# Log24

## Friday, January 6, 2017

### Eightfold Cube at Cornell

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

The assignments page for a graduate algebra course at Cornell

## Saturday, May 23, 2020

### Eightfold Geometry: A Surface Code “Unit Cell”

Filed under: General — Tags: , , — m759 @ 1:50 AM

The resemblance to the eightfold cube  is, of course,
completely coincidental.

Some background from the literature —

## Sunday, March 22, 2020

### Eightfold Site

Filed under: General — Tags: — m759 @ 2:00 AM

A brief summary of the eightfold cube is now at octad.us.

## Sunday, December 22, 2019

### M24 from the Eightfold Cube

Filed under: General — Tags: , , — m759 @ 12:01 PM

Exercise:  Use the Guitart 7-cycles below to relate the 56 triples
in an 8-set (such as the eightfold cube) to the 56 triangles in
a well-known Klein-quartic hyperbolic-plane tiling. Then use
the correspondence of the triples with the 56 spreads of PG(3,2)
to construct M24.

See as well earlier posts also tagged Triangles, Spreads, Mathieu.

## Tuesday, March 5, 2019

### The Eightfold Cube and PSL(2,7)

Filed under: General,Geometry — Tags: , , — m759 @ 10:45 PM

For PSL(2,7), this is ((49-1)(49-7))/((7-1)(2))=168.

The group GL(3,2), also of order 168, acts naturally
on the set of seven cube-slicings below —

Another way to picture the seven natural slicings —

Application of the above images to picturing the
isomorphism of PSL(2,7) with GL(3,2) —

For a more detailed proof, see . . .

## Sunday, September 30, 2018

### Iconology of the Eightfold Cube

Filed under: General,Geometry — Tags: , — m759 @ 11:13 AM

Found today in an Internet image search, from the website of
an anonymous amateur mathematics enthusiast

Forming Gray codes in the eightfold cube with the eight
I Ching  trigrams (bagua ) —

This  journal on Nov. 7, 2016

A different sort of cube, from the makers of the recent
Netflix miniseries "Maniac" —

## Friday, September 14, 2018

### Warburg at Cornell, Continued

Filed under: General — Tags: — m759 @ 9:29 AM

## Monday, July 23, 2018

### Eightfold Cube for Furey*

Filed under: General,Geometry — Tags: — m759 @ 10:31 PM

Click to enlarge:

Above are the 7 frames of an animated gif from a Wikipedia article.

* For the Furey of the title, see a July 20 Quanta Magazine  piece

"Before time began . . . ." — Optimus Prime

## Friday, June 29, 2018

### Triangles in the Eightfold Cube

From a post of July 25, 2008, “56 Triangles,” on the Klein quartic
and the eightfold cube

Baez’s discussion says that the Klein quartic’s 56 triangles
can be partitioned into 7 eight-triangle Egan ‘cubes’ that
correspond to the 7 points of the Fano plane in such a way
that automorphisms of the Klein quartic correspond to
automorphisms of the Fano plane. Show that the
56 triangles within the eightfold cube can also be partitioned
into 7 eight-triangle sets that correspond to the 7 points of the
Fano plane in such a way that (affine) transformations of the
eightfold cube induce (projective) automorphisms of the Fano plane.”

Related material from 1975 —

More recently

## Monday, January 9, 2017

### Analogical Extension at Cornell

Filed under: General,Geometry — Tags: — m759 @ 1:30 PM

Click to enlarge the following (from Cornell U. Press in 1962) —

For a more recent analogical extension at Cornell, see the
Epiphany 2017 post on the eightfold cube and yesterday
evening's post "A Theory of Everything."

## Tuesday, August 30, 2016

### The Eightfold Cube in Oslo

Filed under: General,Geometry — Tags: , — m759 @ 8:04 PM

A KUNSTforum.as article online today (translation by Google) —

Update of Sept. 7, 2016: The corrections have been made,
except for the misspelling "Cullinan," which was caused by

## Thursday, March 17, 2016

### On the Eightfold Cube

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

The following page quotes "Raiders of the Lost Crucible,"
a Log24 post from Halloween 2015.

From KUNSTforum.as, a Norwegian art quarterly, issue no. 1 of 2016.

Related posts — See Lyche Eightfold.

## Friday, October 9, 2015

### Eightfold Cube in Oslo

Filed under: General,Geometry — Tags: , , — m759 @ 2:00 PM

An eightfold cube appears in this detail
of a photo by Josefine Lyche of her
Norwegian Sculpture Biennial 2015

(Detail from private Instagram photo.)

Catalog description of installation —

In a small bedroom to Foredragssalen populate
Josefine Lyche exhibition with a group sculptures
that are part of the work group 4D Ambassador
(2014-2015). Together they form an installation
where she uses light to amplify the feeling of
stepping into a new dimension, for which the title
suggests, this "ambassadors" for a dimension we
physical forms presents nonphysical phenomena.
Lyches works have in recent years been placed
in something one might call an "esoteric direction"
in contemporary art, and defines itself this
sculpture group humorous as "glam-minimalist."
She has in many of his works returned to basic
geometric shapes, with hints to the occult,
"new space-age", mathematics and where
everything in between.

her website page with a 2012 version of that title.

## Monday, April 9, 2012

### Eightfold Cube Revisited

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

A search today (Élie Cartan's birthday) for material related to triality*

yielded references to something that has been called a Bhargava cube .

Two pages from a 2006 paper by Bhargava—

Bhargava's reference [4] above for "the story of the cube" is to…

Higher Composition Laws I:
A New View on Gauss Composition,

Manjul Bhargava

The Annals of Mathematics
Second Series, Vol. 159, No. 1 (Jan., 2004), pp. 217-250
Article Stable URL: http://www.jstor.org/stable/3597249

A brief account in the context of embedding problems (click to enlarge)—

For more ways of slicing a cube,
see The Eightfold Cube —

* Note (1) some remarks by Tony Smith
related to the above Dynkin diagram
and (2) another colorful variation on the diagram.

## Tuesday, March 30, 2010

### Eightfold Symmetries

Filed under: General,Geometry — Tags: , , , — m759 @ 9:48 PM

Deconstructing Design

Reconstructing Design

The phrase “eightfold way” in today’s
previous entry has a certain
graphic resonance…

For instance, an illustration from the
Wikipedia article “Noble Eightfold Path” —

St. Joseph’s Day

Harvard students who view Christian symbols
with fear and loathing may meditate
on the above as a representation of
the Gankyil rather than of the Trinity.

## Saturday, May 1, 2021

### How Deep the Rabbit Hole

Filed under: General — Tags: — m759 @ 9:24 PM

## Sunday, August 29, 2021

### “Before Time Began . . .” — Optimus Prime

Filed under: General — Tags: — m759 @ 1:01 PM

Concepts of Space —

(From the March 2019 post Back to the Annus Mirabilis , 1905 )

Concepts of Space and  Time —

## Saturday, August 28, 2021

### Solomon’s Super*  Cube…

Filed under: General — Tags: — m759 @ 1:33 PM

Geometry for Jews  continues.

The conclusion of Solomon Golomb's
"Rubik's Cube and Quarks,"
American Scientist , May-June 1982 —

Related geometric meditation —
Archimedes at Hiroshima
in posts tagged Aitchison.

* As opposed to Solomon's Cube .

## Wednesday, August 18, 2021

### Eight the Great

Filed under: General — m759 @ 9:03 AM

Starring J. J. Abrams as Leonhard Euler?

Related material —

The Cornell cap in the recent HBO "White Lotus" —

## Monday, August 9, 2021

### The Tune  (Suggested by “Hum: Seek the Void”)

Filed under: General — Tags: , , , , — m759 @ 1:43 AM

"Two years ago . . . ." — Synopsis of the August 3 film "Hum"

The above is an image from
the August 3, 2019,
post "Butterfield's Eight."

