling 7 axes; the cube has 3 pairs of faces + 6 pairs of edges + 4 pairs of corners, totalling 13 axes (the octahedron simply interchanges the roles of faces and corners); and the pentagon dodecahedron has 6 pairs of faces + 15 pairs of edges + 10 pairs of corners, totalling 31 axes (the icosahedron again interchanging roles of faces and corners). This is such a suggestive result, one would expect to find it dealt with in most texts on related subjects; instead, while "well known to those who well know such things" (as Richard Guy likes to quip), it is scarcely to be found in the formal literature [9]. The reason for the common numbers, it turns out, is that the groups of symmetry motions of the regular solids are subgroups of the groups of collineations of the respective finite planes, a face axis being different from an edge axis of a regular solid but all points of a projective plane being alike, so the latter has more symmetries than the former.

[9] I am aware only of a series of in-house publications by Fernand Lemay of the Laboratoire de Didactique, Faculté des Sciences de I 'Éducation, Univ. Laval, Québec, in particular those collectively titled Genèse de la géométrie I-X.

— Stephen Eberhart, Dept. of Mathematics,
California State University, Northridge,
"Pythagorean and Platonic Bridges between
Geometry and Algebra," in BRIDGES: Mathematical
Connections in Art, Music, and Science , 1998,
archive.bridgesmathart.org/1998/bridges1998-121.pdf

Eberhart died of bone cancer in 2003. A memorial by his
high school class includes an Aug. 7, 2003, transcribed
letter from Eberhart to a classmate that ends…

… I earned MA’s in math (UW, Seattle) and history (UM, Missoula) where a math/history PhD program had been announced but canceled. So 1984 to 2002 I taught math (esp. non-Euclidean geometry) at C.S.U. Northridge. It’s been a rich life. I’m grateful.

Steve

See also another informative BRIDGES paper by Eberhart
on mathematics and the seven traditional liberal arts.

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Tuesday, November 25, 2014

Euclidean-Galois Interplay

Filed under: General,Geometry — Tags: Galois-Plane Models, Interplay, Springer — m759 @ 11:00 am

For previous remarks on this topic, as it relates to
symmetry axes of the cube, see previous posts tagged Interplay.

The above posts discuss, among other things, the Galois
projective plane of order 3, with 13 points and 13 lines.

These Galois points and lines may be modeled in Euclidean geometry
by the 13 symmetry axes and the 13 rotation planes
of the Euclidean cube. They may also be modeled in Galois geometry
by subsets of the 3x3x3 Galois cube (vector 3-space over GF(3)).

The 3×3×3 Galois Cube

Exercise: Is there any such analogy between the 31 points of the
order-5 Galois projective plane and the 31 symmetry axes of the
Euclidean dodecahedron and icosahedron? Also, how may the
31 projective points be naturally pictured as lines within the
5x5x5 Galois cube (vector 3-space over GF(5))?

Update of Nov. 30, 2014 —

For background to the above exercise, see
pp. 16-17 of A Geometrical Picture Book ,
by Burkard Polster (Springer, 1998), esp.
the citation to a 1983 article by Lemay.

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Tuesday, October 28, 2014

Figural Processing

Filed under: General,Geometry — Tags: Figural Processing — m759 @ 4:22 am

Part I:

Part II:

Click images for some context.

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Tuesday, October 21, 2014

Tools

Filed under: General,Geometry — Tags: Harvard Miller — m759 @ 12:00 am

(Night at the Museum continues.)

"Strategies for making or acquiring tools

While the creation of new tools marked the route to developing the social sciences,
the question remained: how best to acquire or produce those tools?"

— Jamie Cohen-Cole, “Instituting the Science of Mind: Intellectual Economies
and Disciplinary Exchange at Harvard’s Center for Cognitive Studies,”
British Journal for the History of Science vol. 40, no. 4 (2007): 567-597.

Obituary of a co-founder, in 1960, of the Center for Cognitive Studies at Harvard:

"Disciplinary Exchange" —

In exchange for the free Web tools of HTML and JavaScript,
some free tools for illustrating elementary Galois geometry —

The Kaleidoscope Puzzle, The Diamond 16 Puzzle,
The 2x2x2 Cube, and The 4x4x4 Cube

"Intellectual Economies" —

In exchange for a $10 per month subscription, an excellent
"Quilt Design Tool" —

This illustrates not geometry, but rather creative capitalism.

Related material from the date of the above Harvard death: Art Wars.

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Saturday, October 18, 2014

Educational Series

Filed under: General,Geometry — Tags: Educational Series, Rubik Adoration — m759 @ 1:01 pm

Barron's Educational Series (click to enlarge):

The Tablet of Ahkmenrah:

IMAGE- The Tablet of Ahkmenrah, from 'Night at the Museum'

"With the Tablet of Ahkmenrah and the Cube of Rubik,
my power will know no bounds!"
— Kahmunrah in a novelization of Night at the Museum:
Battle of the Smithsonian , Barron's Educational Series

Another educational series (this journal):

Art theorist Rosalind Krauss and The Ninefold Square

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Wednesday, October 1, 2014

Mathematics for Tromsø

Filed under: General,Geometry — m759 @ 9:45 am

Loren Olson, Harvard ’64, a professor of mathematics
at Norway’s Tromsø University,* died June 22, 2014.

In his memory, a search in this journal for Lie Group.

That search yields a post titled Lie Groups for Holy Week (March 30, 2010).

A quotation related to that post:

* The city of Tromsø hosted some art related to group theory in 2010.
Neither that art nor my own related remarks on group theory are very
relevant to physics (yet).

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Tuesday, September 16, 2014

Where the Joints Are

Filed under: General,Geometry — Tags: Articulation, Eightfold Cube, Lechner's End, Sensibility — m759 @ 10:00 am

An image related to the recent posts Sense and Sensibility:

A quote from yesterday's post The Eight:

A possible source for the above phrase about phenomena "carved at their joints":

See also the carving at the joints of Plato's diamond from the Meno :

Related material: Phaedrus on Kant as a diamond cutter
in Zen and the Art of Motorcycle Maintenance .

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Monday, September 1, 2014

Mathematics, Not Theology

Filed under: General — Tags: Geometric Theology — m759 @ 5:00 pm

(Continued)

“A set having three members is a single thing
wholly constituted by its members but distinct from them.
After this, the theological doctrine of the Trinity as
‘three in one’ should be child’s play.”

— Max Black, Caveats and Critiques: Philosophical Essays
in Language, Logic, and Art , Cornell U. Press, 1975

IMAGE- The Trinity of Max Black (a 3-set, with its eight subsets arranged in a Hasse diagram that is also a cube)

“There is such a thing as a three-set.”
— Saying adapted from a novel by Madeleine L’Engle

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Saturday, July 12, 2014

Mars Package

Filed under: General,Geometry — Tags: Cullinane Cube, Solomon's Cube — m759 @ 8:00 pm

For Ursula K. Le Guin

“For me it is a sign that we have fundamentally different
conceptions of the work of the intelligence services.”

— Germany’s Chancellor Angela Merkel in
theguardian.com, Saturday, 12 July 2014, 14.32 EDT

Another sort of service, thanks to Dan Brown and Tom Hanks:

Friday, July 11, 2014

Spiegel-Spiel des Gevierts

Filed under: Uncategorized — m759 @ 12:00 PM

See Cube Symbology.

Robert Langdon (played by Tom Hanks) and a corner of Solomon's Cube

“Da hats ein Eck“

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Sequel

Filed under: General,Geometry — Tags: Cube School, Cullinane Cube, Design Cube, Fitch, Solomon's Cube — m759 @ 9:00 am

A sequel to the 1974 film
Thunderbolt and Lightfoot :

Contingent and Fluky

Some variations on a thunderbolt theme:

These variations also exemplify the larger
Verbum theme:

Image-- Escher's 'Verbum'

Escher’s Verbum

Image-- Solomon's Cube

Solomon’s Cube

A search today for Verbum in this journal yielded
a Georgetown University Chomskyite, Professor
David W. Lightfoot.

"Dr. Lightfoot writes mainly on syntactic theory,
language acquisition and historical change, which
he views as intimately related. He argues that
internal language change is contingent and fluky,
takes place in a sequence of bursts, and is best
viewed as the cumulative effect of changes in
individual grammars, where a grammar is a
'language organ' represented in a person's
mind/brain and embodying his/her language
faculty."

Some syntactic work by another contingent and fluky author
is related to the visual patterns illustrated above.

See Tecumseh Fitch in this journal.

For other material related to the large Verbum cube,
see posts for the 18th birthday of Harry Potter.

That birthday was also the upload date for the following:

See esp. the comments section.

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Saturday, July 5, 2014

Core Curriculum

Filed under: General — Tags: indigo — m759 @ 2:01 pm

This post was suggested by reviews of the David Hare play “Skylight” at
The New York Times , at WorldSocialism.org, and at ChicagoCritic.com.

Vide Atoms in the Family , by Laura Fermi, a book I read in high school.

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Friday, June 13, 2014

It’s 10 PM

Filed under: General — Tags: Consciousness Growth, Rubik Adoration, Rubik Exorcism — m759 @ 10:00 pm

“The wind of change is blowing throughout the continent.
Whether we like it or not, this growth of national consciousness
is a political fact.”— Prime Minister Harold Macmillan,
South Africa, 1960

“Lord knows when the cold wind blows
it’ll turn your head around.” — James Taylor

From a Log24 post of August 27, 2011:

For related remarks on “national consciousness,” see Frantz Fanon.

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Thursday, June 5, 2014

Twisty Quaternion Symmetry

Filed under: General,Geometry — m759 @ 9:11 pm

The previous post told how user58512 at math.stackexchange.com
sought in 2013 a geometric representation of Q₈, the quaternion group.
He ended up displaying an illustration that very possibly was drawn,
without any acknowledgement of its source, from my own work.

On the date that user58512 published that illustration, he further
pursued his March 1, 2013, goal of a “twisty” quaternion model.

On March 12, 2013, he suggested that the quaternion group might be
the symmetry group of the following twisty-cube coloring:

Illustration by Jim Belk

Here is part of a reply by Jim Belk from Nov. 11, 2013, elaborating on
that suggestion:

Belk argues that the colored cube is preserved under the group
of actions he describes. It is, however, also preserved under a
larger group. (Consider, say, rotation of the entire cube by 180
degrees about the center of any one of its checkered faces.) The
group Belk describes seems therefore to be a symmetry group,
not the symmetry group, of the colored cube.

I do not know if any combination puzzle has a coloring with
precisely the quaternion group as its symmetry group.

(Updated at 12:15 AM June 6 to point out the larger symmetry group
and delete a comment about an arXiv paper on quaternion group models.)

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Saturday, May 31, 2014

Quaternion Group Models:

Filed under: General,Geometry — Tags: Cube Quaternions, Cube School, Eightfold Cube — m759 @ 10:00 am

The ninefold square, the eightfold cube, and monkeys.

IMAGE- Actions of the unit quaternions in finite geometry, on a ninefold square and on an eightfold cube

For posts on the models above, see quaternion
in this journal. For the monkeys, see

"Nothing Is More Fun than a Hypercube of Monkeys,"
Evelyn Lamb's Scientific American weblog, May 19, 2014:

The Scientific American item is about the preprint
"The Quaternion Group as a Symmetry Group,"
by Vi Hart and Henry Segerman (April 26, 2014):

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Wednesday, May 21, 2014

Through the Vanishing Point*

Filed under: General,Geometry — Tags: Logicomix, McLuhan and Mathematics, on140521 — m759 @ 9:48 am

Marshall McLuhan in "Annie Hall" —

"You know nothing of my work."

