(A review)
Friday, August 14, 2015
Discrete Space
Galois space:
Counting symmetries of Galois space:
The reason for these graphic symmetries in affine Galois space —
symmetries of the underlying projective Galois space:
Friday, December 18, 2020
Square Space at Athens
(A sequel to the previous post, Square Space at Wikipedia)
Related remarks: A Dec. 16 Wikipedia revision by Quack5quack,
and posts in this journal tagged Helsinki Math.
Thursday, December 17, 2020
Square Space at Wikipedia
The State of SquareSpace Art at Wikipedia as of December 16, 2020,
after a revision by an anonymous user on that date:
See also Square Space at Squarespace.
Monday, September 14, 2020
Space People Puzzle
Thursday, April 16, 2020
Wednesday, January 1, 2020
Exploring Inner Space* at The New York Times
From Corrections: Jan. 1, 2020 —
The astronomy article, by Dennis Overbye, is dated Dec. 23* (a Monday).
The above reference to "Tuesday" is explained by the fine print
at the bottom of the Science Times article — "A version of this article
appears in print on [Tuesday] , Section D, Page 6 of the
New York edition with the headline: In Battle of Giant Telescopes,
Outlook for the U.S. Dims."
From the article as quoted on Thursday, Dec. 26,
at https://uclafacultyassociation.blogspot.com —
"Now, as the wheels of the academic and government bureaucracy begin to turn, many American astronomers worry that they are following in the footsteps of their physicist colleagues. In 1993, Congress canceled the Superconducting Super Collider, and the United States ceded the exploration of inner space to Europe and CERN, which built the Large Hadron Collider, 27 miles in diameter, where the longsought Higgs boson was eventually discovered. The United States no longer builds particle accelerators. There could come a day, soon, when Americans no longer build giant telescopes. That would be a crushing disappointment to a handful of curious humans stuck on Earth, thirsting for cosmic grandeur. In outer space, nobody can hear you cry." Source: https://www.nytimes.com/2019/12/23/science/telescopesmagellanhawaiiastronomy.html 
Related material from this journal on April 2, 2019 —
Cover design by Greg Stadnyk, available in an animated gif.
* See also this journal on Dec. 23.
Monday, December 2, 2019
Gray Space
See as well a search for Gray Space in this journal.
Related material: The Schwartz Omega .
“Looking carefully at Golay’s code
is like staring into the sun.”
Monday, June 3, 2019
Art Wars for Spaceheads
Friday, May 3, 2019
The Structure of Story Space
Four Quartets
. . . Only by the form, the pattern,
Can words or music reach
The stillness, as a Chinese jar still
Moves perpetually in its stillness.
A Permanent Order of Wondertale Elements
In Vol. I of Structural Anthropology , p. 209, I have shown that this analysis alone can account for the double aspect of time representation in all mythical systems: the narrative is both “in time” (it consists of a succession of events) and “beyond” (its value is permanent). With regard to Propp’s theories my analysis offers another advantage: I can reconcile much better than Propp himself his principle of a permanent order of wondertale elements with the fact that certain functions or groups of functions are shifted from one tale to the next (pp. 9798. p. 108). If my view is accepted, the chronological succession will come to be absorbed into an atemporal matrix structure whose form is indeed constant. The shifting of functions is then no more than a mode of permutation (by vertical columns or fractions of columns). 
… Or by congruent quartersections.
Thursday, May 2, 2019
Story Space
Monday, March 25, 2019
Sunday, March 24, 2019
Espacement: Geometry of the Interstice in Literary Theory
"You said something about the significance of spaces between
elements being repeated. Not only the element itself being repeated,
but the space between. I'm very interested in the space between.
That is where we come together." — Peter Eisenman, 1982
https://www.parrhesiajournal.org/ Parrhesia No. 3 • 2007 • 22–32
(Up) Against the (In) Between: Interstitial Spatiality by Clare Blackburne Blackburne — www.parrhesiajournal.org 24 — "The excessive notion of espacement as the resurgent spatiality of that which is supposedly ‘without space’ (most notably, writing), alerts us to the highly dynamic nature of the interstice – a movement whose discontinuous and ‘aberrant’ nature requires further analysis." Blackburne — www.parrhesiajournal.org 25 — "Espacement also evokes the ambiguous figure of the interstice, and is related to the equally complex derridean notions of chora , différance , the trace and the supplement. Derrida’s reading of the Platonic chora in Chora L Works (a series of discussions with the architect Peter Eisenman) as something which defies the logics of noncontradiction and binarity, implies the internal heterogeneity and instability of all structures, neither ‘sensible’ nor ‘intelligible’ but a third genus which escapes conceptual capture.^{25} Crucially, chora , spacing, dissemination and différance are highly dynamic concepts, involving hybridity, an ongoing ‘corruption’ of categories, and a ‘bastard reasoning.’^{26} Derrida identification of différance in Margins of Philosophy , as an ‘unappropriable excess’ that operates through spacing as ‘the becomingspace of time or the becomingtime of space,’^{27} chimes with his description of chora as an ‘unidentifiable excess’ that is ‘the spacing which is the condition for everything to take place,’ opening up the interval as the plurivocity of writing in defiance of ‘origin’ and ‘essence.’^{28} In this unfolding of différance , spacing ‘insinuates into presence an interval,’^{29} again alerting us to the crucial role of the interstice in deconstruction, and, as Derrida observes in Positions , its impact as ‘a movement, a displacement that indicates an irreducible alterity’: ‘Spacing is the impossibility for an identity to be closed on itself, on the inside of its proper interiority, or on its coincidence with itself. The irreducibility of spacing is the irreducibility of the other.’^{30}"
25. Quoted in Jeffrey Kipnis and Thomas Leeser, eds., 26. Ibid, 25.
27. Derrida, Margins of Philosophy. 28. Derrida, Chora L Works , 19 and 10. 29. Ibid, 203. 30. Derrida, Positions , 94. 
Thursday, January 24, 2019
Name Space
A correction at Wikipedia (Click to enlarge.) —
That this correction is needed indicates that the phrase
"Cullinane space" might be useful. (Click to enlarge.)