"Within the week . . . ."
— The above synopsis of "Hum"

This suggests a review of a post
from August 5, 2019, that might
be retitled . . .

"The void she knows,
the tune she hums."

## Wednesday, April 7, 2021

### Timeless  Capsules

Filed under: General — Tags: , — m759 @ 4:34 AM

Drilling down . . .

My own, more abstract, academic interests are indicated by
a post from this  journal on January 20, 2020
Dyadic Harmonic Analysis: The Fourfold Square and Eightfold Cube.

Those poetically inclined may regard that post as an instance of the
“intersection of the timeless  with time.”

## Friday, March 12, 2021

### Grid

Filed under: General — Tags: — m759 @ 10:45 AM

See Trinity Cube in this  journal and . . .

McDonnell’s illustration is from 9 June 1983.
See as well a less official note from later that June.

## Friday, December 25, 2020

### Change Arises: Mathematical Examples

Filed under: General — Tags: , — m759 @ 12:59 AM

From old posts tagged Change Arises

 From Christmas 2005: Click on image for details. For the eightfold cube as it relates to Klein’s simple group, see “A Reflection Group of Order 168.” For an rather more complicated theory of Klein’s simple group, see Click on image for details.

The phrase “change arises” is from Arkani-Hamed in 2013, describing
calculations in physics related to properties of the positive Grassmannian

A related recent illustration from Quanta Magazine —

The above illustration of seven cells is not unrelated to
the eightfold-cube model of the seven projective points in
the Fano plane.

## Tuesday, December 15, 2020

### Connection

Filed under: General — Tags: , — m759 @ 11:27 AM

Hurt’s dies natalis  (date of death, in the saints’ sense) was,
it now seems, 25  January 2017, not 27.

A connection, for fantasy fans, between the Philosopher’s Stone
(represented by the eightfold cube) and the Deathly Hallows
(represented by the usual Fano-plane figure) —

Images from a Log24 search for “Holocron.”

## Sunday, November 22, 2020

### The Galois-Fano Plane

Filed under: General — Tags: , — m759 @ 9:52 PM

A figure adapted from “Magic Fano Planes,” by
Ben Miesner and David Nash, Pi Mu Epsilon Journal
Vol. 14, No. 1, 1914, CENTENNIAL ISSUE 3 2014
(Fall 2014), pp. 23-29 (7 pages) —

Related material — The Eightfold Cube.

Update at 10:51 PM ET the same day —

Essentially the same figure as above appears also in
the second arXiv version (11 Jan. 2016) of . . .

DAVID A. NASH, and JONATHAN NEEDLEMAN.
“When Are Finite Projective Planes Magic?”
Mathematics Magazine, vol. 89, no. 2, 2016, pp. 83–91.
JSTOR, www.jstor.org/stable/10.4169/math.mag.89.2.83.

## Wednesday, November 18, 2020

Filed under: General — Tags: — m759 @ 11:03 AM

“Principles before personalities.” — AA motto

Related personalities —

Amazon.com review by John Miller :

“… The Metaphysical Club  is not a dry tome for academics.

Related principles —

## Thursday, September 17, 2020

### Structure and Mutability . . .

Continues in The New York Times :

“One day — ‘I don’t know exactly why,’ he writes — he tried to
put together eight cubes so that they could stick together but
also move around, exchanging places. He made the cubes out
of wood, then drilled a hole in the corners of the cubes to link
them together. The object quickly fell apart.

Many iterations later, Rubik figured out the unique design
that allowed him to build something paradoxical:
a solid, static object that is also fluid….” — Alexandra Alter

Another such object: the eightfold cube .

## Tuesday, September 8, 2020

### “The Eight” according to Coleridge

Filed under: General — Tags: — m759 @ 10:32 PM

Metaphysical ruminations of Coleridge that might be applied to
the eightfold cube

## Saturday, September 5, 2020

### Ikonologie des Zwischenraums

Filed under: General — Tags: , — m759 @ 11:28 AM

The title is from a Cornell page in the previous post.

Related material (click to enlarge) —

The above remarks on primitive mentality suggest
a review of Snakes on a Plane.

## Sunday, May 17, 2020

### “The Ultimate Epistemological Fact”

Filed under: General — Tags: — m759 @ 11:49 PM

“Let me say this about that.” — Richard Nixon

Interpenetration in Weyl’s epistemology —

Interpenetration in Mazzola’s music theory —

Interpenetration in the eightfold cube — the three midplanes —

A deeper example of interpenetration:

Aitchison has shown that the Mathieu group M24 has a natural
action on the 24 center points of the subsquares on the eightfold
cube’s six faces (four such points on each of the six faces). Thus
the 759 octads of the Steiner system S(5, 8, 24) interpenetrate
on the surface of the cube.

## Thursday, March 5, 2020

### “Generated by Reflections”

Filed under: General — Tags: — m759 @ 8:42 PM

See the title in this journal.

Such generation occurs both in Euclidean space

… and in some Galois spaces —

.

In Galois spaces, some care must be taken in defining "reflection."

## Sunday, March 1, 2020

### Same Staircase, Different Day

Filed under: General — Tags: , , , — m759 @ 2:18 PM

Freeman Dyson on his staircase at Trinity College
(University of Cambridge) and on Ludwig Wittgenstein:

“I held him in the highest respect and was delighted
to find him living in a room above mine on the same
staircase. I frequently met him walking up or down
the stairs, but I was too shy to start a conversation.”

Frank Close on Ron Shaw:

“Shaw arrived there in 1949 and moved into room K9,
overlooking Jesus Lane. There is nothing particularly
its previous occupant was Freeman Dyson.”

— Close, Frank. The Infinity Puzzle  (p. 78).
Basic Books. Kindle Edition.

Illuminati enthusiasts  may enjoy the following image:

## Saturday, February 29, 2020

### Template

Filed under: General — Tags: , — m759 @ 1:00 PM

BY ALEX GREENBERGER

February 28, 2020 1:04pm

If Minimalist artist Donald Judd is known as a writer at all, it’s likely for one important text— his 1965 essay “Specific Objects,” in which he observed the rise of a new kind of art that collapsed divisions between painting, sculpture, and other mediums. But Judd was a prolific critic, penning shrewd reviews for various publications throughout his career—including ARTnews . With a Judd retrospective going on view this Sunday at the Museum of Modern Art in New York, ARTnews  asked New York Times  co-chief art critic Roberta Smith— who, early in her career, worked for Judd as his assistant— to comment on a few of Judd’s ARTnews  reviews. How would she describe his critical style? “In a word,” she said, “great.” . . . .

And then there is Temple Eight, or Ex Fano Apollinis —

Cicero, In Verrem  II. 1. 46 —

```He reached Delos. There one night he secretly   46
carried off, from the much-revered sanctuary of
Apollo, several ancient and beautiful statues, and
had them put on board his own transport. Next
day, when the inhabitants of Delos saw their sanc-
tuary stripped of its treasures, they were much
distressed . . . .```
```Delum venit. Ibi ex fano Apollinis religiosissimo
noctu clam sustulit signa pulcherrima atque anti-
quissima, eaque in onerariam navem suam conicienda
curavit. Postridie cum fanum spoliatum viderent ii
qui Delum incolebant, graviter ferebant . . . .```

## Thursday, February 27, 2020

### Occult Writings

Filed under: General — Tags: , — m759 @ 12:44 PM

From the author who in 2001 described "God's fingerprint"
(see the previous post) —

From the same publisher —

From other posts tagged Triskele in this journal —

Other geometry for enthusiasts of the esoteric —

Monday, November 4, 2019

### As Above, So Below*

Filed under: General —
Tags:  —
m759 @ 5:43 AM

## Sunday, February 23, 2020

### The Representation of Reality

Filed under: General — Tags: , — m759 @ 1:36 PM

"Although art is fundamentally everywhere and always the same,
nevertheless two main human inclinations, diametrically opposed
to each other, appear in its many and varied expressions. ….
The first aims at representing reality objectively, the second subjectively."