Related material —

"I need a photo opportunity
I want a shot at redemption
Don't want to end up a cartoon
In a cartoon graveyard"

— Paul Simon

It was a dark and stormy night…

— Page 180, Logicomix

A photo opportunity for Whitehead
(from Romancing the Cube, April 20, 2011)—

See also Absolute Ambition (Nov. 19, 2010).

* For the title, see Vanishing Point in this journal.

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Tuesday, May 20, 2014

Play

Filed under: General,Geometry — Tags: Galois October — m759 @ 7:47 pm

From a recreational-mathematics weblog yesterday:

"This appears to be the arts section of the post,
so I’ll leave Martin Probert’s page on
The Survival, Origin and Mathematics of String Figures
here. I’ll be back to pick it up at the end. Maybe it’d like
to play with Steven H. Cullinane’s pages on the
Finite Geometry of the Square and Cube."

I doubt they would play well together.

Perhaps the offensive linking of the purely recreational topic
of string figures to my own work was suggested by the
string figures' resemblance to figures of projective geometry.

A pairing I prefer: Desargues and Galois —

For further details, see posts on Desargues and Galois.

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Saturday, May 3, 2014

Gray Space

Filed under: General,Geometry — Tags: April Awareness 2014 — m759 @ 9:00 am

Or: Three Shades of Gray

(Continued from previous Gray Space posts.)

Click the above image for some related mathematics.

Those who prefer “magic” approaches to mathematics*
may consult the works of Robert J. Stewart and his
mentor William G. Gray.

Robert J. Stewart (left) and a pentagram photo posted yesterday evening
by Oslo artist Josefine Lyche. See also Lyche in this journal.

* See the April 2014 banners displayed at the websites
of the American Mathematical Society and of the
Mathematical Association of America, as well as
a mathematician’s remarks linked to here last evening.

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Wednesday, April 30, 2014

Fast Forward

Filed under: General — m759 @ 9:00 am

Continued from Nov. 21, 2010:

“They always print… the lottery.”

The reader may interpret the lottery numbers
as he or she pleases.

Related material: Jersey City in Log24 posts
of April 25 and April 28, and today’s NY Times
image of another Jersey City landmark.

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Friday, April 25, 2014

Toying

Filed under: General,Geometry — Tags: Rubik vs. Abel, Toying — m759 @ 10:00 pm

IMAGE- 'Another instance of producers toying with artists' and a Rubik's Cube exhibition in Jersey City beginning Saturday, April 26

Related material: Quilt Geometry and Magical Realism Revisited.

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Friday, April 4, 2014

Eight Gate

Filed under: General,Geometry — Tags: Cube School, Eightfold Cube, Field Dream, Perspective — m759 @ 12:00 pm

From a Huffington Post discussion of aesthetics by Colm Mulcahy
of Spelman College, Atlanta:

"The image below on the left… is… overly simplistic, and lacks reality:

It's all a matter of perspective: the problem here is that opposite sides
of the cube, which are parallel in real life, actually look parallel in the
left image! The image on the right is better…."

A related discussion: Eight is a Gate.

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Monday, March 17, 2014

Narratives

Filed under: General — Tags: Glasperlenspiel — m759 @ 12:00 pm

Or: The Confessions of Nat Tate

“A convincing lie is, in its own way, a tiny, perfect narrative.”
— William Boyd, “A Short History of the Short Story” (2006)

“A novel written in the first-person singular has certain powerful
narrative advantages, especially when it takes the form of a ‘confession.'”
— William Boyd, “Memoir of a Plagiarist” (1994)

From a Log24 post yesterday —

For Little Man Tate —

IMAGE- Wechsler block-design cubes and related WAIS-R manual

Related material — Wechsler in this journal and an earlier Siri Hustvedt
art novel, from 2003 —

Mark and Lucille, Bill and Violet, Al and Regina, etc., etc., etc. —

IMAGE- Siri Hustvedt on the name 'Wechsler' in 'What I Loved'

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Saturday, March 8, 2014

Conwell Heptads in Eastern Europe

Filed under: General,Geometry — Tags: Depth — m759 @ 11:07 am

“Charting the Real Four-Qubit Pauli Group
via Ovoids of a Hyperbolic Quadric of PG(7,2),”
by Metod Saniga, Péter Lévay and Petr Pracna,
arXiv:1202.2973v2 [math-ph] 26 Jun 2012 —

P. 4— “It was found that Q ⁺(5,2) (the Klein quadric)
has, up to isomorphism, a unique one — also known,
after its discoverer, as a Conwell heptad [18].
The set of 28 points lying off Q ⁺(5,2) comprises
eight such heptads, any two having exactly one
point in common.”

P. 11— “This split reminds us of a similar split of
63 points of PG(5,2) into 35/28 points lying on/off
a Klein quadric Q ⁺(5,2).”

[18] G. M. Conwell, Ann. Math. 11 (1910) 60–76

A similar split occurs in yesterday’s Kummer Varieties post.
See the 63 = 28 + 35 vectors of R⁸ discussed there.

For more about Conwell heptads, see The Klein Correspondence,
Penrose Space-Time, and a Finite Model.

For my own remarks on the date of the above arXiv paper
by Saniga et. al., click on the image below —

Walter Gropius

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Thursday, February 27, 2014

Secular Space

Filed under: General — m759 @ 11:30 am

This morning's previous post, on sacred space,
linked to "Positively White Cube Revisited,"
an article by one Simon Sheikh.

Sheikh writes well, but he seems to be a disciple
of the damned Marxist lunatic Louis Althusser.

As Pynchon put it in Gravity's Rainbow ,
"For every kind of vampire, there is a kind of cross."

In this case, a video starring Sheikh on the exhibition "All That Fits"
suggests, by its filming date (May 27, 2011), a Maltese cross.

"The stuff that dreams are made of." — Bogart

(See also Oct. 25, 2012.)

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Sacred Space, continued

Filed under: General — Tags: Sacred Space — m759 @ 11:00 am

"An image comes to mind of a white, ideal space
that, more than any single picture, may be the
archetypal image of 20th-century art."

— Brian O'Doherty, "Inside the White Cube"

Cube spaces exist also in mathematics.

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Wednesday, February 26, 2014

Cubist Aesthetics…

Filed under: General — m759 @ 8:00 pm

in Stevens' "The Man with the Blue Guitar"

Author:	Ruszkowska-Buchowska, Dominika
Publication:	Studia Anglica Posnaniensia: An International Review of English Studies
Article Type:	Critical essay
Date:	Jan 1, 2004

See also Blue Guitar
and Cubist Language Game
as well as Dali Cube.

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Thursday, February 13, 2014

Winter’s Game*

Filed under: General — Tags: on140213 — m759 @ 11:00 am

Part I: Continued from January 20 — "Arising Heaven" —

Part II: The Stars My Destination in this journal

'The Stars My Destination,' current edition (with cover slightly changed)

Part III: Ender's Game —

* The title refers to a character, Rogue Winter, in Alfred Bester's
1981 novel The Deceivers .

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Wednesday, January 22, 2014

A Riddle for Davos

Filed under: General,Geometry — Tags: Also Sprach, Crucible Raiders, Snow Dance — m759 @ 9:00 pm

Hexagonale Unwesen

Einstein and Thomas Mann, Princeton, 1938

See also the life of Diogenes Allen, a professor at Princeton
Theological Seminary, a life that reportedly ended on the date—
January 13, 2013— of the above Log24 post.

January 13 was also the dies natalis of St. James Joyce.

Some related reflections —

"Praeterit figura huius mundi " — I Corinthians 7:31 —

Conclusion of of "The Dead," by James Joyce—

The air of the room chilled his shoulders. He stretched himself cautiously along under the sheets and lay down beside his wife. One by one, they were all becoming shades. Better pass boldly into that other world, in the full glory of some passion, than fade and wither dismally with age. He thought of how she who lay beside him had locked in her heart for so many years that image of her lover's eyes when he had told her that he did not wish to live.

Generous tears filled Gabriel's eyes. He had never felt like that himself towards any woman, but he knew that such a feeling must be love. The tears gathered more thickly in his eyes and in the partial darkness he imagined he saw the form of a young man standing under a dripping tree. Other forms were near. His soul had approached that region where dwell the vast hosts of the dead. He was conscious of, but could not apprehend, their wayward and flickering existence. His own identity was fading out into a grey impalpable world: the solid world itself, which these dead had one time reared and lived in, was dissolving and dwindling.

A few light taps upon the pane made him turn to the window. It had begun to snow again. He watched sleepily the flakes, silver and dark, falling obliquely against the lamplight. The time had come for him to set out on his journey westward. Yes, the newspapers were right: snow was general all over Ireland. It was falling on every part of the dark central plain, on the treeless hills, falling softly upon the Bog of Allen and, farther westward, softly falling into the dark mutinous Shannon waves. It was falling, too, upon every part of the lonely churchyard on the hill where Michael Furey lay buried. It lay thickly drifted on the crooked crosses and headstones, on the spears of the little gate, on the barren thorns. His soul swooned slowly as he heard the snow falling faintly through the universe and faintly falling, like the descent of their last end, upon all the living and the dead.

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Friday, January 17, 2014

The 4×4 Relativity Problem

Filed under: General,Geometry — Tags: Coordinatization, Finite Relativity, Geometry, Priority — m759 @ 11:00 pm

The sixteen-dot square array in yesterday’s noon post suggests
the following remarks.

“This is the relativity problem: to fix objectively a class of
equivalent coordinatizations and to ascertain the group of
transformations S mediating between them.”

— Hermann Weyl, The Classical Groups ,
Princeton University Press, 1946, p. 16

The Galois tesseract appeared in an early form in the journal
Computer Graphics and Art , Vol. 2, No. 1, February 1977—

The 1977 matrix Q is echoed in the following from 2002—

IMAGE- Dolgachev and Keum, coordinatization of the 4x4 array in 'Birational Automorphisms of Quartic Hessian Surfaces,' AMS Transactions, 2002

A different representation of Cullinane’s 1977 square model of the
16-point affine geometry over the two-element Galois field GF(2)
is supplied by Conway and Sloane in Sphere Packings, Lattices and Groups
(first published in 1988) :

IMAGE- The Galois tesseract as a four-dimensional vector space, from a diagram by Conway and Sloane in 'Sphere Packings, Lattices, and Groups'

Here a, b, c, d are basis vectors in the vector 4-space over GF(2).
(For a 1979 version of this vector space, see AMS Abstract 79T-A37.)

See also a 2011 publication of the Mathematical Association of America —

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Monday, January 13, 2014

A Prime for Marissa

Filed under: General,Geometry — m759 @ 10:00 pm

"I don't like odd numbers, and I really don't like primes."

— Marissa Mayer

See Cube Symmetry Axes in this journal.

IMAGE- The 13 symmetry axes of the cube

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Saturday, January 4, 2014

For Phil Everly

Filed under: General — m759 @ 10:00 am

A Souther song at YouTube.

See also the lyrics and, in this journal,
synchronicity on the uploading date.