Tuesday, January 22, 2019
Monday, December 17, 2018
Tales from Story Space
"Kiernan Brennan Shipka (born November 10, 1999)
is an American actress. She is best known for starring as
Sabrina Spellman on the Netflix supernatural horror series
Chilling Adventures of Sabrina (2018–present)." — Wikipedia
As noted here earlier, Shipka turned 18 on Nov. 10 last year.
From Log24 on that date —
Another 18th birthday in Story Space —
Sunday, December 9, 2018
Quaternions in a Small Space
The previous post, on the 3×3 square in ancient China,
suggests a review of group actions on that square
that include the quaternion group.
Click to enlarge —
Three links from the above finitegeometry.org webpage on the
quaternion group —

Visualizing GL(2,p) — A 1985 note illustrating group actions
on the 3×3 (ninefold) square. 
Another 1985 note showing group actions on the 3×3 square
transferred to the 2x2x2 (eightfold) cube.  Quaternions in an Affine Galois Plane — A webpage from 2010.
Related material —
See as well the two Log24 posts of December 1st, 2018 —
Character and In Memoriam.
Sunday, November 4, 2018
Kristen vs. the Space Witch*
* We know the former. There is no shortage of candidates for the latter.
Saturday, November 3, 2018
The Space Theory of Truth
Earlier posts have discussed the "story theory of truth"
versus the "diamond theory of truth," as defined by
Richard Trudeau in his 1987 book The NonEuclidean Revolution.
In a New York Times opinion piece for tomorrow's print edition,*
novelist Dara Horn touched on what might be called
"the space theory of truth."
When they return to synagogue, mourners will be greeted
with more ancient words: “May God comfort you
among the mourners of Zion and Jerusalem.”
In that verse, the word used for God is hamakom —
literally, “the place.” May the place comfort you.
[Link added.]
The Source —
See Dara Horn in this journal, as well as Makom.
* "A version of this article appears in print on ,
on Page A23 of the New York edition with the headline:
American Jews Know This Story."
Tuesday, October 30, 2018
Story Structure, Story Space
Constance Grady at Vox today on a new Netflix series —
We don’t yet have a story structure that allows witches to be powerful for long stretches of time without men holding them back. And what makes the new Sabrina so exciting is that it seems to be trying to build that story structure itself, in real time, to find a way to let Sabrina have her power and her freedom. It might fail. But if it does, it will be a glorious and worthwhile failure — the type that comes with trying to pioneer a new kind of story. 
See also Story Space in this journal.
Monday, October 22, 2018
Story Space
Sunday, July 22, 2018
Wednesday, May 2, 2018
Galois’s Space
(A sequel to Foster's Space and Sawyer's Space)
See posts now tagged Galois's Space.
Sunday, March 4, 2018
The Square Inch Space: A Brief History
Sunday, November 19, 2017
Galois Space
This is a sequel to yesterday's post Cube Space Continued.
Friday, September 15, 2017
Space Art
Silas in "Equals" (2015) —
Ever since we were kids it's been drilled into us that …
Our purpose is to explore the universe, you know.
Outer space is where we'll find …
… the answers to why we're here and …
… and where we come from.
Related material —
See also Galois Space in this journal.
Sunday, April 16, 2017
Art Space Paradigm Shift
This post’s title is from the tags of the previous post —
The title’s “shift” is in the combined concepts of …
Space and Number
From Finite Jest (May 27, 2012):
The books pictured above are From Discrete to Continuous ,
by Katherine Neal, and Geometrical Landscapes , by Amir Alexander.
For some details of the shift, see a Log24 search for Boole vs. Galois.
From a post found in that search —
“Benedict Cumberbatch Says
a Journey From Fact to Faith
Is at the Heart of Doctor Strange“
— io9 , July 29, 2016
” ‘This man comes from a binary universe
where it’s all about logic,’ the actor told us
at San Diego ComicCon . . . .
‘And there’s a lot of humor in the collision
between Easter [ sic ] mysticism and
Western scientific, sort of logical binary.’ “
[Typo now corrected, except in a comment.]
Thursday, April 13, 2017
Wednesday, March 29, 2017
Art Space Illustrated
Another view of the previous post's art space —
More generally, see Solomon's Cube in Log24.
See also a remark from Stack Exchange in yesterday's post Backstory,
and the Stack Exchange math logo below, which recalls the above
cube arrangement from "Affine groups on small binary spaces" (1984).
Art Space, Continued
"And as the characters in the meme twitch into the abyss
that is the sky, this meme will disappear into whatever
internet abyss swallowed MySpace."
—Staff writer Kamila Czachorowski, Harvard Crimson today
From Log24 posts tagged Art Space —
From a recent paper on Kummer varieties,
arXiv:1208.1229v3 [math.AG] 12 Jun 2013,
“The Universal Kummer Threefold,” by
Qingchun Ren, Steven V Sam, Gus Schrader, and
Bernd Sturmfels —
Two such considerations —
Monday, September 26, 2016
Tuesday, May 31, 2016
Friday, April 8, 2016
Ogdoads: A Space Odyssey
"Like the Valentinian Ogdoad— a selfcreating theogonic system
of eight Aeons in four begetting pairs— the projected eightfold work
had an esoteric, gnostic quality; much of Frye's formal interest lay in
the 'schematosis' and fearful symmetries of his own presentations."