An image search today (click to enlarge) —

## Wednesday, February 19, 2020

Filed under: General — Tags: , , — m759 @ 11:36 AM

The 759 octads of the Steiner system S(5,8,24) are displayed
rather neatly in the Miracle Octad Generator of R. T. Curtis.

A March 9, 2018, construction by Iain Aitchison* pictures the
759 octads on the faces of a cube , with octad elements the
24 edges of a  cuboctahedron :

The Curtis octads are related to symmetries of the square.

See my webpage "Geometry of the 4×4 square" from March 2004.
Aitchison's p. 42 slide includes an illustration from that page —

Note that essentially the same model as Aitchison's can be pictured
by using, instead of the 24 edges of a cuboctahedron, the 24 outer
faces of subcubes in the eightfold cube .

Image from Christmas Day 2005.

## Wednesday, February 12, 2020

### The Reality Bond

Filed under: General — Tags: , , — m759 @ 3:33 PM

The plane at left is modeled naturally by
seven types of “cuts” in the cube at right.

## Monday, January 20, 2020

The Fourfold Square and Eightfold Cube

Related material:  A Google image search for “field dream” + log24.

## Thursday, January 2, 2020

### Interality

Filed under: General — Tags: , — m759 @ 8:25 PM

## Saturday, December 14, 2019

### Colorful Tale

Filed under: General — Tags: , — m759 @ 9:00 PM

The above image is from

"A Four-Color Theorem:
Function Decomposition Over a Finite Field,"
http://finitegeometry.org/sc/gen/mapsys.html.

These partitions of an 8-set into four 2-sets
occur also in Wednesday night's post

This  post was suggested by a Daily News
story from August 8, 2011, and by a Log24
post from that same date, "Organizing the
Mine Workers
" —

## Monday, October 7, 2019

### Berlekamp Garden vs. Kinder Garten

Filed under: General — Tags: — m759 @ 11:00 PM

Stevens's Omega and Alpha (see previous post) suggest a review.

Omega — The Berlekamp Garden.  See Misère Play (April 8, 2019).
Alpha  —  The Kinder Garten.  See Eighfold Cube.

Illustrations —

The sculpture above illustrates Klein's order-168 simple group.
So does the sculpture below.

Cube Bricks 1984 —

## Sunday, September 29, 2019

### Spiritual Kin

Filed under: General — Tags: , — m759 @ 12:00 PM

"The 15 Puzzle and the Magic Cube
are spiritual kin …."

"Metamagical Themas"  column,
Douglas R. Hofstadter, Scientific American ,
Vol. 244, No. 3 (March 1981), pp. 20-39

As are the 15 Schoolgirls and the Eightfold Cube.

## Tuesday, July 9, 2019

### Schoolgirl Space: 1984 Revisited

Filed under: General — Tags: , , , , — m759 @ 9:24 PM

Cube Bricks 1984 —

From "Tomorrowland" (2015) —

From John Baez (2018) —

and yesterday's Exploring Schoolgirl Space.

## Thursday, June 20, 2019

### The Lively Hallows

Filed under: General — Tags: , — m759 @ 7:11 PM

## Thursday, June 13, 2019

### The Reality Blocks

Filed under: General — Tags: — m759 @ 9:24 PM

The new Log24 tag "Eightfold Metaphysics" used in the previous post
suggests a review of posts that were tagged "The Reality Blocks" on May 24.

Then there is, of course, the May 24 death of Murray Gell-Mann, who
hijacked from Buddhism the phrase "eightfold way."

### Seeing the Seing

Filed under: General — Tags: , , , — m759 @ 2:30 PM

The phrase "experimental metaphysics" appeared in Peter Woit's weblog on June 11.
Google reveals that . . .

Shimony reportedly died on August 8, 2015.  Also on that date —

## Sunday, May 26, 2019

### Nine-Dot Patterns

Filed under: General — Tags: — m759 @ 9:00 PM

Some nine-dot patterns of greater interest:

## Sunday, May 19, 2019

### The Building Blocks of Geometry

From "On the life and scientific work of Gino Fano
by Alberto Collino, Alberto Conte, and Alessandro Verra,
ICCM Notices , July 2014, Vol. 2 No. 1, pp. 43-57 —

 " Indeed, about the Italian debate on foundations of Geometry, it is not rare to read comments in the same spirit of the following one, due to Jeremy Gray13. He is essentially reporting Hans Freudenthal’s point of view: ' When the distinguished mathematician and historian of mathematics Hans Freudenthal analysed Hilbert’s  Grundlagen he argued that the link between reality and geometry appears to be severed for the first time in Hilbert’s work. However, he discovered that Hilbert had been preceded by the Italian mathematician Gino Fano in 1892. . . .' " 13 J. Gray, "The Foundations of Projective Geometry in Italy," Chapter 24 (pp. 269–279) in his book Worlds Out of Nothing , Springer (2010).

Related material —

## Monday, May 6, 2019

### One Stuff

Building blocks?

From a post of May 4

## Saturday, May 4, 2019

### Inside the White Cube

Filed under: General — Tags: , , , , , — m759 @ 8:48 PM

## Monday, March 25, 2019

### Espacement

(Continued from the previous post.)

 (Ch. 2 in Henk Oosterling & Ewa Plonowska Ziarek (Eds.),  Intermedialities: Philosophy, Arts, Politics , Lexington Books, October 14, 2010) "The term 'spacing' ('espacement ') is absolutely central to Derrida's entire corpus, where it is indissociable from those of différance  (characterized, in the text from 1968 bearing this name, as '[at once] spacing [and] temporizing' 1), writing  (of which 'spacing' is said to be 'the fundamental property' 2) and deconstruction (with one of Derrida's last major texts, Le Toucher: Jean-Luc Nancy , specifying 'spacing ' to be 'the first word of any deconstruction' 3)." 1  Jacques Derrida, “La Différance,” in Marges – de la philosophie  (Paris: Minuit, 1972), p. 14. Henceforth cited as  D  . 2  Jacques Derrida, “Freud and the Scene of Writing,” trans. A. Bass, in Writing and  Difference  (Chicago: University of Chicago Press, 1978), p. 217. Henceforth cited as FSW . 3  Jacques Derrida, Le Toucher, Jean-Luc Nancy  (Paris: Galilée, 2000), p. 207. . . . . "… a particularly interesting point is made in this respect by the French philosopher, Michel Haar. After remarking that the force Derrida attributes to différance  consists simply of the series of its effects, and is, for this reason, 'an indefinite process of substitutions or permutations,' Haar specifies that, for this process to be something other than a simple 'actualisation' lacking any real power of effectivity, it would need “a soubassement porteur ' – let’s say a 'conducting underlay' or 'conducting medium' which would not, however, be an absolute base, nor an 'origin' or 'cause.' If then, as Haar concludes, différance  and spacing show themselves to belong to 'a pure Apollonism' 'haunted by the groundless ground,' which they lack and deprive themselves of,16 we can better understand both the threat posed by the 'figures' of space and the mother in the Timaeus  and, as a result, Derrida’s insistent attempts to disqualify them. So great, it would seem, is the menace to différance  that Derrida must, in a 'properly' apotropaic  gesture, ward off these 'figures' of an archaic, chthonic, spatial matrix in any and all ways possible…." 16  Michel Haar, “Le jeu de Nietzsche dans Derrida,” Revue philosophique de la France et de l’Etranger  2 (1990): 207-227. . . . . … "The conclusion to be drawn from Democritus' conception of rhuthmos , as well as from Plato's conception of the chôra , is not, therefore, as Derrida would have it, that a differential field understood as an originary site of inscription would 'produce' the spatiality of space but, on the contrary, that 'differentiation in general' depends upon a certain 'spatial milieu' – what Haar would name a 'groundless ground' – revealed as such to be an 'in-between' more 'originary' than the play of differences it in-forms. As such, this conclusion obviously extends beyond Derrida's conception of 'spacing,' encompassing contemporary philosophy's continual privileging of temporization in its elaboration of a pre-ontological 'opening' – or, shall we say, 'in-between.'