Related art —

End of the Line Blues

IMAGE- YouTube statistics: 384 views, 3 thumbs up, 0 down.

and The Crosswicks Curse

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Sunday, December 29, 2013

Good Question

Filed under: General — Tags: Arcade Fire — m759 @ 3:00 pm

Amy Adams in the new film “Her” —

“You’re dating an OS? What is that like?”

— Question quoted in a Hollywood Reporter
story on the film’s second trailer

From the same story, by Philiana Ng —

” The trailer is set to Arcade Fire’s
mid-tempo ballad ‘Supersymmetry.’ “

Parts of an answer for Amy —

Nov. 26, 2012, as well as

July 19, 2008,

Dec. 18, 2013,

Dec. 24, 2013, and

Dec. 27, 2013.

The Hollywood Reporter story is from Dec. 3, 2013.
See also that date in this journal.

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Saturday, December 14, 2013

Beautiful Mathematics

Filed under: General,Geometry — Tags: Depth, Gitterkrieg, Octad, Octad Generator, on131214 — m759 @ 7:59 pm

The title, which I dislike, is taken from a 2011 publication
of the MAA, also sold by Cambridge University Press.

Some material relevant to the title adjective:

"For those who have learned something of higher mathematics, nothing could be more natural than to use the word 'beautiful' in connection with it. Mathematical beauty, like the beauty of, say, a late Beethoven quartet, arises from a combination of strangeness and inevitability. Simply defined abstractions disclose hidden quirks and complexities. Seemingly unrelated structures turn out to have mysterious correspondences. Uncanny patterns emerge, and they remain uncanny even after being underwritten by the rigor of logic."— Jim Holt, opening of a book review in the Dec. 5, 2013, issue of The New York Review of Books

Some relevant links—

The above list was updated on Jan. 31, 2014, to include the
"Strangeness" and "Hidden quirks" links. See also a post of
Jan. 31, 2014.

Update of March 9, 2014 —

The link "Simply defined abstractions" is to the construction of the Steiner
system S(5, 8, 24) described by R. T. Curtis in his 1976 paper defining the
Miracle Octad Generator. It should be noted that this construction is due
to Richard J. Turyn, in a 1967 Sylvania research report. (See Emily Jennings's
talk of 1 Nov. 2012.) Compare the Curtis construction, written in 1974,
with the Turyn construction of 1967 as described in Sphere Packings, Lattices
and Groups , by J. H. Conway and N. J. A. Sloane (first published in 1988).

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Saturday, November 30, 2013

Waiting for Ogdoad

Filed under: General,Geometry — Tags: Change Arises, Cube School, Eightfold Cube, Strange Change — m759 @ 10:30 am

Continued from October 30 (Devil’s Night), 2013.

“In a sense, we would see that change
arises from the structure of the object.”

— Theoretical physicist quoted in a
Simons Foundation article of Sept. 17, 2013

This suggests a review of mathematics and the
“Classic of Change ,” the I Ching .

The physicist quoted above was discussing a rather
complicated object. His words apply to a much simpler
object, an embodiment of the eight trigrams underlying
the I Ching as the corners of a cube.

Friday, November 29, 2013

Odd Facts

Filed under: General — m759 @ 7:01 pm

"These are odd facts…." — G. H. Hardy,
quoted in the previous post, "Centered"

Other odd facts:

If n is odd, then the object at the center
of the n×n square is a square.
Similarly for the n×n×n cube.

Related meditation:

“In a sense, we would see that change
arises from the structure of the object,” he said.
“But it’s not from the object changing.
The object is basically timeless.”

— Theoretical physicist quoted in a
Simons Foundation article of Sept. 17, 2013,
"A Jewel at the Heart of Quantum Physics"

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Sunday, November 17, 2013

The X-Men Tree

Filed under: General — Tags: Eichler, Frenkel-Metaphors, Wormhole — m759 @ 7:59 am

Continued from November 12, 2013. A post on that date
showed the tree from Waiting for Godot along with the two
X-Men patriarchs. See also last night's Chapel post,
which shows a more interesting tree—

A recent book on the Langlands program by Edward Frenkel
repeats a metaphor about building a bridge between unrelated
worlds within mathematics. A review of the Frenkel book by
Marcus du Sautoy replaces the bridge metaphor with a wormhole .
Some users of such metaphors seem to feel they are justified,
for maximum rhetorical effect, in lying about the unrelatedness of
the worlds being connected. The connections they discuss are
surprising (see the Eichler function discussed by Frenkel and
du Sautoy), but the connections occur, at least in the case of
elliptic curves and modular forms, between areas of mathematics
long known to be, in less subtle ways, related. See remarks
from 2005 by Diamond and Shurman below.

Related material:

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Saturday, November 16, 2013

Mathematics and Rhetoric

Filed under: General,Geometry — Tags: Frenkel-Metaphors — m759 @ 12:00 pm

Jim Holt in the current (Dec. 5) New York Review of Books—

One form of Eros is the sexual desire aroused by the physical beauty of a particular beloved person. That, according to Diotima, is the lowest form. With philosophical refinement, however, Eros can be made to ascend toward loftier and loftier objects. The penultimate of these—just short of the Platonic idea of beauty itself—is the perfect and timeless beauty discovered by the mathematical sciences. Such beauty evokes in those able to grasp it a desire to reproduce—not biologically, but intellectually, by begetting additional “gloriously beautiful ideas and theories.” For Diotima, and presumably for Plato as well, the fitting response to mathematical beauty is the form of Eros we call love.

Consider (for example) the beauty of the rolling donut—

(Animation source: MIQEL.com)

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Monday, August 12, 2013

Form

Filed under: General,Geometry — Tags: Diamond-Theory-tesseract, Natural Diagram, Octad, Octad Generator — m759 @ 12:00 pm

The Galois tesseract appeared in an early form in the journal
Computer Graphics and Art , Vol. 2, No. 1, February 1977—

The Galois tesseract is the basis for a representation of the smallest
projective 3-space, PG(3,2), that differs from the representation at
Wolfram Demonstrations Project. For the latter, see yesterday’s post.

The tesseract representation underlies the diamond theorem, illustrated
below in its earliest form, also from the above February 1977 article—

As noted in a more recent version, the group described by
the diamond theorem is also the group of the 35 square
patterns within the 1976 Miracle Octad Generator (MOG) of
R. T. Curtis.

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Sunday, July 21, 2013

Comic-Con

Filed under: General,Geometry — m759 @ 5:18 am

This is the weekend for Comic-Con International in San Diego.

The convention includes an art show. (Click above image to enlarge.)

Related material from Norway…

Suggested nominations for a Kavli Prize:

1. Josefine Lyche's highly imaginative catalog page for
the current Norwegian art exhibition I de lange nætter,
which mentions her interest in sacred geometry

2. Sacred Geometry: Drawing a Metatron Cube

… and from San Diego—

The Kavli Institutes logo:

IMAGE- Logo of the Kavli institutes

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Tuesday, July 9, 2013

Vril Chick

Filed under: General,Geometry — Tags: Fields, Octad, Octad Generator — m759 @ 4:30 am

Profile picture of "Jo Lyxe" (Josefine Lyche) at Vimeo—

Profile picture for "Jo Lyxe" (Josefine Lyche) at Vimeo

Compare to an image of Vril muse Maria Orsitsch.

From the catalog of a current art exhibition
(25 May – 31 August, 2013) in Norway,
I DE LANGE NÆTTER —

Josefine Lyche
Born in 1973 in Bergen, Norway.
Lives and works in Oslo and Berlin.

Keywords (to help place my artwork in the
proper context): Aliens, affine geometry, affine
planes, affine spaces, automorphisms, binary
codes, block designs, classical groups, codes,
coding theory, collineations, combinatorial,
combinatorics, conjugacy classes, the Conwell
correspondence, correlations, Cullinane,
R. T. Curtis, design theory, the diamond theorem,
diamond theory, duads, duality, error correcting
codes, esoteric, exceptional groups,
extraterrestrials, finite fields, finite geometry, finite
groups, finite rings, Galois fields, generalized
quadrangles, generators, geometry, GF(2),
GF(4), the (24,12) Golay code, group actions,
group theory, Hadamard matrices, hypercube,
hyperplanes, hyperspace, incidence structures,
invariance, Karnaugh maps, Kirkman’s schoolgirls
problem, Latin squares, Leech lattice, linear
groups, linear spaces, linear transformations,
Magick, Mathieu groups, matrix theory, Meno,
Miracle Octad Generator, MOG, multiply transitive
groups, occultism, octahedron, the octahedral
group, Orsic, orthogonal arrays, outer automorphisms,
parallelisms, partial geometries,
permutation groups, PG(3,2), Plato, Platonic
solids, polarities, Polya-Burnside theorem, projective
geometry, projective planes, projective
spaces, projectivities, Pythagoras, reincarnation,
Reed-Muller codes, the relativity problem,
reverse engineering, sacred geometry, Singer
cycle, skew lines, Socrates, sporadic simple
groups, Steiner systems, Sylvester, symmetric,
symmetry, symplectic, synthemes, synthematic,
Theosophical Society tesseract, Tessla, transvections,
Venn diagrams, Vril society, Walsh
functions, Witt designs.

(See also the original catalog page.)

Clearly most of this (the non-highlighted parts) was taken
from my webpage Diamond Theory. I suppose I should be
flattered, but I am not thrilled to be associated with the
(apparently fictional) Vril Society.

For some background, see (for instance)
Conspiracy Theories and Secret Societies for Dummies .

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Tuesday, June 25, 2013

Lexicon (continued)

Filed under: General,Geometry — m759 @ 7:20 pm

Online biography of author Cormac McCarthy—

"… he left America on the liner Sylvania, intending to visit
the home of his Irish ancestors (a King Cormac McCarthy
built Blarney Castle)."

Two Years Ago:

Blarney in The Harvard Crimson—

Melissa C. Wong, illustration for "Atlas to the Text,"
by Nicholas T. Rinehart:

Thirty Years Ago:

Non-Blarney from a rural outpost—

Illustration for the generalized diamond theorem,
by Steven H. Cullinane:

See also Barry's Lexicon .

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Sunday, June 23, 2013

Random Dudes

Filed under: General — m759 @ 10:00 pm

Here is the link to an MIT Scratch project from the above comment.

See also a comment by a Random Norwegian Dude:

For related art, see
"4D AMBASSADOR (HYPERCUBE)" for Steven H. Cullinane
by the Norwegian artist Josefine Lyche.

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Friday, June 14, 2013

Object of Beauty

Filed under: General,Geometry — Tags: Device Narratives, Eightfold Cube, on130614 — m759 @ 2:01 am

This journal on July 5, 2007 —

“It is not clear why MySpace China will be successful."

— The Chinese magazine Caijing in 2007, quoted in
Asia Sentinel on July 12, 2011

This journal on that same date, July 12, 2011 —

See also the eightfold cube and kindergarten blocks
at finitegeometry.org/sc.

Friedrich Froebel, Froebel's Chief Writings on Education ,
Part II, "The Kindergarten," Ch. III, "The Third Play":

"The little ones, who always long for novelty and change,
love this simple plaything in its unvarying form and in its
constant number, even as they love their fairy tales with
the ever-recurring dwarfs…."