— From p. 61 of James C. Nohrnberg's "The Master of the Myth
of Literature: An Interpenetrative Ogdoad for Northrop Frye,"
Comparative Literature , Vol. 53 No. 1, pp. 5882, Duke University
Press (quarterly, January 2001)
See also Two by Four in this journal.
Tuesday, January 12, 2016
Harmonic Analysis and Galois Spaces
The above sketch indicates, in a vague, handwaving, fashion,
a connection between Galois spaces and harmonic analysis.
For more details of the connection, see (for instance) yesterday
afternoon's post Space Oddity.
Monday, January 11, 2016
Space Oddity
It is an odd fact that the close relationship between some
small Galois spaces and small Boolean spaces has gone
unremarked by mathematicians.
A Google search today for “Galois spaces” + “Boolean spaces”
yielded, apart from merely terminological sources, only some
introductory material I have put on the Web myself.
Some more sophisticated searches, however led to a few
documents from the years 1971 – 1981 …
“Harmonic Analysis of Switching Functions” ,
by Robert J. Lechner, Ch. 5 in A. Mukhopadhyay, editor,
Recent Developments in Switching Theory , Academic Press, 1971.
“Galois Switching Functions and Their Applications,”
by B. Benjauthrit and I. S. Reed,
JPL Deep Space Network Progress Report 4227 , 1975
D.K. Pradhan, “A Theory of Galois Switching Functions,”
IEEE Trans. Computers , vol. 27, no. 3, pp. 239249, Mar. 1978
“Switching functions constructed by Galois extension fields,”
by Iwaro Takahashi, Information and Control ,
Volume 48, Issue 2, pp. 95–108, February 1981
An illustration from the Lechner paper above —
“There is such a thing as harmonic analysis of switching functions.”
— Saying adapted from a youngadult novel
Wednesday, October 21, 2015
Algebra and Space
"Perhaps an insane conceit …." Perhaps.
Related remarks on algebra and space —
"The Quality Without a Name" (Log24, August 26, 2015).
Friday, August 14, 2015
Space Station 2015
(A sequel to Space Station 1976)
For Kathleen Gibbons* —
* Note Gibbons's work on "Discrete phase space based on finite fields."
Monday, September 22, 2014
Thursday, February 27, 2014
Sacred Space, continued
"An image comes to mind of a white, ideal space
that, more than any single picture, may be the
archetypal image of 20thcentury art."
— Brian O'Doherty, "Inside the White Cube"
Cube spaces exist also in mathematics.
Tuesday, December 3, 2013
Diamond Space
A new website illustrates its URL.
See DiamondSpace.net.
Tuesday, July 16, 2013
Space Itself
"How do you get young people excited
about space? How do you get them interested
not just in watching movies about space,
or in playing video games set in space …
but in space itself?"
— Megan Garber in The Atlantic , Aug. 16, 2012
One approach:
"There is such a thing as a tesseract" and
Diamond Theory in 1937.
See, too, Baez in this journal.
Sunday, March 10, 2013
Galois Space
The 16point affine Galois space:
Further properties of this space:
In Configurations and Squares, see the
discusssion of the Kummer 16_{6} configuration.
Some closely related material:
 Wolfgang Kühnel,
"Minimal Triangulations of Kummer Varieties,"
Abh. Math. Sem. Univ. Hamburg 57, 720 (1986).For the first two pages, click here.
 Jonathan Spreer and Wolfgang Kühnel,
"Combinatorial Properties of the K 3 Surface:
Simplicial Blowups and Slicings,"
preprint, 26 pages. (2009/10) (pdf).
(Published in Experimental Math. 20,
issue 2, 201–216 (2011).)
Monday, March 4, 2013
Occupy Galois Space
Continued from February 27, the day Joseph Frank died…
"Throughout the 1940s, he published essays
and criticism in literary journals, and one,
'Spatial Form in Modern Literature'—
a discussion of experimental treatments
of space and time by Eliot, Joyce, Proust,
Pound and others— published in
The Sewanee Review in 1945, propelled him
to prominence as a theoretician."
— Bruce Weber in this morning's print copy
of The New York Times (p. A15, NY edition)
That essay is reprinted in a 1991 collection
of Frank's work from Rutgers University Press:
See also Galois Space and Occupy Space in this journal.
Frank was best known as a biographer of Dostoevsky.
A very loosely related reference… in a recent Log24 post,
Freeman Dyson's praise of a book on the history of
mathematics and religion in Russia:
"The intellectual drama will attract readers
who are interested in mystical religion
and the foundations of mathematics.
The personal drama will attract readers
who are interested in a human tragedy
with characters who met their fates with
exceptional courage."
Frank is survived by, among others, his wife, a mathematician.
Thursday, February 21, 2013
Galois Space
The previous post suggests two sayings:
"There is such a thing as a Galois space."
— Adapted from Madeleine L'Engle
"For every kind of vampire, there is a kind of cross."
Illustrations—
Saturday, September 22, 2012
Occupy Space
"The word 'space' has, as you suggest, a large number of different meanings."
— Nanavira Thera in [Early Letters. 136] 10.xii.1958
From that same letter (links added to relevant Wikipedia articles)—
Space (ākāsa) is undoubtedly used in the Suttas
Your second letter seems to suggest that the space 
A simpler metaphysical system along the same lines—
The theory, he had explained, was that the persona
— The Gameplayers of Zan , 
"I am glad you have discovered that the situation is comical:
ever since studying Kummer I have been, with some difficulty,
refraining from making that remark."
— Nanavira Thera, [Early Letters, 131] 17.vii.1958
Thursday, July 12, 2012
Galois Space
An example of lines in a Galois space * —
The 35 lines in the 3dimensional Galois projective space PG(3,2)—
There are 15 different individual linear diagrams in the figure above.