For permutations and a possible "groundless ground," see
the eightfold cube and group actions both on a set of eight
building blocks arranged in a cube (a "conducting base") and
on the set of seven natural interstices (espacements )  between
the blocks. Such group actions provide an elementary picture of
the isomorphism between the groups PSL(2,7) (acting on the
eight blocks) and GL(3,2) (acting on the seven interstices).

Espacements

For the Church of Synchronology

Intermedialities , the Log24 post Synchronicity.

## Saturday, March 16, 2019

### Grundlagen

Filed under: General — Tags: , , , — m759 @ 12:25 PM

## Thursday, February 21, 2019

### A Tale of Eight Building Blocks*

Filed under: General — Tags: , — m759 @ 4:53 PM

* For another such tale, see Eightfold Cube in this  journal.

## Thursday, December 6, 2018

### The Mathieu Cube of Iain Aitchison

This journal ten years ago today —

Surprise Package

From a talk by a Melbourne mathematician on March 9, 2018 —

The source — Talk II below —

# Search Results

### pdf of talk I(March 8, 2018)

www.math.sci.hiroshima-u.ac.jp/branched/…/Aitchison-Hiroshima-2018-Talk1-2.pdf

Iain Aitchison. Hiroshima  University March 2018 … Immediate: Talk given last year at Hiroshima  (originally Caltech 2010).

### pdf of talk II(March 9, 2018)  (with model for M24)

www.math.sci.hiroshima-u.ac.jp/branched/files/…/Aitchison-Hiroshima-2-2018.pdf

Iain Aitchison. Hiroshima  University March 2018. (IRA: Hiroshima  03-2018). Highly symmetric objects II.

### Abstract

www.math.sci.hiroshima-u.ac.jp/branched/files/2018/abstract/Aitchison.txt

Iain AITCHISON  Title: Construction of highly symmetric Riemann surfaces , related manifolds, and some exceptional objects, I, II Abstract: Since antiquity, some …

Related material —

The 56 triangles of  the eightfold cube . . .

Image from Christmas Day 2005.

## Wednesday, November 28, 2018

### Geometry and Experience

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 9:18 AM

Einstein, "Geometry and Experience," lecture before the
Prussian Academy of Sciences, January 27, 1921–

 … This view of axioms, advocated by modern axiomatics, purges mathematics of all extraneous elements, and thus dispels the mystic obscurity, which formerly surrounded the basis of mathematics. But such an expurgated exposition of mathematics makes it also evident that mathematics as such cannot predicate anything about objects of our intuition or real objects. In axiomatic geometry the words "point," "straight line," etc., stand only for empty conceptual schemata. That which gives them content is not relevant to mathematics. Yet on the other hand it is certain that mathematics generally, and particularly geometry, owes its existence to the need which was felt of learning something about the behavior of real objects. The very word geometry, which, of course, means earth-measuring, proves this. For earth-measuring has to do with the possibilities of the disposition of certain natural objects with respect to one another, namely, with parts of the earth, measuring-lines, measuring-wands, etc. It is clear that the system of concepts of axiomatic geometry alone cannot make any assertions as to the behavior of real objects of this kind, which we will call practically-rigid bodies. To be able to make such assertions, geometry must be stripped of its merely logical-formal character by the coordination of real objects of experience with the empty conceptual schemata of axiomatic geometry. To accomplish this, we need only add the proposition: solid bodies are related, with respect to their possible dispositions, as are bodies in Euclidean geometry of three dimensions. Then the propositions of Euclid contain affirmations as to the behavior of practically-rigid bodies. Geometry thus completed is evidently a natural science; we may in fact regard it as the most ancient branch of physics. Its affirmations rest essentially on induction from experience, but not on logical inferences only. We will call this completed geometry "practical geometry," and shall distinguish it in what follows from "purely axiomatic geometry." The question whether the practical geometry of the universe is Euclidean or not has a clear meaning, and its answer can only be furnished by experience.  ….

Later in the same lecture, Einstein discusses "the theory of a finite
universe." Of course he is not using "finite" in the sense of the field
of mathematics known as "finite geometry " — geometry with only finitely
many points.

Nevertheless, his remarks seem relevant to the Fano plane , an
axiomatically defined entity from finite geometry, and the eightfold cube ,
a physical object embodying the properties of the Fano plane.

 I want to show that without any extraordinary difficulty we can illustrate the theory of a finite universe by means of a mental picture to which, with some practice, we shall soon grow accustomed. First of all, an observation of epistemological nature. A geometrical-physical theory as such is incapable of being directly pictured, being merely a system of concepts. But these concepts serve the purpose of bringing a multiplicity of real or imaginary sensory experiences into connection in the mind. To "visualize" a theory therefore means to bring to mind that abundance of sensible experiences for which the theory supplies the schematic arrangement. In the present case we have to ask ourselves how we can represent that behavior of solid bodies with respect to their mutual disposition (contact) that corresponds to the theory of a finite universe.

## Thursday, November 8, 2018

### Reality vs. Axiomatic Thinking

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 11:16 PM
 A  Few  of  My  Favorite  Spaces: The Fano Plane The intuition-challenging Fano plane may be the smallest interesting configuration of points and lines. By Evelyn Lamb on October 24, 2015 "…finite projective planes seem like a triumph of purely axiomatic thinking over any hint of reality. . . ."

For Fano's axiomatic  approach, see the Nov. 3 Log24 post
"Foundations of Geometry."

For the Fano plane's basis in reality , see the eightfold cube
at finitegeometry.org/sc/ and in this journal.

See as well "Two Views of Finite Space" (in this journal on the date
of Lamb's remarks — Oct. 24, 2015).

Some context:  Gödel's Platonic realism vs. Hilbert's axiomatics
in remarks by Manuel Alfonseca on June 7, 2018. (See too remarks
in this journal on that date, in posts tagged "Road to Hell.")

## Saturday, September 15, 2018

### Eidetic Reduction in Geometry

Filed under: G-Notes,General,Geometry — Tags: , , , — m759 @ 1:23 AM

 "Husserl is not the greatest philosopher of all times. He is the greatest philosopher since Leibniz." Some results from a Google search — Eidetic reduction, in phenomenology, a method by which the philosopher moves from the consciousness of individual and concrete objects to the transempirical realm of pure essences and thus achieves an intuition of the eidos (Greek: “shape”) of a thing—i.e., of what it is in its invariable and essential structure, apart … The eidetic reduction: eidos. Method: Bracket all incidental meaning and ask: what are some of the possible invariate aspects of this experience? The research … Sep 19, 2017 – Eidetic reduction is a technique in Husserlian phenomenology, used to identify the essential components of the given phenomenon or experience.

For example —

The reduction of two-colorings and four-colorings of a square or cubic
array of subsquares or subcubes to lines, sets of lines, cuts, or sets of
cuts between the subsquares or subcubes.

See the diamond theorem and the eightfold cube.