This journal, Group Actions, Nov. 14, 2012:

"Those who insist on vulgarizing their mathematics
may regard linear and affine group actions on the eight
cubes as the dance of Snow White (representing (0,0,0))
and the Seven Dwarfs—

Edwin M. Knowles Fine China Company, 1991

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Thursday, June 13, 2013

Gate

Filed under: General,Geometry — Tags: Cube School, Field Dream, Field Theology, Vanishing Point — m759 @ 2:13 pm

"Eight is a Gate." — Mnemonic rhyme

Today's previous post, Window, showed a version
of the Chinese character for "field"—

This suggests a related image—

The related image in turn suggests…

Unlike linear perspective, axonometry has no vanishing point,
and hence it does not assume a fixed position by the viewer.
This makes axonometry 'scrollable'. Art historians often speak of
the 'moving' or 'shifting' perspective in Chinese paintings.

Axonometry was introduced to Europe in the 17th century by
Jesuits returning from China.

— Jan Krikke

As was the I Ching. A related structure:

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Saturday, June 8, 2013

Multispeech (continued)

Filed under: General — m759 @ 2:01 pm

For this, the dies natalis of poet Gerard Manley Hopkins,
it seems apt to cite a 1973 master's thesis on what the author
calls multiguity in Hopkins.

Tuesday, May 28, 2013

Codes

Filed under: General,Geometry — Tags: Coding Theory, Diamond-Theory-tesseract, Geometry, Natural Diagram, Octad, Octad Generator, Priority — m759 @ 12:00 pm

The hypercube model of the 4-space over the 2-element Galois field GF(2):

IMAGE- A hyperspace model of the 4D vector space over GF(2)

The phrase Galois tesseract may be used to denote a different model
of the above 4-space: the 4×4 square.

MacWilliams and Sloane discussed the Miracle Octad Generator
(MOG) of R. T. Curtis further on in their book (see below), but did not
seem to realize in 1977 that the 4×4 structures within the MOG are
based on the Galois-tesseract model of the 4-space over GF(2).

IMAGE- Octads within the Curtis MOG, which uses a 4x4-array model of the 4D vector space over GF(2)

The thirty-five 4×4 structures within the MOG:

IMAGE- The 35 square patterns within the Curtis MOG

Curtis himself first described these 35 square MOG patterns
combinatorially, (as his title indicated) rather than
algebraically or geometrically:

IMAGE- R. T. Curtis's combinatorial construction of 4x4 patterns within the Miracle Octad Generator

A later book co-authored by Sloane, first published in 1988,
did recognize the 4×4 MOG patterns as based on the 4×4
Galois-tesseract model.

Between the 1977 and 1988 Sloane books came the diamond theorem.

Update of May 29, 2013:

The Galois tesseract appeared in an early form in the journal
Computer Graphics and Art , Vol. 2, No. 1, February 1977
(the year the above MacWilliams-Sloane book was first published):

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Sunday, May 19, 2013

Priority Claim

Filed under: General,Geometry — Tags: Geometry, Mathieu Moonshine, May 19 Gestalt, Octad, Octad Generator, Overarching, Overarching Group, Priority, Rosenhain and Göpel — m759 @ 9:00 am

From an arXiv preprint submitted July 18, 2011,
and last revised on March 11, 2013 (version 4):

"By our construction, this vector space is the dual
of our hypercube F₂⁴ built on I \ O₉. The vector space
structure of the latter, to our knowledge, is first
mentioned by Curtis in [Cur89]. Hence altogether
our proposition 2.3.4 gives a novel geometric
meaning in terms of Kummer geometry to the known
vector space structure on I \ O₉."

[Cur89] reference:
R. T. Curtis, "Further elementary techniques using
the miracle octad generator," Proc. Edinburgh
Math. Soc. 32 (1989), 345-353 (received on
July 20, 1987).

— Anne Taormina and Katrin Wendland,
"The overarching finite symmetry group of Kummer
surfaces in the Mathieu group M ₂₄,"
arXiv.org > hep-th > arXiv:1107.3834

"First mentioned by Curtis…."

No. I claim that to the best of my knowledge, the
vector space structure was first mentioned by me,
Steven H. Cullinane, in an AMS abstract submitted
in October 1978, some nine years before the
Curtis article.

Update of the above paragraph on July 6, 2013—

No. The vector space structure was described by
(for instance) Peter J. Cameron in a 1976
Cambridge University Press book —
Parallelisms of Complete Designs .
See the proof of Theorem 3A.13 on pages 59 and 60.

The vector space structure as it occurs in a 4×4 array
of the sort that appears in the Curtis Miracle Octad
Generator may first have been pointed out by me,
Steven H. Cullinane, in an AMS abstract submitted in
October 1978, some nine years before the Curtis article.

See Notes on Finite Geometry for some background.

See in particular The Galois Tesseract.

For the relationship of the 1978 abstract to Kummer
geometry, see Rosenhain and Göpel Tetrads in PG(3,2).

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Saturday, May 11, 2013

Core

Filed under: General,Geometry — Tags: Cube School, Eightfold Cube, Hofstadter-Cube, Psycho History, Rubik Adoration — m759 @ 12:00 pm

Promotional description of a new book:

“Like Gödel, Escher, Bach before it, Surfaces and Essences will profoundly enrich our understanding of our own minds. By plunging the reader into an extraordinary variety of colorful situations involving language, thought, and memory, by revealing bit by bit the constantly churning cognitive mechanisms normally completely hidden from view, and by discovering in them one central, invariant core— the incessant, unconscious quest for strong analogical links to past experiences— this book puts forth a radical and deeply surprising new vision of the act of thinking.”

“Like Gödel, Escher, Bach before it….”

Or like Metamagical Themas .

Rubik core:

Swarthmore Cube Project, 2008

Non- Rubik cores:

Of the odd nxnxn cube:

Of the even nxnxn cube:

Related material: The Eightfold Cube and…

“A core component in the construction
is a 3-dimensional vector space V over F₂.”

— Page 29 of “A twist in the M₂₄ moonshine story,”
by Anne Taormina and Katrin Wendland.
(Submitted to the arXiv on 13 Mar 2013.)

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Saturday, April 13, 2013

Narrative

Filed under: General — m759 @ 6:16 pm

Two friends from Brooklyn —

"… both marveled at early Ingmar Bergman movies."

One of the friends' "humor was inspired by
surrealist painters and Franz Kafka."

Wikipedia—

"Most of Marvel's fictional characters operate in
a single reality known as the Marvel Universe…."

This journal yesterday —

Related material: The Cosmic Cube.

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Friday, April 12, 2013

Midnight in Paris

Filed under: General,Geometry — m759 @ 6:00 pm

Surreal requiem for the late Jonathan Winters:

"They 'burn, burn, burn like fabulous yellow roman candles
exploding like spiders across the stars,'
as Jack Kerouac once wrote. It was such a powerful
image that Wal-Mart sells it as a jigsaw puzzle."

— "When the Village Was the Vanguard,"
by Henry Allen, in today's Wall Street Journal

See also Damnation Morning and the picture in
yesterday evening's remarks on art:

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Tuesday, March 26, 2013

Blockheads

Filed under: General,Geometry — Tags: Eightfold Cube — m759 @ 12:00 pm

(Continued)

"It should be emphasized that block models are physical models, the elements of which can be physically manipulated. Their manipulation differs in obvious and fundamental ways from the manipulation of symbols in formal axiomatic systems and in mathematics. For example the transformations described above, in which two linear arrays are joined together to form one array, or a rectangle of blocks is re-assembled into a linear array, are physical transformations not symbolic transformations. …"

— Storrs McCall, Department of Philosophy, McGill University, "The Consistency of Arithmetic"

"It should be emphasized…."

OK:

Storrs McCall at a 2008 philosophy conference .

His blocks talk was at 2:50 PM July 21, 2008.
See also this journal at noon that same day:

Froebel's Third Gift and the Eightfold Cube

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Monday, February 18, 2013

Permanence

Filed under: General,Geometry — m759 @ 2:00 pm

Inscribed hexagon (1984)

The well-known fact that a regular hexagon
may be inscribed in a cube was the basis
in 1984 for two ways of coloring the faces
of a cube that serve to illustrate some graphic
aspects of embodied Galois geometry—

Inscribed hexagon (2013)

A redefinition of the term "symmetry plane"
also uses the well-known inscription
of a regular hexagon in the cube—

Related material

"Here is another way to present the deep question 1984 raises…."

— "The Quest for Permanent Novelty," by Michael W. Clune,
The Chronicle of Higher Education , Feb. 11, 2013

“What we do may be small, but it has a certain character of permanence.”

— G. H. Hardy, A Mathematician’s Apology

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Sunday, February 10, 2013

The Cleaning

Filed under: General,Geometry — m759 @ 12:00 am

Arthur Jaffe CV:

"In 2005 Arthur Jaffe succeeded Sir Michael Atiyah as
Chair of the Board of the Dublin Institute for Advanced Study,
School of Theoretical Physics."

Related material:

Biddies in this journal and…

Detail:

An early version of quaternions.

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Wednesday, January 9, 2013

Bad Idea

Filed under: General,Geometry — Tags: on130109 — m759 @ 12:00 pm

For the 2013 Joint Mathematics Meetings in San Diego,
which start today, a cartoon by Andrew Spann—

(Click for larger image.)

Related remarks:

This journal on the Feast of Epiphany, 2013—

"The Tesseract is where it belongs: out of our reach."

— The Avengers' Nick Fury, played by Samuel L. Jackson

Today's New York Times —

"You never know what could happen.
If you have Sam, you’re going to be cool."

— The late David R. Ellis, film director

If anyone in San Diego cares about the relationship
of Spann's plane to Fury's Tesseract, he or she may
consult Finite Geometry of the Square and Cube.

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Saturday, January 5, 2013

Vector Addition in a Finite Field

Filed under: General,Geometry — Tags: Geometry, Priority — m759 @ 10:18 am

The finite (i.e., Galois) field GF(16),
according to J. J. Seidel in 1974—

The same field according to Steven H. Cullinane in 1986,
in its guise as the affine 4-space over GF(2)—

The same field, again disguised as an affine 4-space,
according to John H. Conway and N.J.A. Sloane in
Sphere Packings, Lattices, and Groups , first published in 1988—

The above figure by Conway and Sloane summarizes, using
a 4×4 array, the additive vector-space structure of the finite
field GF(16).

This structure embodies what in Euclidean space is called
the parallelogram rule for vector addition—

(Thanks to June Lester for the 3D (uvw) part of the above figure.)

For the transition from this colored Euclidean hypercube
(used above to illustrate the parallelogram rule) to the
4×4 Galois space (illustrated by Cullinane in 1979 and
Conway and Sloane in 1988— or later… I do not have
their book’s first edition), see Diamond Theory in 1937,
Vertex Adjacency in a Tesseract and in a 4×4 Array,
Spaces as Hypercubes, and The Galois Tesseract.

For some related narrative, see tesseract in this journal.

(This post has been added to finitegeometry.org.)

Update of August 9, 2013—

Coordinates for hypercube vertices derived from the
parallelogram rule in four dimensions were better
illustrated by Jürgen Köller in a web page archived in 2002.

Update of August 13, 2013—

The four basis vectors in the 2002 Köller hypercube figure
are also visible at the bottom of the hypercube figure on
page 7 of “Diamond Theory,” excerpts from a 1976 preprint
in Computer Graphics and Art , Vol. 2, No. 1, February 1977.
A predecessor: Coxeter’s 1950 hypercube figure from
“Self-Dual Configurations and Regular Graphs.”