These are the points of the Galois space PG(3,2). Each 3set of linear diagrams
represents the structure of one of the 35 4×4 arrays and also represents a line
of the projective space.
The symmetry of the linear diagrams accounts for the symmetry of the
840 possible images in the kaleidoscope puzzle.
* For further details on the phrase "Galois space," see
Beniamino Segre's "On Galois Geometries," Proceedings of the
International Congress of Mathematicians, 1958 [Edinburgh].
(Cambridge U. Press, 1960, 488499.)
(Update of Jan. 5, 2013— This post has been added to finitegeometry.org.)
Saturday, February 18, 2012
Symmetry
From the current Wikipedia article "Symmetry (physics)"—
"In physics, symmetry includes all features of a physical system that exhibit the property of symmetry—that is, under certain transformations, aspects of these systems are 'unchanged', according to a particular observation. A symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is 'preserved' under some change.
A family of particular transformations may be continuous (such as rotation of a circle) or discrete (e.g., reflection of a bilaterally symmetric figure, or rotation of a regular polygon). Continuous and discrete transformations give rise to corresponding types of symmetries. Continuous symmetries can be described by Lie groups while discrete symmetries are described by finite groups (see Symmetry group)."….
"A discrete symmetry is a symmetry that describes noncontinuous changes in a system. For example, a square possesses discrete rotational symmetry, as only rotations by multiples of right angles will preserve the square's original appearance."
Note the confusion here between continuous (or discontinuous) transformations and "continuous" (or "discontinuous," i.e. "discrete") groups .
This confusion may impede efforts to think clearly about some pure mathematics related to current physics— in particular, about the geometry of spaces made up of individual units ("points") that are not joined together in a continuous manifold.
For an attempt to forestall such confusion, see Noncontinuous Groups.
For related material, see Erlanger and Galois as well as the opening paragraphs of Diamond Theory—
Symmetry is often described as invariance under a group of transformations. An unspoken assumption about symmetry in Euclidean 3space is that the transformations involved are continuous.
Diamond theory rejects this assumption, and in so doing reveals that Euclidean symmetry may itself be invariant under rather interesting groups of noncontinuous (and asymmetric) transformations. (These might be called noncontinuous groups, as opposed to socalled discontinuous (or discrete ) symmetry groups. See Weyl's Symmetry .)
For example, the affine group A on the 4space over the 2element field has a natural noncontinuous and asymmetric but symmetrypreserving action on the elements of a 4×4 array. (Details)
(Version first archived on March 27, 2002)
Update of Sunday, February 19, 2012—
The abuse of language by the anonymous authors
of the above Wikipedia article occurs also in more
reputable sources. For instance—
Some transformations referred to by Brading and Castellani
and their editees as "discrete symmetries" are, in fact, as
linear transformations of continuous spaces, themselves
continuous transformations.
This unfortunate abuse of language is at least made explicit
in a 2003 text, Mathematical Perspectives on Theoretical
Physics (Nirmala Prakash, Imperial College Press)—
"… associated[*] with any given symmetry there always exists
a continuous or a discrete group of transformations….
A symmetry whose associated group is continuous (discrete)
is called a continuous (discrete ) symmetry ." — Pp. 235, 236
[* Associated how?]
Monday, August 29, 2005
Monday August 29, 2005
Date: Sun, 28 Aug 2005 12:30:40 0400 From: Alf van der Poorten AM Subject: Vale George Szekeres and Esther Klein Szekeres Members of the Number Theory List will be sad to learn that George and Esther Szekeres both died this morning. George, 94, had been quite ill for the last 23 days, barely conscious, and died first at 06:30. Esther, 95, died a half hour later. Both George Szekeres and Esther Klein will be recalled by number theorists as members of the group of young Hungarian mathematicians of the 1930s including Turan and Erdos. George and Esther's coming to Australia in the late 40s played an important role in the invigoration of Australian Mathematics. George was also an expert in group theory and relativity; he was my PhD supervisor. Emeritus Professor 
AVE
"Hello! Kinch here. Put me on to Edenville. Aleph, alpha: nought, nought, one."
"A very short space of time through very short times of space…. — James Joyce, Ulysses, Proteus chapter A very short space of time through very short times of space…. "It is demonstrated that spacetime should possess a discrete structure on Planck scales." — Peter Szekeres, abstract of Discrete SpaceTime 
Peter Szekeres is the son of George and Esther Szekeres.
"At present, such relationships can at best be heuristically described in terms that invoke some notion of an 'intelligent user standing outside the system.'"
— GianCarlo Rota in Indiscrete Thoughts, p. 152
Thursday, August 25, 2005
Thursday August 25, 2005
Train of Thought
Part I: The 24Cell
From S. H. Cullinane,
Visualizing GL(2,p),
March 26, 1985–
From John Baez, “This Week’s Finds in Mathematical Physics (Week 198),” September 6, 2003: Noam Elkies writes to John Baez:
The enrapturing discoveries of our field systematically conceal, like footprints erased in the sand, the analogical train of thought that is the authentic life of mathematics – GianCarlo Rota 
Like footprints erased in the sand….
“Hello! Kinch here. Put me on to Edenville. Aleph, alpha: nought, nought, one.”
“A very short space of time through very short times of space….
Am I walking into eternity along Sandymount strand?”
— James Joyce, Ulysses, Proteus chapter
A very short space of time through very short times of space….
“It is demonstrated that spacetime should possess a discrete structure on Planck scales.”
— Peter Szekeres, abstract of Discrete SpaceTime
“A theory…. predicts that space and time are indeed made of discrete pieces.”
— Lee Smolin in Atoms of Space and Time (pdf), Scientific American, Jan. 2004
“… a fundamental discreteness of spacetime seems to be a prediction of the theory….”