* Cf. posts tagged Interality and Interstice.

## Friday, August 31, 2018

### Perception of Number

Filed under: General,Geometry — Tags: , — m759 @ 8:31 PM

Review of yesterday's post Perception of Space

From Harry Potter and the Philosopher's Stone  (1997),
republished as "… and the Sorcerer's Stone ," Kindle edition:

In a print edition from Bloomsbury (2004), and perhaps in the
earliest editions, the above word "movements" is the first word
on page 168:

Click the above ellipse for some Log24 posts on the eightfold cube,
the source of the 168 automorphisms ("movements") of the Fano plane.

"Refined interpretation requires that you know that
someone once said the offspring of reality and illusion
is only a staggering confusion."

— Poem, "The Game of Roles," by Mary Jo Bang

Related material on reality and illusion
an ad on the back cover of the current New Yorker

"Hey, the stars might lie, but the numbers never do." — Song lyric

## Thursday, August 30, 2018

### Perception* of Space

Filed under: General,Geometry — Tags: — m759 @ 2:12 PM

* A footnote in memory of a dancer who reportedly died
yesterday, August 29 —  See posts tagged Paradigm Shift.

"Birthday, death-day — what day is not both?" — John Updike

## Saturday, August 25, 2018

### “Waugh, Orwell. Orwell, Waugh.”

Filed under: General,Geometry — Tags: , , , — m759 @ 4:00 PM

Suggested by a review of Curl on Modernism —

Related material —

Waugh + Orwell in this journal and

## Sunday, July 29, 2018

### The Materialization

Filed under: General,Geometry — Tags: — m759 @ 11:01 PM

McCarthy's "materialization of plot and character" does not,
for me, constitute a proof that "there is  being, after all,
beyond the arbitrary flux of existence."

Neither does the above materialization of 281 as the page
number of her philosophical remark.

the book Witchcraft  by Charles Williams —

The materialization of 168 as a page number in a
Stephen King novel is somewhat more convincing,
but less convincing than the materialization of Klein's
simple group of of 168 elements in the eightfold cube.

## Sunday, July 22, 2018

### Space

Filed under: General,Geometry — Tags: , , , — m759 @ 10:29 AM

## Saturday, July 21, 2018

### Building-Block Theory

Filed under: General,Geometry — Tags: , — m759 @ 10:56 AM

(A sequel to yesterday’s Geometry for Jews)

From this journal on the above UCI posting  date — April 6, 2018 —

From this journal on the above lecture  date — April 26, 2018 —
illustrations in a post titled Defining Form

For the relevance of the above material to building blocks,
see Eightfold Cube in this journal.

## Sunday, July 1, 2018

### Deutsche Ordnung

The title is from a phrase spoken, notably, by Yul Brynner
to Christopher Plummer in the 1966 film “Triple Cross.”

Related structures —

Greg Egan’s animated image of the Klein quartic —

For a smaller tetrahedral arrangement, within the Steiner quadruple
system of order 8 modeled by the eightfold cube, see a book chapter
by Michael Huber of Tübingen

For further details, see the June 29 post Triangles in the Eightfold Cube.

 Abstract for a talk at the City University of New York: The Experience of Meaning Jan Zwicky, University of Victoria 09:00-09:40 Friday, April 5, 2013 Once the question of truth is settled, and often prior to it, what we value in a mathematical proof or conjecture is what we value in a work of lyric art: potency of meaning. An absence of clutter is a feature of such artifacts: they possess a resonant clarity that allows their meaning to break on our inner eye like light. But this absence of clutter is not tantamount to ‘being simple’: consider Eliot’s Four Quartets  or Mozart’s late symphonies. Some truths are complex, and they are simplified  at the cost of distortion, at the cost of ceasing to be  truths. Nonetheless, it’s often possible to express a complex truth in a way that precipitates a powerful experience of meaning. It is that experience we seek — not simplicity per se , but the flash of insight, the sense we’ve seen into the heart of things. I’ll first try to say something about what is involved in such recognitions; and then something about why an absence of clutter matters to them.

For the talk itself, see a YouTube video.

The conference talks also appear in a book.

The book begins with an epigraph by Hilbert

## Sunday, June 10, 2018

### Number Concept

Filed under: General,Geometry — Tags: — m759 @ 2:45 PM

The previous post was suggested by some April 17, 2016, remarks
by James Propp on the eightfold cube.

Propp's remarks included the following:

"Here’s a caveat about my glib earlier remark that
'There are only finitely many numbers ' in a finite field.
It’s a bit of a stretch to call the elements of finite fields
'numbers'. Elements of GF() can be thought of as
the integers mod q  when q  is prime, and they can be
represented by 0, 1, 2, …, q–1; but when  is a prime
raised to the 2nd power or higher, describing the
elements of GF() is more complicated, and the word
'number' isn’t apt."

Related material —

See also this  journal on the date of Propp's remarks — April 17, 2016.

## Wednesday, June 6, 2018

### Geometry for Goyim

Filed under: General,Geometry — Tags: , , — m759 @ 12:45 PM

Mystery box  merchandise from the 2011  J. J. Abrams film  Super 8  —

A mystery box that I prefer —

Click image for some background.

## Monday, June 4, 2018

### The Trinity Stone Defined

Filed under: General,Geometry — Tags: , , — m759 @ 8:56 PM

“Unsheathe your dagger definitions.” — James Joyce, Ulysses

The “triple cross” link in the previous post referenced the eightfold cube
as a structure that might be called the trinity stone .

## Sunday, April 1, 2018

### Logos

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

Happy April 1.

## Thursday, March 29, 2018

### To Imagine (or, Better, to Construct)

Filed under: General,Geometry — Tags: — m759 @ 8:10 PM

The title reverses a phrase of Fano —
costruire (o, dirò meglio immaginare).

Illustrations of imagining (the Fano plane) and of constructing (the eightfold cube) —

## Tuesday, March 27, 2018

### Compare and Contrast

Filed under: General,Geometry — Tags: , — m759 @ 4:28 PM

Related material on automorphism groups —

The "Eightfold Cube" structure shown above with Weyl
competes rather directly with the "Eightfold Way" sculpture
shown above with Bryant. The structure and the sculpture
each illustrate Klein's order-168 simple group.

Perhaps in part because of this competition, fans of the Mathematical
Sciences Research Institute (MSRI, pronounced "Misery') are less likely
to enjoy, and discuss, the eight-cube mathematical structure  above
than they are an eight-cube mechanical puzzle  like the one below.

Note also the earlier (2006) "Design Cube 2x2x2" webpage
illustrating graphic designs on the eightfold cube. This is visually,
if not mathematically, related to the (2010) "Expert's Cube."

## Wednesday, March 7, 2018

### Unite the Seven.

Filed under: General,Geometry — Tags: , , , — m759 @ 10:31 AM

Related material —

The seven points of the Fano plane within

"Before time began . . . ."

— Optimus Prime

## Saturday, January 6, 2018

### Report from Red Mountain

Filed under: General,Geometry — Tags: , , — m759 @ 4:00 PM

Tom Wolfe in The Painted Word  (1975):

“It is important to repeat that Greenberg and Rosenberg
did not create their theories in a vacuum or simply turn up
with them one day like tablets brought down from atop
Green Mountain or Red Mountain (as B. H. Friedman once
called the two men). As tout le monde  understood, they
were not only theories but … hot news,
straight from the studios, from the scene.”

Harold Rosenberg in The New Yorker  (click to enlarge)

## Friday, January 5, 2018

### Seven Types of Interality*

Filed under: General,Geometry — Tags: , , — m759 @ 1:00 AM

* See the term interality  in this journal.
For many synonyms, see
The Human Seriousness of Interality,”
by Peter Zhang, Grand Valley State University,
China Media Research  11(2), 2015, 93-103.