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Thursday, December 27, 2012

Object Lesson

Filed under: General,Geometry — Tags: Crucible Raiders, Cube School, The Building-Block Metaphor, White Cube — m759 @ 3:17 am

Yesterday's post on the current Museum of Modern Art exhibition
"Inventing Abstraction: 1910-1925" suggests a renewed look at
abstraction and a fundamental building block: the cube.

From a recent Harvard University Press philosophical treatise on symmetry—

The treatise corrects Nozick's error of not crediting Weyl's 1952 remarks
on objectivity and symmetry, but repeats Weyl's error of not crediting
Cassirer's extensive 1910 (and later) remarks on this subject.

For greater depth see Cassirer's 1910 passage on Vorstellung :

IMAGE- Ernst Cassirer on 'representation' or 'Vorstellung' in 'Substance and Function' as 'the riddle of knowledge'

This of course echoes Schopenhauer, as do discussions of "Will and Idea" in this journal.

For the relationship of all this to MoMA and abstraction, see Cube Space and Inside the White Cube.

"The sacramental nature of the space becomes clear…." — Brian O'Doherty

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Monday, December 24, 2012

Eternal Recreation

Filed under: General,Geometry — Tags: ArkLaTex, Cube Descending, Cube School, Intake Manifold, Khora, Sacred Space, White Cube — m759 @ 3:17 am

Memories, Dreams, Reflections
by C. G. Jung

Recorded and edited By Aniela Jaffé, translated from the German
by Richard and Clara Winston, Vintage Books edition of April 1989

From pages 195-196:

"Only gradually did I discover what the mandala really is:
'Formation, Transformation, Eternal Mind's eternal recreation.'*
And that is the self, the wholeness of the personality, which if all
goes well is harmonious, but which cannot tolerate self-deceptions."

* Faust , Part Two, trans. by Philip Wayne (Harmondsworth,
England, Penguin Books Ltd., 1959), p. 79. The original:

… Gestaltung, Umgestaltung,
Des ewigen Sinnes ewige Unterhaltung….

Jung's "Formation, Transformation" quote is from the realm of
the Mothers (Faust , Part Two, Act 1, Scene 5: A Dark Gallery).
The speaker is Mephistopheles.

See also Prof. Bruce J. MacLennan on this realm
in a Web page from his Spring 2005 seminar on Faust:

"In alchemical terms, F is descending into the dark, formless
primary matter from which all things are born. Psychologically
he is descending into the deepest regions of the
collective unconscious, to the source of life and all creation.
Mater (mother), matrix (womb, generative substance), and matter
all come from the same root. This is Faust's next encounter with
the feminine, but it's obviously of a very different kind than his
relationship with Gretchen."

The phrase "Gestaltung, Umgestaltung " suggests a more mathematical
approach to the Unterhaltung . Hence…

Part I: Mothers

"The ultimate, deep symbol of motherhood raised to
the universal and the cosmic, of the birth, sending forth,
death, and return of all things in an eternal cycle,
is expressed in the Mothers, the matrices of all forms,
at the timeless, placeless originating womb or hearth
where chaos is transmuted into cosmos and whence
the forms of creation issue forth into the world of
place and time."

— Harold Stein Jantz, The Mothers in Faust:
The Myth of Time and Creativity ,
Johns Hopkins Press, 1969, page 37

Part II: Matrices

Part III: Spaces and Hypercubes

Click image for some background.

Part IV: Forms

Forms from the I Ching :

Click image for some background.

Forms from Diamond Theory :

Click image for some background.

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Saturday, November 24, 2012

Will and Representation*

Filed under: General,Geometry — Tags: Eightfold Cube — m759 @ 2:56 pm

Robert A. Wilson, in an inaugural lecture in April 2008—

Representation theory

A group always arises in nature as the symmetry group of some object, and group
theory in large part consists of studying in detail the symmetry group of some
object, in order to throw light on the structure of the object itself (which in some
sense is the “real” object of study).

But if you look carefully at how groups are used in other areas such as physics
and chemistry, you will see that the real power of the method comes from turning
the whole procedure round: instead of starting from an object and abstracting
its group of symmetries, we start from a group and ask for all possible objects
that it can be the symmetry group of .

This is essentially what we call Representation theory . We think of it as taking a
group, and representing it concretely in terms of a symmetrical object.

Now imagine what you can do if you combine the two processes: we start with a
symmetrical object, and find its group of symmetries. We now look this group up
in a work of reference, such as our big red book (The ATLAS of Finite Groups),
and find out about all (well, perhaps not all) other objects that have the same
group as their group of symmetries.

We now have lots of objects all looking completely different, but all with the same
symmetry group. By translating from the first object to the group, and then to
the second object, we can use everything we know about the first object to tell
us things about the second, and vice versa.

As Poincaré said,

Mathematicians do not study objects, but relations between objects.
Thus they are free to replace some objects by others, so long as the
relations remain unchanged.

Par exemple—

Fano plane transformed to eightfold cube,
and partitions of the latter as points of the former:

* For the "Will" part, see the PyrE link at Talk Amongst Yourselves.

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Monday, November 19, 2012

Poetry and Truth

Filed under: General,Geometry — Tags: Bard Rock, Geometry, Octad, Octad Generator, Poetry, Stevens Rock, The Poet's Pineapple — m759 @ 7:59 pm

From today's noon post—

"In all his poems with all their enchantments
for the poet himself, there is the final enchantment
that they are true. The significance of the poetic act
then is that it is evidence. It is instance and illustration.
It is an illumination of a surface,
the movement of a self in the rock.
Above all it is a new engagement with life.
It is that miracle to which the true faith of the poet
attaches itself."

— Wallace Stevens at Bard College, March 30, 1951

Stevens also said at Bard that

"When Joan of Arc said:

Have no fear: what I do, I do by command.
My brothers of Paradise tell me what I have to do.

these words were the words of an hallucination.
No matter what her brothers of Paradise drove her to do,
what she did was never a poetic act of faith in reality
because it could not be."

There are those who would dispute this.

Some related material:

"Ageometretos me eisito."—
"Let no one ignorant of geometry enter."—
Said to be a saying of Plato, part of the
seal of the American Mathematical Society—

A poetic approach to geometry—

"A surface" and "the rock," from All Saints' Day, 2012—

— and from 1981—

Some mathematical background for poets in Purgatory—

"… the Klein correspondence underlies Conwell's discussion
of eight heptads. These play an important role in another
correspondence, illustrated in the Miracle Octad Generator
of R. T. Curtis, that may be used to picture actions
of the large Mathieu group M₂₄."

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Wednesday, November 14, 2012

Group Actions

Filed under: General,Geometry — Tags: Cube Quaternions, Cube School, Eightfold Cube, Time in Space — m759 @ 4:30 pm

The December 2012 Notices of the American
Mathematical Society has an ad on page 1564
(in a review of two books on vulgarized mathematics)
for three workshops next year on “Low-dimensional
Topology, Geometry, and Dynamics”—

(Only the top part of the ad is shown; for further details
see an ICERM page.)

(ICERM stands for Institute for Computational
and Experimental Research in Mathematics.)

The ICERM logo displays seven subcubes of
a 2x2x2 eight-cube array with one cube missing—

The logo, apparently a stylized image of the architecture
of the Providence building housing ICERM, is not unlike
a picture of Froebel’s Third Gift—

Froebel's third gift, the eightfold cube

Photo by Norman Brosterman from the Inventing Kindergarten
exhibit at The Institute for Figuring (co-founded by Margaret Wertheim)

The eighth cube, missing in the ICERM logo and detached in the
Froebel Cubes photo, may be regarded as representing the origin
(0,0,0) in a coordinatized version of the 2x2x2 array—
in other words the cube invariant under linear , as opposed to
more general affine , permutations of the cubes in the array.

These cubes are not without relevance to the workshops’ topics—
low-dimensional exotic geometric structures, group theory, and dynamics.

See The Eightfold Cube, A Simple Reflection Group of Order 168, and
The Quaternion Group Acting on an Eightfold Cube.

Those who insist on vulgarizing their mathematics may regard linear
and affine group actions on the eight cubes as the dance of
Snow White (representing (0,0,0)) and the Seven Dwarfs—

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Monday, November 5, 2012

For the Hollow Men

Filed under: General — Tags: Bitspace, Fawkes — m759 @ 12:00 pm

"Remember, remember the Fifth of November."

Very well. See a post of Nov. 5, 2005, and the related posts
Shadows, Cuber, Cube Partitions, and Cube Review.

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Saturday, November 3, 2012

Rigor

Filed under: General,Geometry — m759 @ 11:01 am

A New Yorker weblog post from yesterday, All Souls' Day—

"As the mathematician Terence Tao has written,
math study has three stages:
the 'pre-rigorous,' in which basic rules are learned,
the theoretical 'rigorous' stage, and, last and most intriguing,
'the post-rigorous,' in which intuition suddenly starts to play a part."

Related material—

Rigor in a Log24 post of Sunday evening, May 25, 2008: "Hall of Mirrors."

Note in that post the tesseract viewed as the lattice of
the 16 subsets of a 4-element set.

Some further material related to tesseracts and time, in three stages
(roughly corresponding to Tao's, but not in chronological order):

See also a recent Log24 post on remarks from Four Quartets .

(The vertices of a tesseract form, in various natural ways, four quartets.)

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Thursday, September 13, 2012

Backstory

Filed under: General — m759 @ 2:00 am

Yesterday's online Los Angeles Times
on a film that inspired recent protests in Cairo—

The film… was shown on June 23
to an audience of less than 10
at a theater on Hollywood Boulevard,
a source familiar with the screening said….
The screening was at The Vine Theater,
which rents itself out for private screenings,
said one person involved in the theater.

An image from this journal on that same day, June 23—

IMAGE- Rudolf Koch's version of the 'double cross' symbol

Source: Rudolf Koch, The Book of Signs

For some background on the symbol, see Damnation Morning.

See also Don Henley's Hollywood hymn "Garden of Allah."

Update of 8 PM Sept. 13, 2012—

Other sources give the film's screening date not as June 23,
2012, but rather as June 30, 2012. (BBC News, LAWEEKLY blogs)

The following post from this journal on that date may or
may not have some religious relevance.

Saturday, June 30, 2012

Snares

Filed under: Uncategorized — m759 @ 7:20 PM

"… to snare the spirits of mankind in nets of magic"

— The aim of the artist, according to Thomas Wolfe

Related entertainment—

High-minded— Many Dimensions .

Not so high-minded— The Cosmic Cube .

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Tuesday, September 11, 2012

Symmetry and Hierarchy

Filed under: General,Geometry — Tags: Crowe, Fitch, Pattern Theory, Wechsler Patterns — m759 @ 1:00 pm

A followup to Intelligence Test (April 2, 2012).

Philosophical Transactions of the Royal Society
B (2012) 367, 2007–2022
(theme issue of July 19, 2012) —

Production and Perception Rules

Underlying Visual Patterns: Effects of

Symmetry and Hierarchy

Gesche Westphal-Fitch [1], Ludwig Huber [2],
Juan Carlos Gómez [3], and W. Tecumseh Fitch [1]

[1] Department of Cognitive Biology, University of Vienna,
Althanstrasse 14, 1090 Vienna, Austria

[2] Messerli Research Institute, University of Veterinary Medicine Vienna,
Medical University of Vienna and University of Vienna,
Veterinärplatz 1, 1210 Vienna, Austria

[3] School of Psychology, St Mary’s College, University of St Andrews,
South Street, St Andrews, KY16 9JP, UK

Excerpt (added in an update on Dec. 8, 2012) —

Conclusion —

"… We believe that applying the theoretical

framework of formal language theory to two-dimensional

patterns offers a rich new perspective on the

human capacity for producing regular, hierarchically

organized structures. Such visual patterns may actually

prove more flexible than music or language for probing

the full extent of human pattern processing abilities.