— Thomas Thiemann, abstract of Introduction to Modern Canonical Quantum General Relativity
“Theories of discrete spacetime structure are being studied from a variety of perspectives.”
— Quantum Gravity and the Foundations of Quantum Mechanics at Imperial College, London
The above speculations by physicists
are offered as curiosities.
I have no idea whether
any of them are correct.
Related material:
Stephen Wolfram offers a brief
History of Discrete Space.
For a discussion of space as discrete
by a nonphysicist, see John Bigelow‘s
Space and Timaeus.
in a Discrete Space
physics, there are of course many
purely mathematical discrete spaces.
See Visible Mathematics, continued
(Aug. 4, 2005):
Saturday, December 19, 2020
Quackers
Friday, December 18, 2020
Notes Towards the Redefinition of Culture
In the altered headline above, ” Q******* ” may, if you like,
be interpreted as ” Quellers ,” an invented term for scholars
who investigate the origins of Christianity.
See the Log24 post “Q is for Quelle ” (November 7, 2020).
Dan Brown, like the earlier novelist who wrote The Source ,
is such an investigator (of sorts), though not a scholar .
(For an example of actual scholarship , see the webpage
https://quod.lib.umich.edu/m/
middleenglishdictionary/dictionary/MED35525.
That page may be interpreted as putting the “hit” in “s***.”)
Monday, September 14, 2020
Shades (Of London Bondage continues)
“Loitering in Lara’s dressing room, she tries on
the fauxbondage harness she picked up in London….”
See as well . . .
Socrates in the Marketplace
“The 2×2 matrix is commonly used in business strategy
as a representational tool to show conflicting concepts and
for decision making. This fourquadrant matrix diagram
is perfect to be used for business or marketing matrices
like BCG, SWOT, Ansoff, risk assessment…
Additionally, it will also be suitable to illustrate 4 ideas or
concepts.” [Link on “illustrate” added.]
See also a Log24 search for “Resplendent.”
Thursday, August 6, 2020
Structure and Mutability . . .
See a Log24 search for Beadgame Space.
This post might be regarded as a sort of “checked cell”
for the above concepts listed as tags . . .
Related material from a Log24 search for Structuralism —
Wednesday, June 3, 2020
The Ghost Writer
See also Litsky’s obituary from All Saints’ Day, 2018.
Litsky reportedly died on October 30, 2018 — Devil’s Night.
Sunday, May 17, 2020
Thursday, April 16, 2020
A FourColor Epic
“A love story of epic, epic, epic proportion” — Kristen Stewart
See also the following letter to Knuth on fourcolor enthusiast
SpencerBrown, as well as Tim Robinson on the same subject
in his book My Time in Space .
Wednesday, April 15, 2020
Death Warmed Over
In memory of the author of My Time in Space * —
Tim Robinson, who reportedly died on April 3 —
See also an image from a Log24 post, Gray Space —
Related material from Robinson’s reported date of death —
* First edition, hardcover, Lilliput Press, Ireland, April 1, 2001.
Saturday, March 14, 2020
Friday, January 3, 2020
Valhalla Requiem
Spectral Woo
"… during that spell between the feasts of Christmas and Epiphany
when ghosts and specters are supposed to be abroad . . . ."
Heinrich Zimmer on Sir Gawain and the Green Knight
Times Literary Supplement , January 3, 2020
Sciences  Book Review
The world is not enough:
Guessing at the game God is playing
By Samuel Graydon
See as well …
Thursday, January 2, 2020
Nada for Hemingway
See Nada + Hemingway in this journal.
The above upload date suggests a look at
other posts now tagged Red to Green.
Wednesday, January 1, 2020
Águila
"We learned so much about singing from each other because you get to sort of be them for a second when you're shadowing them in harmony. It's like getting on an eagle and getting to see the world through that eagle's experience." Read more: https://www.springfieldspringfield.co.uk/ movie_script.php?movie=lindaronstadtthesoundofmyvoice
See also Aguila de Oro.
Monday, December 23, 2019
Orbit
"December 22, the birth anniversary of India’s famed mathematician
Srinivasa Ramanujan, is celebrated as National Mathematics Day."
— Indian Express yesterday
"Orbits and stabilizers are closely related." — Wikipedia
Symmetries by Plato and R. T. Curtis —
In the above, 322,560 is the order
of the octad stabilizer group .
Tuesday, December 10, 2019
Eternal Color
For the above title, see posts tagged Eternal Color.
From this evening's online New York Times —
Related imaterial —
A scene from the film of the above book —
Monday, December 2, 2019
Aesthetics at Harvard
"What the piece of art is about is the gray space in the middle."
— David Bowie, as quoted in the above Crimson piece.
Bowie's "gray space" is the space between the art and the beholder.
I prefer the gray space in the following figure —
Context: The Trinity Stone (Log24, June 4, 2018).
Thursday, June 13, 2019
Tuesday, June 4, 2019
Zen and the Art
Or: Burning Bright
A post in memory of Chicago architect Stanley Tigerman,
who reportedly died at 88 on Monday.
Inside Out
For fans of Space Fleet and of "reclusive but gifted" programmers—
“Hello the Camp”
The title is a quotation from the 2015 film "Mojave."
Monday, June 3, 2019
Jar Story
“. . . Only by the form, the pattern,
Can words or music reach
The stillness, as a Chinese jar still
Moves perpetually in its stillness.”
— T. S. Eliot, Four Quartets
From Writing Chinese Characters:
“It is practical to think of a character centered
within an imaginary square grid . . . .
The grid can… be… subdivided, usually to
9 or 16 squares. . . .“
These “Chinese jars” (as opposed to their contents)
are as follows:
.