## Wednesday, November 22, 2017

### Goethe on All Souls’ Day

Filed under: General,Geometry — Tags: , — m759 @ 11:00 PM

David E. Wellbery on Goethe

From an interview published on 2 November 2017 at

as later republished in

The logo at left above is that of The Point .
The menu icon at right above is perhaps better
suited to illustrate Verwandlungslehre .

## Saturday, November 18, 2017

### Cube Space Continued

Filed under: General,Geometry — Tags: , — m759 @ 4:44 AM

James Propp in the current Math Horizons  on the eightfold cube

For another puerile approach to the eightfold cube,
see Cube Space, 1984-2003 (Oct. 24, 2008).

## Sunday, October 29, 2017

### File System… Unlocked

Filed under: General,Geometry — Tags: — m759 @ 2:16 PM

Logo from the above webpage

Related dialogue from the new film "Unlocked"

1057
01:31:59,926 –> 01:32:01,301
Nice to have you back, Alice.

1058
01:32:04,009 –> 01:32:05,467
Don't be a stranger.

## Thursday, October 19, 2017

### Graphic Design: Fast Forward

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 8:18 PM

Typographical: »

Eightfold Cube:

## Saturday, October 7, 2017

### Byte Space

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 7:59 PM

"Before time began,
there was the Cube."

Optimus Prime

## Wednesday, September 13, 2017

### Summer of 1984

The previous two posts dealt, rather indirectly, with
the notion of "cube bricks" (Cullinane, 1984) —

Group actions on partitions —

Cube Bricks 1984 —

Another mathematical remark from 1984 —

For further details, see Triangles Are Square.

## Tuesday, August 8, 2017

### Cube Quaternions

Filed under: General,Geometry — Tags: — m759 @ 9:36 AM

See posts now tagged with the above title.

## Saturday, July 29, 2017

### MSRI Program

Filed under: General,Geometry — Tags: , — m759 @ 8:29 PM

"The field of geometric group theory emerged from Gromov’s insight
that even mathematical objects such as groups, which are defined
completely in algebraic terms, can be profitably viewed as geometric
objects and studied with geometric techniques."

— Mathematical Sciences Research Institute, 2016:

For a simpler example than those discussed at MSRI
of both algebraic and geometric techniques applied to
the same group, see a post of May 19, 2017,
"From Algebra to Geometry." That post reviews
an earlier illustration —

For greater depth, see "Eightfold Cube" in this journal.

## Tuesday, June 20, 2017

### Epic

Continuing the previous post's theme

Group actions on partitions

Cube Bricks 1984 —

Related material — Posts now tagged Device Narratives.

## Wednesday, June 7, 2017

### Three Things at Once

Filed under: General,Geometry — Tags: , — m759 @ 1:28 PM

Cambridge University Press in 1999 —

## Tuesday, May 2, 2017

### Image Albums

Filed under: General,Geometry — Tags: , — m759 @ 1:05 PM

Pinterest boards uploaded to the new m759.net/piwigo

Update of May 2 —

Update of May 3 —

Update of May 8 —

Art Space board created at Pinterest

## Wednesday, April 12, 2017

### Contracting the Spielraum

The contraction of the title is from group actions on
the ninefold square  (with the center subsquare fixed)
to group actions on the eightfold cube.

From a post of June 4, 2014

At math.stackexchange.com on March 1-12, 2013:

The above illustration, though neatly drawn, appeared under the
cloak of anonymity.  No source was given for the illustrated group actions.
Possibly they stem from my Log24 posts or notes such as the Jan. 4, 2012,
note on quaternion actions at finitegeometry.org/sc (hence ultimately
from my note “GL(2,3) actions on a cube” of April 5, 1985).

## Thursday, March 9, 2017

### One Eighth

Filed under: General,Geometry — Tags: — m759 @ 1:16 PM

From Wikipedia's Iceberg Theory

Related material:

## Saturday, January 14, 2017

### 1984: A Space Odyssey

Filed under: General,Geometry — m759 @ 2:40 PM

See Eightfold 1984 in this journal.

Related material —

"… the object sets up a kind of
frame or space or field
within which there can be epiphany."

"… Instead of an epiphany of being,
we have something like
an epiphany of interspaces."

— Charles Taylor, "Epiphanies of Modernism,"
Chapter 24 of Sources of the Self ,
Cambridge University Press, 1989

and end with algebra; and perhaps without the metaphor
there would never have been any algebra."

— Max Black, Models and Metaphors ,
Cornell University Press, Ithaca, NY, 1962

Click to enlarge:

## Sunday, January 8, 2017

### A Theory of Everything

Filed under: General,Geometry — Tags: , — m759 @ 7:11 PM

The title refers to the Chinese book the I Ching ,
the Classic of Changes .

The 64 hexagrams of the I Ching  may be arranged
naturally in a 4x4x4 cube. The natural form of transformations
("changes") of this cube is given by the diamond theorem.

A related post —

## Saturday, January 7, 2017

### Conceptualist Minimalism

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

"Clearly, there is a spirit of openhandedness in post-conceptual art
uses of the term 'Conceptualism.' We can now endow it with a
capital letter because it has grown in scale from its initial designation
of an avant-garde grouping, or various groups in various places, and
has evolved in two further phases. It became something like a movement,
on par with and evolving at the same time as Minimalism. Thus the sense
it has in a book such as Tony Godfrey’s Conceptual Art.  Beyond that,
it has in recent years spread to become a tendency, a resonance within
art practice that is nearly ubiquitous." — Terry Smith, 2011

## Sunday, November 27, 2016

### A Machine That Will Fit

Filed under: General,Geometry — Tags: , , — m759 @ 8:00 AM

Or:  Notes for the Metaphysical Club

"He… stands in contrast to the the dualistic
approach of Eliot, who so often speaks of poetry
as though it were an emotional and sensational
soul looking for a 'correlative' skeleton of
thought to be provided by a philosopher, a
Cartesian ghost trying to find a machine that
will fit."

Ralph Waldo Emerson on "vacant and vain" knowledge:

"The new position of the advancing man has all
the powers of the old, yet has them all new. It
carries in its bosom all the energies of the past,
yet is itself an exhalation of the morning. I cast
away in this new moment all my once hoarded
knowledge, as vacant and vain."

Harold Bloom on Emerson:

"Emerson may not have invented the American
Sublime, yet he took eternal possession of it."

Wallace Stevens on the American Sublime:

"And the sublime comes down
To the spirit itself,

The spirit and space,
The empty spirit
In vacant space."

A founding member of the Metaphysical Club:

## Thursday, November 3, 2016

### Triple Cross

(Continued See the title in this journal, as well as Cube Bricks.)

Cube Bricks 1984 —

Related material —

Dirac and Geometry in this journal,
Kummer’s Quartic Surface in this journal,
Nanavira Thera in this journal, and
The Razor’s Edge  and Nanavira Thera.

See as well Bill Murray’s 1984 film “The Razor’s Edge”

Movie poster from 1984 —

“A thin line separates
love from hate,
success from failure,
life from death.”

Three other dualities, from Nanavira Thera in 1959 —

“I find that there are, in every situation,
three independent dualities….”

(Click to enlarge.)

## Sunday, October 23, 2016

### Quartet

Filed under: General,Geometry — Tags: , — m759 @ 12:48 PM

“The man who lives in contact with what he believes to be a living Church
is a man always expecting to meet Plato and Shakespeare to-morrow
at breakfast.”

— G. K. Chesterton

Or Sunday dinner.

 Platonic solid Natasha Wescoat, 2004 Shakespearean Fool Not to mention Euclid and Picasso. In the above pictures, Euclid is represented by  Alexander Bogomolny, Picasso by Robert Foote.