With the results presented here, we have taken the

first steps in decoding the uniquely human fascination

with visual patterns, what Gombrich termed our

‘sense of order’.

Although the patterns we studied are most similar

to tilings or mosaics, they are examples of a much

broader type of abstract plane pattern, a type found

in virtually all of the world’s cultures [4]. Given that

such abstract visual patterns seem to represent

human universals, they have received astonishingly

little attention from psychologists. This neglect is particularly

unfortunate given their democratic nature,

their popular appeal and the ease with which they

can be generated and analysed in the laboratory.

With the current research, we hope to spark renewed

scientific interest in these ‘unregarded arts’, which

we believe have much to teach us about the nature of
the human mind."

[4] Washburn, D. K. & Crowe, D. W.,1988
Symmetries of Culture:
Theory and Practice of Plane Pattern Analysis.
Seattle, WA: University of Washington Press.

Commentary —

For symmetry , see Finite Geometry of the Square and Cube.

For hierarchy , see my assessment of Gombrich.
For culture , see T. S. Eliot and Russell Kirk on Eliot.

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Wednesday, August 22, 2012

For John Denver*

Filed under: General — m759 @ 1:01 am

A followup—

* Of "Country Roads"

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Thursday, July 26, 2012

Solomon’s Seal

Filed under: General,Geometry — Tags: Cullinane Cube, Ring and Stone, Solomon's Cube, Solomon's Mines — m759 @ 12:00 pm

(Mathematics and Narrative, continued)

Narrative—

The Ring and The Stone from yesterday’s post, and…

“In Medieval Jewish, Christian and Islamic legends,
the Seal of Solomon was a magical signet ring
said to have been possessed by King Solomon….”

— Wikipedia article, Seal of Solomon

Mathematics—

A fact related to the mathematical
“Solomon’s seal” described above by Bell:

The reference to Edge is as follows—

[3] Edge, W. L., Quadrics over GF(2) and
their relevance for the cubic surface group,
Canadian J. Maths. 11 (1959) ….

(This reference relates Hirschfeld’s remarks
quoted above to the 64-point affine space
illustrated below (via the associated
63-point projective space PG (5, 2)).

As for the narrative’s Stone…

See Solomon’s Cube.

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Saturday, June 30, 2012

Snares

Filed under: General — m759 @ 7:20 pm

"… to snare the spirits of mankind in nets of magic"

— The aim of the artist, according to Thomas Wolfe

Related entertainment—

High-minded— Many Dimensions .

Not so high-minded— The Cosmic Cube .

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Tuesday, June 26, 2012

Looking Deeply

Filed under: General,Geometry — Tags: Eightfold Cube, Looking Deeply, Triskele — m759 @ 3:48 pm

Last night's post on The Trinity of Max Black and the use of
the term "eightfold" by the Mathematical Sciences Research Institute
at Berkeley suggest a review of an image from Sept. 22, 2011—

The triskele detail above echoes a Buddhist symbol found,
for instance, on the Internet in an ad for meditation supplies—

Related remarks—

http://www.spencerart.ku.edu/about/dialogue/fdpt.shtml—

Mary Dusenbury (Radcliffe '64)—

"… I think a textile, like any work of art, holds a tremendous amount of information— technical, material, historical, social, philosophical— but beyond that, many works of art are very beautiful and they speak to us on many layers— our intellect, our heart, our emotions. I've been going to museums since I was a very small child, thinking about what I saw, and going back to discover new things, to see pieces that spoke very deeply to me, to look at them again, and to find more and more meaning relevant to me in different ways and at different times of my life. …

… I think I would suggest to people that first of all they just look. Linger by pieces they find intriguing and beautiful, and look deeply. Then, if something interests them, we have tried to put a little information around the galleries to give a bit of history, a bit of context, for each piece. But the most important is just to look very deeply."

http://en.wikipedia.org/wiki/Nikaya_Buddhism—

According to Robert Thurman, the term "Nikāya Buddhism" was coined by Professor Masatoshi Nagatomi of Harvard University, as a way to avoid the usage of the term Hinayana.^[12] "Nikaya Buddhism" is thus an attempt to find a more neutral way of referring to Buddhists who follow one of the early Buddhist schools, and their practice.

12. The Emptiness That is Compassion:
An Essay on Buddhist Ethics, Robert A. F. Thurman, 1980
[Religious Traditions , Vol. 4 No. 2, Oct.-Nov. 1981, pp. 11-34]

http://dsal.uchicago.edu/cgi-bin/philologic/getobject.pl?c.2:1:6.pali—

Nikāya [Sk. nikāya, ni+kāya]
collection ("body") assemblage, class, group

http://en.wiktionary.org/wiki/नि—

Sanskrit etymology for नि (ni)

1 From Proto-Indo-European *ni …

Prefix

नि (ni)

down
back
in, into

http://www.rigpawiki.org/index.php?title=Kaya—

Kaya (Skt. kāya ; སྐུ་, Tib. ku ; Wyl. sku ) —
the Sanskrit word kaya literally means ‘body’
but can also signify dimension, field or basis.

སྐུ། (Wyl. sku ) n. Pron.: ku

• structure, existentiality, founding stratum ▷HVG KBEU

• gestalt ▷HVG LD

Note that The Trinity of Max Black is a picture of a set—
i.e., of an "assemblage, class, group."

Note also the reference above to the word "gestalt."

"Was ist Raum, wie können wir ihn
erfassen und gestalten?"

— Walter Gropius

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Bright Black

Filed under: General,Geometry — Tags: Cube School, Looking Deeply — m759 @ 12:12 am

“‘In the dictionary next to [the] word “bright,” you should see Paula’s picture,’ he said. ‘She was super smart, with a sparkling wit. … She had a beautiful sense of style and color.'”

— Elinor J. Brecher in The Miami Herald on June 8, quoting Palm Beach Post writer John Lantigua on the late art historian Paula Hays Harper

This journal on the date of her death—

IMAGE- The Trinity of Max Black (a 3-set, with its eight subsets arranged in a Hasse diagram that is also a cube)

For some simpleminded commentary, see László Lovász on the cube space.

Some less simpleminded commentary—

“Was ist Raum, wie können wir ihn
erfassen und gestalten?”

Walter Gropius,

The Theory and
Organization of the
Bauhaus (1923)

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Saturday, June 16, 2012

Chiral Problem

Filed under: General,Geometry — Tags: Cube Quaternions, Cube School, Eightfold Cube, Wheeler — m759 @ 1:06 am

In memory of William S. Knowles, chiral chemist, who died last Wednesday (June 13, 2012)—

Detail from the Harvard Divinity School 1910 bookplate in yesterday morning's post—

"ANDOVER–HARVARD THEOLOGICAL LIBRARY"

Detail from Knowles's obituary in this morning's New York Times—

William Standish Knowles was born in Taunton, Mass., on June 1, 1917. He graduated a year early from the Berkshire School, a boarding school in western Massachusetts, and was admitted to Harvard. But after being strongly advised that he was not socially mature enough for college, he did a second senior year of high school at another boarding school, Phillips Academy in Andover, N.H.

Dr. Knowles graduated from Harvard with a bachelor’s degree in chemistry in 1939….

"This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them."

— Hermann Weyl, The Classical Groups, Princeton University Press, 1946, p. 16

From Pilate Goes to Kindergarten—

The six congruent quaternion actions illustrated above are based on the following coordinatization of the eightfold cube—

Problem: Is there a different coordinatization
that yields greater symmetry in the pictures of
quaternion group actions?

A paper written in a somewhat similar spirit—

"Chiral Tetrahedrons as Unitary Quaternions"—

ABSTRACT: Chiral tetrahedral molecules can be dealt [with] under the standard of quaternionic algebra. Specifically, non-commutativity of quaternions is a feature directly related to the chirality of molecules….

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Sunday, June 3, 2012

Child’s Play

Filed under: General,Geometry — Tags: Cube Descending, Cube School, Eightfold Cube, Geometric Theology — m759 @ 2:56 pm

(Continued)

– Max Black, Caveats and Critiques: Philosophical Essays
in Language, Logic, and Art , Cornell U. Press, 1975

IMAGE- The Trinity of Max Black (a 3-set, with its eight subsets arranged in a Hasse diagram that is also a cube)

Related material—

The Trinity Cube

Comments (2)

Saturday, May 19, 2012

G8

Filed under: General,Geometry — Tags: Cube Quaternions, Cube School, Eightfold Cube — m759 @ 8:00 pm

"The group of 8" is a phrase from politics, not mathematics.
Of the five groups of order 8 (see today's noon post),

the one pictured* in the center, Z₂ × Z₂ × Z₂ , is of particular
interest. See The Eightfold Cube. For a connection of this
group of 8 to the last of the five pictured at noon, the
quaternion group, see Finite Geometry and Physical Space.

* The picture is of the group's cycle graph.

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Friday, April 27, 2012

References

Filed under: General — m759 @ 12:00 pm

"These lowbrow, popular cultural references bring the possibilities
of interpretation down to an everyday level, forcing us to acknowledge
that not every painting that looks like a splatter is necessarily a
homage/anti-homage to Jackson Pollock."

— Stina Högqvist, review of the art of Josefine Lyche

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Wednesday, April 18, 2012

Adam in Eden

Filed under: General,Geometry — Tags: Cullinane Cube, Eightfold Cube, on0418, Solomon's Cube — m759 @ 9:00 pm

…. and John Golding, an authority on Cubism who "courted abstraction"—

"Adam in Eden was the father of Descartes." — Wallace Stevens

Fictional symbologist Robert Langdon and a cube—

From a Log24 post, "Eightfold Cube Revisited,"
on the date of Golding's death—

Dynkin diagram D4 for triality

A related quotation—

"… quaternions provide a useful paradigm
for studying the phenomenon of 'triality.'"

— David A. Richter's webpage Zometool Triality

See also quaternions in another Log24 post
from the date of Golding's death— Easter Act.

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Thursday, April 12, 2012

Mythopoetic*

Filed under: General,Geometry — m759 @ 9:29 pm

"Is Space Digital?"

— Cover story, Scientific American magazine, February 2012

"The idea that space may be digital
is a fringe idea of a fringe idea
of a speculative subfield of a subfield."

— Physicist Sabine Hossenfelder
at her weblog on Feb. 5, 2012

"A quantization of space/time
is a holy grail for many theorists…."

— Peter Woit in a comment at his physics weblog today

Thursday, March 1, 2012

Block That Metaphor:

Filed under: General,Geometry — Tags: Eightfold Cube, Gopnik, The Building-Block Metaphor — m759 @ 11:09 pm

The Cube Model and Peano Arithmetic

The eightfold cube model of the Fano plane may or may not have influenced a new paper (with the date Feb. 10, 2011, in its URL) on an attempted consistency proof of Peano arithmetic—

The Consistency of Arithmetic, by Storrs McCall

"Is Peano arithmetic (PA) consistent? This paper contains a proof that it is. …

Axiomatic proofs we may categorize as 'syntactic', meaning that they concern only symbols and the derivation of one string of symbols from another, according to set rules. 'Semantic' proofs, on the other hand, differ from syntactic proofs in being based not only on symbols but on a non-symbolic, non-linguistic component, a domain of objects. If the sole paradigm of 'proof ' in mathematics is 'axiomatic proof ', in which to prove a formula means to deduce it from axioms using specified rules of inference, then Gödel indeed appears to have had the last word on the question of PA-consistency. But in addition to axiomatic proofs there is another kind of proof. In this paper I give a proof of PA's consistency based on a formal semantics for PA. To my knowledge, no semantic consistency proof of Peano arithmetic has yet been constructed.