See as well Eliot’s 1922 remarks on “extinction of personality”
and the phrase “egoextinction” in Weyl’s Philosophy of Mathematics —
Monday, May 20, 2019
The Bond with Reality
Tuesday, May 7, 2019
Breach
"Honored in the Breach:
Graham Bader on Absence as Memorial"
Artforum International , April 2012
. . . . "In the wake of a century marked by inconceivable atrocity, the use of emptiness as a commemorative trope has arguably become a standard tactic, a default style of public memory. The power of the voids at and around Ground Zero is generated by their origin in real historical circumstance rather than such purely commemorative intent: They are indices as well as icons of the losses they mark.
Nowhere is the negotiation between these two possibilities–on the one hand, the cooptation of absence as tasteful mnemonic trope; on the other, absence's disruptive potential as brute historical scar–more evident than in Berlin, a city whose history, as Andreas Huyssen has argued, can be seen as a 'narrative of voids.' Writing in 1997, Huyssen saw this tale culminating in Berlin's postwall development, defined equally by an obsessive coveringover of the city's lacunae–above all in the elaborate commercial projects then proliferating in the mileslong stretch occupied until 1989 by the Berlin Wall–and a carefully orchestrated deployment of absence as memorial device, particularly in the 'voids' integrated by architect Daniel Libeskind into his addition to the Berlin Museum, now known as the Jewish Museum Berlin." 
See also Breach in this journal.
Monday, May 6, 2019
Saturday, May 4, 2019
Inside the White Cube
See also Espacement and The Thing and I.
Friday, May 3, 2019
Wednesday, April 24, 2019
Critical Visibility
Correction — "Death has 'the whole spirit sparkling…'"
should be "Peace after death has 'the whole spirit sparkling….'"
The page number, 373, is a reference to Wallace Stevens:
Collected Poetry and Prose , Library of America, 1997.
See also the previous post, "Critical Invisibility."
Tuesday, April 23, 2019
Critical Invisibility
From Gotay and Isenberg, "The Symplectization of Science,"
Gazette des Mathématiciens 54, 5979 (1992):
"… what is the origin of the unusual name 'symplectic'? ….
Its mathematical usage is due to Hermann Weyl who,
in an effort to avoid a certain semantic confusion, renamed
the then obscure 'line complex group' the 'symplectic group.'
… the adjective 'symplectic' means 'plaited together' or 'woven.'
This is wonderfully apt…."
On "The Emperor's New Clothes" —
Andersen’s weavers, as one commentator points out, are merely insisting that “the value of their labor be recognized apart from its material embodiment.” The invisible cloth they weave may never manifest itself in material terms, but the description of its beauty (“as light as spiderwebs” and “exquisite”) turns it into one of the many wondrous objects found in Andersen’s fairy tales. It is that cloth that captivates us, making us do the imaginative work of seeing something beautiful even when it has no material reality. Deeply resonant with meaning and of rare aesthetic beauty—even if they never become real—the cloth and other wondrous objets d’art have attained a certain degree of critical invisibility. — Maria Tatar, The Annotated Hans Christian Andersen (W. W. Norton & Company, 2007). Kindle Edition. 
A Certain Dramatic Artfulness
Saturday, March 9, 2019
Weapons of Mass Distraction
"Back to the Future" and . . .
I prefer another presentation from the above
Universal Pictures date — June 28, 2018 —
Friday, March 8, 2019
Photo Opportunity
"I need a photo opportunity . . . ." — Paul Simon
Thursday, February 28, 2019
Previn’s Wake
A search for Previn in this journal yields . . .
"whyse Salmonson set his seel on a hexengown,"
Finnegans Wake , Book II, Episode 2, pp. 296297
Fooling
The two books pictured above are From Discrete to Continuous ,
by Katherine Neal, and Geometrical Landscapes , by Amir Alexander.
Note: There is no Galois (i.e., finite) field with six elements, but
the theory of finite fields underlies applications of sixset geometry.
Monday, January 21, 2019
Meditation for the Champ de Mors
"his onesidemissing for an allblind alley
leading to an Irish plot in the Champ de Mors"
— James Joyce, Finnegans Wake
Sunday, January 20, 2019
Scope Resolution
Wikipedia on a programming term —
The scope resolution operator helps to identify
and specify the context to which an identifier refers,
particularly by specifying a namespace. The specific
uses vary across different programming languages
with the notions of scoping. In many languages
the scope resolution operator is written
"::".
In a completely different context, these four dots might represent
a geometric object — the fourpoint plane .
Saturday, January 19, 2019
Friday, January 18, 2019
Location, Location, Location
The Woke Grids …
… as opposed to The Dreaming Jewels .
A July 2014 Amsterdam master's thesis on the Golay code
and Mathieu group —
"The properties of G_{24} and M_{24} are visualized by
four geometric objects: the icosahedron, dodecahedron,
dodecadodecahedron, and the cubicuboctahedron."
Some "geometric objects" — rectangular, square, and cubic arrays —
are even more fundamental than the above polyhedra.
A related image from a post of Dec. 1, 2018 —
Tuesday, December 18, 2018
Shadowhunter Tales
The recent post "Tales from Story Space," about the 18th birthday
of the protagonist in the TV series "Shadowhunters" (2016),
suggests a review of the actual 18th birthday of actress Lily Collins.
Collins is shown below warding off evil with a magical rune as
a shadowhunter in the 2013 film "City of Bones" —
She turned 18 on March 18, 2007. A paper on symmetry and logic
referenced here on that date displays the following "runes" of
philosopher Charles Sanders Peirce —
See also Adamantine Meditation (Log24, Oct. 3, 2018)
and the webpage Geometry of the I Ching.