## Sunday, September 25, 2016

### Introduction to Pragmatism

Filed under: General — Tags: , — m759 @ 7:29 AM

Stanford Encyclopedia of Philosophy
on the origins of Pragmatism:

"Pragmatism had been born in the discussions at
a ‘metaphysical club’ in Harvard around 1870
(see Menand…*). Peirce and James participated
in these discussions along with some other philosophers
and philosophically inclined lawyers. As we have
already noted, Peirce developed these ideas in his
publications from the 1870s."

 From "How to Make Our Ideas Clear," by Charles Sanders Peirce in 1878 — "The very first lesson that we have a right to demand that logic shall teach us is, how to make our ideas clear; and a most important one it is, depreciated only by minds who stand in need of it. To know what we think, to be masters of our own meaning, will make a solid foundation for great and weighty thought. It is most easily learned by those whose ideas are meagre and restricted; and far happier they than such as wallow helplessly in a rich mud of conceptions. A nation, it is true, may, in the course of generations, overcome the disadvantage of an excessive wealth of language and its natural concomitant, a vast, unfathomable deep of ideas. We may see it in history, slowly perfecting its literary forms, sloughing at length its metaphysics, and, by virtue of the untirable patience which is often a compensation, attaining great excellence in every branch of mental acquirement. The page of history is not yet unrolled which is to tell us whether such a people will or will not in the long-run prevail over one whose ideas (like the words of their language) are few, but which possesses a wonderful mastery over those which it has. For an individual, however, there can be no question that a few clear ideas are worth more than many confused ones. A young man would hardly be persuaded to sacrifice the greater part of his thoughts to save the rest; and the muddled head is the least apt to see the necessity of such a sacrifice. Him we can usually only commiserate, as a person with a congenital defect. Time will help him, but intellectual maturity with regard to clearness comes rather late, an unfortunate arrangement of Nature, inasmuch as clearness is of less use to a man settled in life, whose errors have in great measure had their effect, than it would be to one whose path lies before him. It is terrible to see how a single unclear idea, a single formula without meaning, lurking in a young man's head, will sometimes act like an obstruction of inert matter in an artery, hindering the nutrition of the brain, and condemning its victim to pine away in the fullness of his intellectual vigor and in the midst of intellectual plenty. Many a man has cherished for years as his hobby some vague shadow of an idea, too meaningless to be positively false; he has, nevertheless, passionately loved it, has made it his companion by day and by night, and has given to it his strength and his life, leaving all other occupations for its sake, and in short has lived with it and for it, until it has become, as it were, flesh of his flesh and bone of his bone; and then he has waked up some bright morning to find it gone, clean vanished away like the beautiful Melusina of the fable, and the essence of his life gone with it. I have myself known such a man; and who can tell how many histories of circle-squarers, metaphysicians, astrologers, and what not, may not be told in the old German story?"

Peirce himself may or may not have been entirely successful
in making his ideas clear.  See Where Credit Is Due  (Log24,
June 11, 2016) and the Wikipedia article Categories (Peirce).

* Menand, L., 2001. The Metaphysical Club A Story of
Ideas in America

, New York:  Farrar, Straus and Giroux

## Saturday, September 24, 2016

### Core Structure

Filed under: General,Geometry — Tags: , — m759 @ 6:40 AM

For the director of "Interstellar" and "Inception"

At the core of the 4x4x4 cube is …

Cover modified.

## Thursday, September 22, 2016

### Binary Opposition Illustrated

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

Click the above image for remarks on
"deep structure" and binary opposition.

## Thursday, September 15, 2016

### Metaphysics at Notre Dame

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

"When Analogies Fail," by Alexander Stern,
a doctoral candidate in philosophy at Notre Dame, in
The Chronicle of Higher Education  online September 11, 2016.

Related material —

That same Alexander Stern in this  journal on April 17, 2016:

Metaphysics at Scientific American:

## Wednesday, August 31, 2016

### The Lost Crucible

Filed under: General,Geometry — Tags: — m759 @ 4:56 PM

Yesterday's post The Eightfold Cube in Oslo suggests a review of
posts that mention The Lost Crucible.

(The crucible in question is from a book by Katherine Neville,
The Eight . Any connection with Arthur Miller's play  "The Crucible"
is purely coincidental.)

## Saturday, August 27, 2016

### Incarnation

Filed under: General,Geometry — Tags: , — m759 @ 1:06 AM

See a search for the title in this journal.

Related material:

The incarnation of three permutations,
named A, B, and C,
on the 7-set of digits {1, 2, 3, 4, 5, 6, 7}
as  permutations on the eightfold cube.

See Minimal ABC Art, a post of August 22, 2016.

## Monday, April 25, 2016

### Peirce’s Accounts of the Universe

Filed under: General,Geometry — Tags: , — 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 —

From my commentary on Carter's question —

## Wednesday, April 20, 2016

### Symmetric Generation of a Simple Group

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

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.

## Tuesday, April 19, 2016

### The Folding

Filed under: General,Geometry — Tags: — m759 @ 2:00 PM

A recent post about the eightfold cube  suggests a review of two
April 8, 2015, posts on what Northrop Frye called the ogdoad :

As noted on April 8, each 2×4 "brick" in the 1974 Miracle Octad Generator
of R. T. Curtis may be constructed by folding  a 1×8 array from Turyn's
1967 construction of the Golay code.

Folding a 2×4 Curtis array yet again  yields the 2x2x2 eightfold cube .

Those who prefer an entertainment  approach to concepts of space
may enjoy a video (embedded yesterday in a story on theverge.com) —
"Ghost in the Shell: Identity in Space."

## Sunday, April 17, 2016

### The Thing and I

Filed under: General,Geometry — Tags: — 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 —

## Monday, April 4, 2016

### Cube for Berlin

Filed under: General,Geometry — Tags: , , — m759 @ 11:00 AM

Foreword by Sir Michael Atiyah —

“Poincaré said that science is no more a collection of facts
than a house is a collection of bricks. The facts have to be
ordered or structured, they have to fit a theory, a construct
(often mathematical) in the human mind. . . .

Mathematics may be art, but to the general public it is
a black art, more akin to magic and mystery. This presents
a constant challenge to the mathematical community: to
explain how art fits into our subject and what we mean by beauty.

In attempting to bridge this divide I have always found that
architecture is the best of the arts to compare with mathematics.
The analogy between the two subjects is not hard to describe
and enables abstract ideas to be exemplified by bricks and mortar,
in the spirit of the Poincaré quotation I used earlier.”

— Sir Michael Atiyah, “The Art of Mathematics”
in the AMS Notices , January 2010

Judy Bass, Los Angeles Times , March 12, 1989 —

“Like Rubik’s Cube, The Eight  demands to be pondered.”

As does a figure from 1984, Cullinane’s Cube —

For natural group actions on the Cullinane cube,
see “The Eightfold Cube” and
A Simple Reflection Group of Order 168.”

Related remark from the literature —

Note that only the static structure is described by Felsner, not the
168 group actions discussed by Cullinane. For remarks on such
group actions in the literature, see “Cube Space, 1984-2003.”

(From Anatomy of a Cube, Sept. 18, 2011.)

## Tuesday, March 15, 2016

### 15 Projective Points Revisited

Filed under: General,Geometry — Tags: , — 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."

## Thursday, December 17, 2015

### Hint of Reality

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,
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

## Thursday, December 3, 2015

### Design Wars

Filed under: General,Geometry — Tags: , — m759 @ 4:04 PM

"… if your requirement for success is to be like Steve Jobs,
good luck to you."