The difference between 'semantic' and 'syntactic' theories is described by van Fraassen in his book The Scientific Image :

"The syntactic picture of a theory identifies it with a body of theorems, stated in one particular language chosen for the expression of that theory. This should be contrasted with the alternative of presenting a theory in the first instance by identifying a class of structures as its models. In this second, semantic, approach the language used to express the theory is neither basic nor unique; the same class of structures could well be described in radically different ways, each with its own limitations. The models occupy centre stage." (1980, p. 44)

Van Fraassen gives the example on p. 42 of a consistency proof in formal geometry that is based on a non-linguistic model. Suppose we wish to prove the consistency of the following geometric axioms:

A1. For any two lines, there is at most one point that lies on both.
A2. For any two points, there is exactly one line that lies on both.
A3. On every line there lie at least two points.

The following diagram shows the axioms to be consistent:

Figure 1

The consistency proof is not a 'syntactic' one, in which the consistency of A1-A3 is derived as a theorem of a deductive system, but is based on a non-linguistic structure. It is a semantic as opposed to a syntactic proof. The proof constructed in this paper, like van Fraassen's, is based on a non-linguistic component, not a diagram in this case but a physical domain of three-dimensional cube-shaped blocks. ….

… The semantics presented in this paper I call 'block semantics', for reasons that will become clear…. Block semantics is based on domains consisting of cube-shaped objects of the same size, e.g. children's wooden building blocks. These can be arranged either in a linear array or in a rectangular array, i.e. either in a row with no space between the blocks, or in a rectangle composed of rows and columns. A linear array can consist of a single block, and the order of individual blocks in a linear or rectangular array is irrelevant. Given three blocks A, B and C, the linear arrays ABC and BCA are indistinguishable. Two linear arrays can be joined together or concatenated into a single linear array, and a rectangle can be re-arranged or transformed into a linear array by successive concatenation of its rows. The result is called the 'linear transformation' of the rectangle. An essential characteristic of block semantics is that every domain of every block model is finite. In this respect it differs from Tarski’s semantics for first-order logic, which permits infinite domains. But although every block model is finite, there is no upper limit to the number of such models, nor to the size of their domains.

It should be emphasized that block models are physical models, the elements of which can be physically manipulated. Their manipulation differs in obvious and fundamental ways from the manipulation of symbols in formal axiomatic systems and in mathematics. For example the transformations described above, in which two linear arrays are joined together to form one array, or a rectangle of blocks is re-assembled into a linear array, are physical transformations not symbolic transformations. …"

— Storrs McCall, Department of Philosophy, McGill University

Friday, February 17, 2012

Pregeometry and Finite Geometry

Filed under: General,Geometry — Tags: Wheeler — m759 @ 7:35 pm

Today's previous post, on the Feb. 2012 Scientific American
article "Is Space Digital?", suggested a review of a notion
that the theoretical physicist John Archibald Wheeler called
pregeometry .

From a paper on that topic—

"… the idea that geometry should constitute
'the magic building material of the universe'
had to collapse on behalf of what Wheeler
has called pregeometry (see Misner et al. 1973,
pp. 1203-1212; Wheeler 1980), a somewhat
indefinite term which expresses “a combination
of hope and need, of philosophy and physics
and mathematics and logic” (Misner et al. 1973,
p. 1203)."

— Jacques Demaret, Michael Heller, and
Dominique Lambert, "Local and Global Properties
of the World," preprint of paper published in
Foundations of Science 2 (1): 137-176

Misner, C. W., Thorne, K. S. and Wheeler, J. A.
1973, Gravitation , W.H. Freeman and Company:
San Francisco.

Wheeler, J.A. 1980, "Pregeometry: Motivations
and Prospects," in: Quantum Theory and Gravitation ,
ed. A.R. Marlow, Academic Press: New York, pp. 1-11.

Some related material from pure mathematics—

Click image for further details.

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Tuesday, February 14, 2012

The Ninth Configuration

Filed under: General,Geometry — m759 @ 2:01 pm

The showmanship of Nicki Minaj at Sunday's
Grammy Awards suggested the above title,
that of a novel by the author of The Exorcist .

The Ninth Configuration —

The ninth* in a list of configurations—

"There is a (2^d-1)_d configuration
known as the Cox configuration."

— MathWorld article on "Configuration"

For further details on the Cox 32₆ configuration's Levi graph,
a model of the 64 vertices of the six-dimensional hypercube γ₆,
see Coxeter, "Self-Dual Configurations and Regular Graphs,"
Bull. Amer. Math. Soc. Vol. 56, pages 413-455, 1950.
This contains a discussion of Kummer's 16₆ as it
relates to γ₆, another form of the 4×4×4 Galois cube.

See also Solomon's Cube.

* Or tenth, if the fleeting reference to 11₃ configurations is counted as the seventh—
and then the ninth would be a 15₃ and some related material would be Inscapes.

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Friday, January 27, 2012

Mathematics and Narrative (continued)

Filed under: General,Geometry — Tags: January Narratives — m759 @ 1:44 am

Princeton University Press on a book it will publish in March—

Circles Disturbed: The Interplay of Mathematics and Narrative

"Circles Disturbed brings together important thinkers in mathematics, history, and philosophy to explore the relationship between mathematics and narrative. The book's title recalls the last words of the great Greek mathematician Archimedes before he was slain by a Roman soldier— 'Don't disturb my circles'— words that seem to refer to two radically different concerns: that of the practical person living in the concrete world of reality, and that of the theoretician lost in a world of abstraction. Stories and theorems are, in a sense, the natural languages of these two worlds–stories representing the way we act and interact, and theorems giving us pure thought, distilled from the hustle and bustle of reality. Yet, though the voices of stories and theorems seem totally different, they share profound connections and similarities."

Timeline of the Marvel Cinematic Universe — Norway, March 1942—

"The Red Skull finds the Tesseract, a cube of strange power,
said to be the jewel of Odin’s treasure room, in Tonsberg Norway.
(Captain America: The First Avenger)"

Tesseracts Disturbed — (Click to enlarge)

Detail of Tesseracts Disturbed —

Narrative of the detail—

See Tesseract in this journal and Norway, May 2010—

The Oslo Version and Annals of Conceptual Art.

"Oh, what a tangled web we weave…"

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Tuesday, January 3, 2012

Theorum

Filed under: General,Geometry — Tags: Cube Quaternions, Cube School, Eightfold Cube — m759 @ 7:48 am

In memory of artist Ronald Searle—

Searle reportedly died at 91 on December 30th.

From Log24 on that date—

Click the above image for some context.

Update of 9:29 PM EST Jan. 3, 2012—

Theorum

From RationalWiki

Theorum (rhymes with decorum, apparently) is a neologism proposed by Richard Dawkins in The Greatest Show on Earth to distinguish the scientific meaning of theory from the colloquial meaning. In most of the opening introduction to the show, he substitutes "theorum" for "theory" when referring to the major scientific theories such as evolution.

Problems with "theory"

Dawkins notes two general meanings for theory; the scientific one and the general sense that means a wild conjecture made up by someone as an explanation. The point of Dawkins inventing a new word is to get around the fact that the lay audience may not thoroughly understand what scientists mean when they say "theory of evolution". As many people see the phrase "I have a theory" as practically synonymous with "I have a wild guess I pulled out of my backside", there is often confusion about how thoroughly understood certain scientific ideas are. Hence the well known creationist argument that evolution is "just a theory" – and the often cited response of "but gravity is also just a theory".

To convey the special sense of thoroughness implied by the word theory in science, Dawkins borrowed the mathematical word "theorem". This is used to describe a well understood mathematical concept, for instance Pythagoras' Theorem regarding right angled triangles. However, Dawkins also wanted to avoid the absolute meaning of proof associated with that word, as used and understood by mathematicians. So he came up with something that looks like a spelling error. This would remove any person's emotional attachment or preconceptions of what the word "theory" means if it cropped up in the text of The Greatest Show on Earth , and so people would (in "theory ") have no other choice but to associate it with only the definition Dawkins gives.

This phrase has completely failed to catch on, that is, if Dawkins intended it to catch on rather than just be a device for use in The Greatest Show on Earth . When googled, Google will automatically correct the spelling to theorem instead, depriving this very page its rightful spot at the top of the results.

Saturday, November 19, 2011

Star Wars (continued)

Filed under: General — m759 @ 1:14 pm

From Thursday's post All Things Fashion—

From today's online New York Times—

The nuclear symbol beneath the op-ed headline
is the most interesting part of this afternoon's front page—

Jung on projections—

It is possible to project certain characteristics onto another person who does not possess them at all, but the one being projected upon may unconsciously encourage it.

"It frequently happens that the object offers a hook to the projection, and even lures it out. This is generally the case when the object himself (or herself) is not conscious of the quality in question: in that way it works directly upon the unconscious of the projicient. For all projections provoke counter-projections when the object is unconscious of the quality projected upon it by the subject." ["General Aspects of Dream Psychology," CW 8, par. 519.]

For an object that "offers a hook to the projection," see yesterday's Hypercube Rotations—

Central projection
of the hypercube

See also Stallion Gate in this journal.

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Saturday, November 12, 2011

Professor Dodge

Filed under: General,Geometry — m759 @ 11:30 pm

From today's previous post, a fragmentary thought—

"Professor Dodge and the underground artists
whose work he helped save are the subjects of a book…"

Jim Dodge, Stone Junction

(a novel first published in 1989)

From pages 206-208, Kindle Edition—

`Have you seen it?'

Volta hesitated. `Well, I've dreamed it.'

Daniel shook his head. `I'm getting lost. You want me to vanish into your dreams?'

`Good Lord, no,' Volta blanched. `That's exactly what I don't want you to do.'

`So, what is it exactly you do want me to do?'

`Steal the diamond.'

`So, it's a diamond?'

`Yes, though it's a bit like saying the ocean is water. The diamond is perfectly spherical,* perfectly clear— though it seems to glow— and it's about two-thirds the size of a bowling ball. I think of it as the Diamond. Capital D.'

`Who owns it?'

`No one. The United States government has it at the moment. We want it. And to be honest with you, Daniel, I particularly want it, want it dearly. I want to look at it, into it, hold it in my hands. I had a vision involving a spherical diamond, a vision that changed my life, and I want to confirm that it was a vision of something real, the spirit embodied, the circuit complete.'

Daniel was smiling. `You're going to love this. That dream I wanted to talk to you about, my first since the explosion? It just happened to feature a raven with a spherical diamond in its beak. Obviously, it wasn't as big as a bowling ball, and there was a thin spiral flame running edge to edge through its center, which made it seem more coldly brilliant than warmly glowing, but it sounds like the same basic diamond to me.'

`And what do you think it is?'

`I think it's beautiful.'

Volta gave him a thin smile. `If I were more perverse than I already lamentably am, I would say it is the Eye of the Beholder. In fact, I don't know what it is.'