Sunday, December 16, 2018
Tuesday, November 6, 2018
Bait and Tackle
Monday, November 5, 2018
Sunday, November 4, 2018
“Look Up” — The Breakthrough Prize* Theme This Evening
Looking up images for "The Space Theory of Truth" this evening —
Detail (from the post "Logos" of Oct. 14) —
Saturday, November 3, 2018
For St. Anselm
"… at his home in San Anselmo . . . ."
See also Anselm in this journal, as well as the Devil's Night post Ojos.
Tuesday, October 30, 2018
Sunday, September 9, 2018
Plan 9 Continues.
"The role of Desargues's theorem was not understood until
the Desargues configuration was discovered. For example,
the fundamental role of Desargues's theorem in the coordinatization
of synthetic projective geometry can only be understood in the light
of the Desargues configuration.
Thus, even as simple a formal statement as Desargues's theorem
is not quite what it purports to be. The statement of Desargues's theorem
pretends to be definitive, but in reality it is only the tip of an iceberg
of connections with other facts of mathematics."
— From p. 192 of "The Phenomenology of Mathematical Proof,"
by GianCarlo Rota, in Synthese , Vol. 111, No. 2, Proof and Progress
in Mathematics (May, 1997), pp. 183196. Published by: Springer.
Stable URL: https://www.jstor.org/stable/20117627.
Related figures —
Note the 3×3 subsquare containing the triangles ABC, etc.
"That in which space itself is contained" — Wallace Stevens
Friday, July 6, 2018
Something
"… Only by the form, the pattern,
Can words or music reach
The stillness, as a Chinese jar still
Moves perpetually in its stillness."
— T. S. Eliot, "Burnt Norton," 1936
"Read something that means something."
— Advertising slogan for The New Yorker
The previous post quoted some mystic meditations of Octavio Paz
from 1974. I prefer some less mystic remarks of Eddington from
1938 (the Tanner Lectures) published by Cambridge U. Press in 1939 —
"… we have sixteen elements with which to form a groupstructure" —
See as well posts tagged Dirac and Geometry.
Tuesday, May 1, 2018
Wake
Remarks on space from 1998 by scifi author Robert J. Sawyer quoted
here on Sunday (see the tag "Sawyer's Space") suggest a review of
rather similar remarks on space from 1977 by scifi author M. A. Foster
(see the tag "Foster's Space"):
Quoted here on September 26, 2012 —
"All she had to do was kick off and flow."
"I'se so silly to be flowing but I no canna stay."
Another work by Sawyer —
Sunday, April 29, 2018
Amusement
From the online New York Times this afternoon:
Disney now holds nine of the top 10
domestic openings of all time —
six of which are part of the Marvel
Cinematic Universe. “The result is
a reflection of 10 years of work:
of developing this universe, creating
stakes as big as they were, characters
that matter and stories and worlds that
people have come to love,” Dave Hollis,
Disney’s president of distribution, said
in a phone interview.
From this journal this morning:
"But she felt there must be more to this
than just the sensation of folding space
over on itself. Surely the Centaurs hadn't
spent ten years telling humanity how to
make a fancy amusementpark ride.
There had to be more—"
— Factoring Humanity , by Robert J. Sawyer,
Tom Doherty Associates, 2004 Orb edition,
page 168
"The sensation of folding space . . . ."
Or unfolding:
Click the above unfolded space for some background.
Thursday, April 19, 2018
Something to Behold
From a review of a Joyce Carol Oates novel
at firstthings.com on August 23, 2013 —
"Though the Curse is eventually exorcised,
it is through an act of wit and guile,
not an act of repentance or reconciliation.
And so we may wonder if Oates has put this story
to rest, or if it simply lays dormant. A twentyfirst
century eruption of the 'Crosswicks Curse'
would be something to behold." [Link added.]
Related material —
A film version of A Wrinkle in Time —
The Hamilton watch from "Interstellar" (2014) —
See also a post, Vacant Space, from 8/23/13 (the date
of the above review), and posts tagged Space Writer.
Saturday, April 7, 2018
Sides
The FBI holding cube in "The Blacklist" —
" 'The Front' is not the whole story . . . ."
— Vincent Canby, New York Times film review, 1976,
as quoted in Wikipedia.
See also Solomon's Cube in this journal.
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 allmeaningful, 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 4x4x4 design cube —
"I saw that in the alternation between front and back,
between top and bottom, between left and right,
symmetry is forever being created."
See also a related remark by LéviStrauss in 1955:
"…three different readings become possible:
left to right, top to bottom, front to back."
Monday, March 12, 2018
“Quantum Tesseract Theorem?”
Remarks related to a recent film and a notsorecent film.
For some historical background, see Dirac and Geometry in this journal.
Also (as Thas mentions) after Saniga and Planat —
The SanigaPlanat paper was submitted on December 21, 2006.
Excerpts from this journal on that date —
"Open the pod bay doors, HAL."
Wednesday, March 7, 2018
Excited
"How do you get young people excited about space?"
— Megan Garber in The Atlantic , Aug. 16, 2012
The above quote is from this journal on 9/11, 2014.
Related material —
Synchronology for the above date — 9/11, 2014 —
A BuzzFeed article with that date, and in reply
"A Personal Statement from Michael Shermer" with that date.
Wednesday, February 28, 2018
A Girl’s Guide to Chaos
The title is that of a play mentioned last night in
a New York Times obituary .
Related recent film lines —
 Thor: How do I escape?

Heimdall: You're on a planet surrounded by doorways.
Go through one.  Thor: Which one?
 Heimdall: The big one!
Related material from this journal on Jan. 20, 2018 —
Mathematics and Narrative
Excerpts from a post of May 25, 2005 —
Above is an example I like of mathematics….