— "Transformation at Yahoo Foiled by Marissa Mayer’s
Inability to Bet the Farm," New York Times  online yesterday

"Design is how it works." — Steve Jobs

## Thursday, November 5, 2015

### ABC Art or: Guitart Solo

Filed under: General,Geometry — Tags: , , — m759 @ 5:55 PM

“… the A B C of being….” — Wallace Stevens

Scholia —

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.” — Borromean Generators (Log24, Oct. 19)

and “Guitar Solo” from the 2015 CMA Awards on ABC.

## Saturday, October 31, 2015

### Raiders of the Lost Crucible

Filed under: General,Geometry — Tags: , , — m759 @ 10:15 AM

Stanford Encyclopedia of Philosophy
on the date Friday, April 5, 2013 —

Paraconsistent Logic

“First published Tue Sep 24, 1996;
substantive revision Fri Apr 5, 2013”

This  journal on the date Friday, April 5, 2013 —

The object most closely resembling a “philosophers’ stone”
that I know of is the eightfold cube .

For some related philosophical remarks that may appeal
to a general Internet audience, see (for instance) a website
by I Ching  enthusiast Andreas Schöter that displays a labeled
eightfold cube in the form of a lattice diagram —

Related material by Schöter —

A 20-page PDF, “Boolean Algebra and the Yi Jing.”
(First published in The Oracle: The Journal of Yijing Studies ,
Vol 2, No 7, Summer 1998, pp. 19–34.)

I differ with Schöter’s emphasis on Boolean algebra.
The appropriate mathematics for I Ching  studies is,
I maintain, not Boolean algebra  but rather Galois geometry.

See last Saturday’s post Two Views of Finite Space.
Unfortunately, that post is, unlike Schöter’s work, not
suitable for a general Internet audience.

## Monday, October 19, 2015

### Symmetric Generation of the Simple Order-168 Group

Filed under: General,Geometry — Tags: , , — 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

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.

### Borromean Generators

Filed under: General,Geometry — Tags: , — 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  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 ,

## Saturday, October 10, 2015

### Nonphysical Entities

Filed under: General,Geometry — Tags: , — m759 @ 9:00 PM

" 'Ambassadørene' er fysiske former som presenterer
ikk-fysiske fenomener. "

nonphysical phenomena. "

Related definition —

Are the "line diagrams" of the diamond theorem and
the analogous "plane diagrams" of the eightfold cube
nonphysical entities? Discuss.

## Monday, July 13, 2015

### Block Designs Illustrated

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

The Fano Plane —

"A balanced incomplete block design , or BIBD
with parameters , , , , and λ  is an arrangement
of b  blocks, taken from a set of v  objects (known
for historical reasons as varieties ), such that every
variety appears in exactly r  blocks, every block
contains exactly k  varieties, and every pair of
varieties appears together in exactly λ  blocks.
Such an arrangement is also called a
(, v , r , k , λ ) design. Thus, (7, 3, 1) [the Fano plane]
is a (7, 7, 3, 3, 1) design."

— Ezra Brown, "The Many Names of (7, 3, 1),"
Mathematics Magazine , Vol. 75, No. 2, April 2002

W. Cherowitzo uses the notation (v, b, r, k, λ) instead of
Brown's (b , v , r , k , λ ).  Cherowitzo has described,
without mentioning its close connection with the
Fano-plane design, the following —

"the (8,14,7,4,3)-design on the set
X = {1,2,3,4,5,6,7,8} with blocks:

{1,3,7,8} {1,2,4,8} {2,3,5,8} {3,4,6,8} {4,5,7,8}
{1,5,6,8} {2,6,7,8} {1,2,3,6} {1,2,5,7} {1,3,4,5}
{1,4,6,7} {2,3,4,7} {2,4,5,6} {3,5,6,7}."

We can arrange these 14 blocks in complementary pairs:

{1,2,3,6} {4,5,7,8}
{1,2,4,8} {3,5,6,7}
{1,2,5,7} {3,4,6,8}
{1,3,4,5} {2,6,7,8}
{1,3,7,8} {2,4,5,6}
{1,4,6,7} {2,3,5,8}
{1,5,6,8} {2,3,4,7}.

These pairs correspond to the seven natural slicings
of the following eightfold cube —

Another representation of these seven natural slicings —

These seven slicings represent the seven
planes through the origin in the vector
3-space over the two-element field GF(2).
In a standard construction, these seven
planes  provide one way of defining the
seven projective lines  of the Fano plane.

A more colorful illustration —

## Saturday, June 27, 2015

### A Single Finite Structure

Filed under: General,Geometry — Tags: — m759 @ 12:15 PM

"It is as if one were to condense
all trends of present day mathematics
onto a single finite structure…."

— Gian-Carlo Rota, foreword to
A Source Book in Matroid Theory ,
Joseph P.S. Kung, Birkhäuser, 1986

"There is  such a thing as a matroid."

Related remarks from Mathematics Magazine  in 2009 —

.

## Thursday, June 11, 2015

### Omega

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

Omega is a Greek letter, Ω , used in mathematics to denote
a set on which a group acts.

For instance, the affine group AGL(3,2) is a group of 1,344
actions on the eight elements of the vector 3-space over the
two-element Galois field GF(2), or, if you prefer, on the Galois
field  Ω = GF(8).

Related fiction:  The Eight , by Katherine Neville.

Related non-fiction:  A remark by Werner Heisenberg
in this journal on Saturday, June 6, 2015, the eightfold cube ,
and the illustrations below —

 Mathematics The Fano plane block design Magic The Deathly Hallows symbol— Two blocks short of  a design.

## Friday, June 5, 2015

### Narratives

Filed under: General,Geometry — Tags: — m759 @ 11:09 PM

.

For those who prefer mathematics to narrative:

## Thursday, February 26, 2015

### A Simple Group

Filed under: General,Geometry — Tags: — m759 @ 7:59 PM

The previous post's
illustration was
rather complicated.

This is a simpler
algebraic figure.

## Tuesday, February 10, 2015

### In Memoriam…

Filed under: General,Geometry — Tags: , — m759 @ 12:25 PM

industrial designer Kenji Ekuan —

The adjective "eightfold," intrinsic to Buddhist
thought, was hijacked by Gell-Mann and later
by the Mathematical Sciences Research Institute
application to a 2x2x2 cube consisting of eight
subcubes, "the eightfold cube," is not intended to
have either Buddhist or Semitic overtones.
It is pure mathematics.

## Sunday, November 30, 2014

### Two Physical Models of the Fano Plane

Filed under: General,Geometry — Tags: , , — m759 @ 1:23 AM

The seven symmetry axes of the regular tetrahedron
are of two types: vertex-to-face and edge-to-edge.
Take these axes as the "points" of a Fano plane.
Each of the tetrahedron's six reflection planes contains
two vertex-to-face axes and one edge-to-edge axis.
Take these six planes as six of the "lines" of a Fano
plane. Then the seventh line is the set of three
edge-to-edge axes.

(The Fano tetrahedron is not original with me.
See Polster's 1998 A Geometrical Picture Book pp. 16-17.)

There are three reflection planes parallel to faces
of the cube. Take the seven nonempty subsets of
the set of these three planes as the "points" of a
Fano plane. Define the Fano "lines" as those triples
of these seven subsets in which each member of
the triple is the symmetric-difference sum of the
other two members.

(This is the eightfold cube  discussed at finitegeometry.org.)

## Wednesday, September 17, 2014

### Raiders of the Lost Articulation

Tom Hanks as Indiana Langdon in Raiders of the Lost Articulation :

An unarticulated (but colored) cube:

A 2x2x2 articulated cube:

A 4x4x4 articulated cube built from subcubes like
the one viewed by Tom Hanks above:

Solomon’s Cube

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