`It might be a dream,' Daniel said.

`Very possibly,' Volta agreed, `but I don't think so. I think— feel , to be exact— that the Diamond is an interior force given exterior density, the transfigured metaphor of the prima materia , the primordial mass, the Spiritus Mundi . I'm assuming you're familiar with the widely held supposition that the entire universe was created from a tiny ball of dense matter which exploded, sending pieces hurtling into space, expanding from the center. The spherical diamond is the memory, the echo, the ghost of that generative cataclysm; the emblematic point of origin. Or if, as some astrophysicists believe, the universe will reach some entropic point in its expansion and begin to collapse back into itself, in that case the Diamond may be a homing point, the seed crystal, to which it will all come hurtling back together— and perhaps through itself, into another dimension entirely. Or it might be the literal Philosopher's Stone we alchemists speak of so fondly. Or I might be completely wrong. That's why I want to see it. If I could actually stand in its presence, I'm convinced I'd know what it is. I would even venture to say, at the risk of rabid projection, that it wants to be seen and known.'

`But you're not even sure it exists,' Daniel said. `Right? And hey, it's tough to steal something that doesn't exist, even if you can be invisible. The more I think about this the less sense it makes.'

* Here Dodge's mystical vision seems akin to that of Anthony Judge in "Embodying the Sphere of Change" (St. Stephen's Day, 2001). Actually, the cube, not the sphere, is the best embodiment of Judge's vision.

See also Tuesday's "Stoned" and the 47 references
to the term "bowling" in the Kindle Stone Junction .

Furthermore… Live from New York, it's Saturday Night!

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Wednesday, September 21, 2011

Symmetric Generation

Filed under: General,Geometry — Tags: Aitchison, Crucible Raiders, Eightfold Cube, Generation, Geometry, Symmetric Generation — m759 @ 2:00 pm

Suggested by yesterday's Relativity Problem Revisited and by Cassirer on Objectivity—

From Symmetric Generation of Groups , by R.T. Curtis (Cambridge U. Press, 2007)—

"… we are saying much more than that G ≅ M₂₄ is generated by
some set of seven involutions, which would be a very weak
requirement. We are asserting that M₂₄ is generated by a set
of seven involutions which possesses all the symmetries of L₃(2)
acting on the points of the 7-point projective plane…."
— Symmetric Generation , p. 41

"It turns out that this approach is particularly revealing and that
many simple groups, both sporadic and classical, have surprisingly
simple definitions of this type."
— Symmetric Generation , p. 42

Sunday, July 31, 2011

Short Stories

Filed under: General — Tags: Facets — m759 @ 2:02 am

An Amazon.com reader review of Algis Budrys's Writing to the Point: A Complete Guide to Selling Fiction—

"Mr. Budrys claims to have the secret to writing fiction that will sell. His secret is very useful but short enough to include here:

Beginning: Must consist of introducing a character, in a particular context, with a problem. And if there are important yet unique/unusual aspects of the character that will be revealed later in the story they must be foreshadowed in the beginning.

Middle: Must involve the character attempting to solve the problem and encountering unexpected failure. During this attempt he begins to learn more about the problem and himself. The character must undergo stress which causes hitherto concealed facets of him to be revealed-that must fit in. The character must try to overcome the problem a total of 3 times on a rising scale of effort, commitment, and depth of knowledge of the problem and one's self. At the last possible moment, with maximum effort and staking everything, he achieves victory. This must be done by wagering everything in a do-or-die situation. Conversely the villain, coming closer to his goal experiences defeat snatched from the jaws of victory-because of some flaw in character.

End: Validation and foreclosure by someone who has no other vested interest in the story. They step forward and say 'He's dead, Jim' or 'Who was that masked man?' This serves to close the story in the reader's mind."

Here are two parallel stories suggested by yesterday's New York Lottery numbers:

Evening: 003 and 8997—

From an author born on 8/9/97:

For the 003, see

7/11.

Midday: 004 and 1931—

From an author born on 1/9/31:

For the 004, see the ideogram

in Beyond the Limits.

See also the day of the author's
death and the next day.

Happy Feast of St. Ignatius of Loyola.

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Sunday, June 26, 2011

Sunday Dinner

Filed under: General,Geometry — Tags: Fields, Security Story — m759 @ 2:22 pm

From "Sunday Dinner" in this journal—

"'If Jesus were to visit us, it would have been
the Sunday dinner he would have insisted on
being a part of, not the worship service at the church.'"

—Judith Shulevitz at The New York Times
on Sunday, July 18, 2010

Some table topics—

Today's midday New York Lottery numbers were 027 and 7002.

The former suggests a Galois cube, the latter a course syllabus—

CSC 7002
Graduate Computer Security (Spring 2011)
University of Colorado at Denver
Department of Computer Science

An item from that syllabus:

Six

22 February 2011

DES

History of DES; Encryption process; Decryption; Expander function; S-boxes and their output; Key; the function f that takes the modified key and part of the text as input; mulitple Rounds of DES; Present-day lack of Security in DES, which led to the new Encryption Standard, namely AES. Warmup for AES: the mathematics of Fields: Galois Fields, particularly the one of order 256 and its relation to the irreducible polynomial x^8 + x^4 + x^3 + x + 1 with coefficients from the field Z_2.

Related material: A novel, PopCo , was required reading for the course.

Discuss a different novel by the same author—

The End of Mr. Y .

Discuss the author herself, Scarlett Thomas.

Background for the discussion—

Derrida in this journal versus Charles Williams in this journal.

Related topics from the above syllabus date—

Metaphor and Gestell and Quadrat.

Some context— Midsummer Eve's Dream.

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Paradigms Lost

Filed under: General,Geometry — m759 @ 7:20 am

Continued from March 10, 2011 — A post that says

"If Galois geometry is thought of as a paradigm shift
from Euclidean geometry, both… the Kuhn cover
and the nine-point affine plane may be viewed…
as illustrating the shift."

Yesterday's posts The Fano Entity and Theology for Antichristmas,
together with this morning's New York Times obituaries (below)—

—suggest a Sunday School review from last year's
Devil's Night (October 30-31, 2010)—

Sunday, October 31, 2010

ART WARS –

m759 @ 2:00 AM

… There is a Cave
Within the Mount of God, fast by his Throne,
Where light and darkness in perpetual round
Lodge and dislodge by turns, which makes through Heav'n
Grateful vicissitude, like Day and Night….

– Paradise Lost , by John Milton

Click on figure for details.

Al Pacino in Devil's Advocate
as attorney John Milton

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Saturday, June 25, 2011

The Fano Entity

Filed under: General,Geometry — Tags: Eightfold Cube — m759 @ 2:02 am

The New York Times at 9 PM ET June 23, 2011—

ROBERT FANO: I’m trying to think briefly how to put it.

GINO FANO: "On the Fundamental Postulates"—

"E la prova di questo si ha precisamente nel fatto che si è potuto costruire (o, dirò meglio immaginare) un ente per cui sono verificati tutti i postulati precedenti…."

"The proof of this is precisely the fact that you could build (or, to say it better, imagine) an entity by which are verified all previous assumptions…."

Also from the Times article quoted above…

"… like working on a cathedral. We laid our bricks and knew that others might later replace them with better bricks. We believed in the cause even if we didn’t completely understand the implications.”

— Tom Van Vleck

Some art that is related, if only by a shared metaphor, to Van Vleck's cathedral—

The art is also related to the mathematics of Gino Fano.

For an explanation of this relationship (implicit in the above note from 1984),
see "The Fano plane revisualized—or: the eIghtfold cube."

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Tuesday, May 10, 2011

Groups Acting

Filed under: General,Geometry — Tags: Groups Act, Sacred Space, The Coxeter Aleph — m759 @ 10:10 am

The LA Times on last weekend's film "Thor"—

"… the film… attempts to bridge director Kenneth Branagh's high-minded Shakespearean intentions with Marvel Entertainment's bottom-line-oriented need to crank out entertainment product."

Those averse to Nordic religion may contemplate a different approach to entertainment (such as Taymor's recent approach to Spider-Man).

A high-minded— if not Shakespearean— non-Nordic approach to groups acting—

"What was wrong? I had taken almost four semesters of algebra in college. I had read every page of Herstein, tried every exercise. Somehow, a message had been lost on me. Groups act . The elements of a group do not have to just sit there, abstract and implacable; they can do things, they can 'produce changes.' In particular, groups arise naturally as the symmetries of a set with structure. And if a group is given abstractly, such as the fundamental group of a simplical complex or a presentation in terms of generators and relators, then it might be a good idea to find something for the group to act on, such as the universal covering space or a graph."

— Thomas W. Tucker, review of Lyndon's Groups and Geometry in The American Mathematical Monthly , Vol. 94, No. 4 (April 1987), pp. 392-394

"Groups act "… For some examples, see

Related entertainment—

High-minded— Many Dimensions—

Not so high-minded— The Cosmic Cube—

One way of blending high and low—

The high-minded Charles Williams tells a story
in his novel Many Dimensions about a cosmically
significant cube inscribed with the Tetragrammaton—
the name, in Hebrew, of God.

The following figure can be interpreted as
the Hebrew letter Aleph inscribed in a 3×3 square—

The above illustration is from undated software by Ed Pegg Jr.

For mathematical background, see a 1985 note, "Visualizing GL(2,p)."

For entertainment purposes, that note can be generalized from square to cube
(as Pegg does with his "GL(3,3)" software button).

For the Nordic-averse, some background on the Hebrew connection—

Coxeter and the Aleph and
Ayn Sof.

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Thursday, May 5, 2011

Beyond Forgetfulness

Filed under: General,Geometry — Tags: Cube School, Hofstadter-Cube, Rubik Adoration, Stevens Rock — m759 @ 10:10 am

From this journal on July 23, 2007—

It is not enough to cover the rock with leaves.
We must be cured of it by a cure of the ground
Or a cure of ourselves, that is equal to a cure

Of the ground, a cure beyond forgetfulness.
And yet the leaves, if they broke into bud,
If they broke into bloom, if they bore fruit,

And if we ate the incipient colorings
Of their fresh culls might be a cure of the ground.

– Wallace Stevens, “The Rock”

This quotation from Stevens (Harvard class of 1901) was posted here on when Daniel Radcliffe (i.e., Harry Potter) turned 18 in July 2007.

Other material from that post suggests it is time for a review of magic at Harvard.

On September 9, 2007, President Faust of Harvard

“encouraged the incoming class to explore Harvard’s many opportunities.

‘Think of it as a treasure room of hidden objects Harry discovers at Hogwarts,’ Faust said.”

That class is now about to graduate.

It is not clear what “hidden objects” it will take from four years in the Harvard treasure room.

Perhaps the following from a book published in 1985 will help…

The March 8, 2011, Harvard Crimson illustrates a central topic of Metamagical Themas , the Rubik’s Cube—

Hofstadter in 1985 offered a similar picture—

Hofstadter asks in his Metamagical introduction, “How can both Rubik’s Cube and nuclear Armageddon be discussed at equal length in one book by one author?”

For a different approach to such a discussion, see Paradigms Lost, a post made here a few hours before the March 11, 2011, Japanese earthquake, tsunami, and nuclear disaster—

Whether Paradigms Lost is beyond forgetfulness is open to question.

Perhaps a later post, in the lighthearted spirit of Faust, will help. See April 20th’s “Ready When You Are, C.B.“

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