Here is an example I like of narrative:
Kate felt quite dizzy. She didn't know exactly what it was that had just happened, but she felt pretty damn certain that it was the sort of experience that her mother would not have approved of on a first date. "Is this all part of what we have to do to go to Asgard?" she said. "Or are you just fooling around?" "We will go to Asgard...now," he said. At that moment he raised his hand as if to pluck an apple, but instead of plucking he made a tiny, sharp turning movement. The effect was as if he had twisted the entire world through a billionth part of a billionth part of a degree. Everything shifted, was for a moment minutely out of focus, and then snapped back again as a suddenly different world.
— Douglas Adams, The Long Dark TeaTime of the Soul
Image from a different different world —
Hattip to a related Feb. 26 weblog post
at the American Mathematical Society.
Thursday, January 25, 2018
Beware of Analogical Extension
"By an archetype I mean a systematic repertoire
of ideas by means of which a given thinker describes,
by analogical extension , some domain to which
those ideas do not immediately and literally apply."
— Max Black in Models and Metaphors
(Cornell, 1962, p. 241)
"Others … spoke of 'ultimate frames of reference' …."
— Ibid.
A "frame of reference" for the concept four quartets —
A less reputable analogical extension of the same
frame of reference —
Madeleine L'Engle in A Swiftly Tilting Planet :
"… deep in concentration, bent over the model
they were building of a tesseract:
the square squared, and squared again…."
See also the phrase Galois tesseract .
Thursday, November 23, 2017
LéviStrauss vs. Propp
Claude LéviStrauss
From his “Structure and Form: To maintain. as I have done. that the permutability of contents is not arbitrary amounts to saying that, if the analysis is carried to a sufficiently deep level, behind diversity we will discover constancy. And, of course. the avowed constancy of form must not hide from us that functions are also permutable. The structure of the folktale as it is illustrated by Propp presents a chronological succession of qualitatively distinct functions. each constituting an independent genre. One can wonder whether—as with dramatis personae and their attributes— Propp does not stop too soon, seeking the form too close to the level of empirical observation. Among the thirtyone functions that he distinguishes, several are reducible to the same function reappearing at different moments of the narrative but after undergoing one or a number of transformations . I have already suggested that this could be true of the false hero (a transformation of the villain), of assigning a difficult task (a transformation of the test), etc. (see p. 181 above), and that in this case the two parties constituting the fundamental tale would themselves be transformations of each other. Nothing prevents pushing this reduction even further and analyzing each separate partie into a small number of recurrent functions, so that several of Propp’s functions would constitute groups of transformations of one and the same function. We could treat the “violation” as the reverse of the “prohibition” and the latter as a negative transformation of the “injunction.” The “departure” of the hero and his “return” would appear as the negative and positive expressions of the same disjunctive function. The “quest” of the hero (hero pursues someone or something) would become the opposite of “pursuit” (hero is pursued by something or someone), etc. In Vol. I of Structural Anthropology , p. 209, I have shown that this analysis alone can account for the double aspect of time representation in all mythical systems: the narrative is both “in time” (it consists of a succession of events) and “beyond” (its value is permanent). With regard to Propp’s theories my analysis offers another advantage: I can reconcile much better than Propp himself his principle of a permanent order of wondertale elements with the fact that certain functions or groups of functions are shifted from one tale to the next (pp. 9798. p. 108) If my view is accepted, the chronological succession will come to be absorbed into an atemporal matrix structure whose form is indeed constant. The shifting of functions is then no more than a mode of permutation (by vertical columns or fractions of columns). These critical remarks are certainly valid for the method used by Propp and for his conclusions. However. it cannot be stressed enough that Propp envisioned them and in several places formulated with perfect clarity the solutions I have just suggested. Let us take up again from this viewpoint the two essential themes of our discussion: constancy of the content (in spite of its permutability) and permutability of functions (in spite of their constancy). * Translated from a 1960 work in French. It appeared in English as Chapter VIII 
See also “LéviStrauss” + Formula in this journal.
Some background related to the previous post —
Wednesday, October 25, 2017
To the Egress
The New York Times at 8:22 PM ET —
"Knight Landesman, a longtime publisher of Artforum magazine
and a power broker in the art world, resigned on Wednesday
afternoon, hours after a lawsuit was filed in New York accusing
him of sexually harassing at least nine women in episodes that
stretched back almost a decade."
See as well, in this journal, Way to the Egress.
The Palo Alto Edge
From Stanford — The death on October 9, 2017, of a man who
“always wanted to be at the most cutting of cuttingedge technology.”
Related material from Log24 on April 26, 2017 —
A sketch, adapted from Girl Scouts of Palo Alto —
Click the sketch for further details.
Saturday, September 23, 2017
The Turn of the Frame
"With respect to the story's content, the frame thus acts
both as an inclusion of the exterior and as an exclusion
of the interior: it is a perturbation of the outside at the
very core of the story's inside, and as such, it is a blurring
of the very difference between inside and outside."
— Shoshana Felman on a Henry James story, p. 123 in
"Turning the Screw of Interpretation,"
Yale French Studies No. 55/56 (1977), pp. 94207.
Published by Yale University Press.
See also the previous post and The Galois Tesseract.
Sunday, August 27, 2017
Black Well
The “Black” of the title refers to the previous post.
For the “Well,” see Hexagram 48.
Related material —
The Galois Tesseract and, more generally, Binary Coordinate Systems.
Friday, June 16, 2017
Chalkroom Jungle
At MASS MoCA, the installation "Chalkroom" quotes a lyric —
Oh beauty in all its forms funny how hatred can also be a beautiful thing When it's as sharp as a knife as hard as a diamond Perfect 
— From "One Beautiful Evening," by Laurie Anderson.
See also the previous post and "Smallest Perfect" in this journal.
Thursday, June 15, 2017
Building Six
Berkshire tales of May 25, 2017 —
See also, in this journal from May 25 and earlier, posts now tagged
"The Story of Six."