Log24

Sunday, September 4, 2022

Dice and the Eightfold Cube

Filed under: General — Tags: , — m759 @ 4:47 pm

At Hiroshima on March 9, 2018, Aitchison discussed another 
"hexagonal array" with two added points… not at the center, but
rather at the ends  of a cube's diagonal axis of symmetry.

See some related illustrations below. 

Fans of the fictional "Transfiguration College" in the play
"Heroes of the Fourth Turning" may recall that August 6,
another Hiroshima date, was the Feast of the Transfiguration.

Iain Aitchison's 'dice-labelled' cuboctahedron at Hiroshima, March 2018

The exceptional role of  0 and  in Aitchison's diagram is echoed
by the occurence of these symbols in the "knight" labeling of a 
Miracle Octad Generator octad —

Transposition of  0 and  in the knight coordinatization 
induces the symplectic polarity of PG(3,2) discussed by 
(for instance) Anne Duncan in 1968.

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.

Click image below to download a Guitart PowerPoint presentation.

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) —

Why PSL(2,7) is isomorphic to 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 ) —

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" —

See also Rubik in this  journal.

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

See also the eightfold cube in this  journal.

"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

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
last fall had a link to the eightfold cube:

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) —

The eightfold cube at the Vigeland Museum in Oslo

Update of Sept. 7, 2016: The corrections have been made,
except for the misspelling "Cullinan," which was caused by 
Google translation, not by KUNSTforum.

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.

Discussion of Cullinane's eightfold cube as exhibited by Josefine Lyche at the Vigeland Museum in Oslo

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
installation "4D Ambassador" at the 
Norwegian Sculpture Biennial 2015

Sculpture by Josefine Lyche of Cullinane's eightfold cube at Vigeland Museum in Oslo

(Detail from private Instagram photo.)

Catalog description of installation —

Google Translate version —

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
normally do not have access to. "Ambassadors"
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.

See also Lyche + "4D Ambassador" in this journal and
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*

Dynkin diagram D4 for 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,
and Quadratic Generalizations

Manjul Bhargava

The Annals of Mathematics
Second Series, Vol. 159, No. 1 (Jan., 2004), pp. 217-250
Published by: Annals of Mathematics
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.

Saturday, November 12, 2022

Inside a White Cube

Filed under: General — Tags: — m759 @ 12:09 pm

For the late Brian O'Doherty, from posts now tagged "Pless Birthday 2022" —

A Mathieu Puzzle: 24 Diamond Facets of the Eightfold Cube

This post was suggested by an obituary of O'Doherty and by
"The Life and Work of Vera Stepen Pless" in
Notices of the American Mathematical Society , December 2022.

Wednesday, March 9, 2022

Supercube Space

Filed under: General — Tags: , — m759 @ 12:31 am

The new URL supercube.space forwards to http://box759.wordpress.com/.

The term supercube  is from a 1982 article by Solomon W. Golomb.

The related new URL supercube.group forwards to a page that
describes how the 2x2x2 (or eightfold, or "super") cube's natural
underlying automorphism group is Klein's simple group of order 168.

For further context, see the new URL supercube.art.

For some background, see the phrase Cube Space in this journal. 

Monday, March 7, 2022

Saturday, February 5, 2022

Mathieu Cube Labeling

Filed under: General — Tags: , , , — m759 @ 2:08 pm

Shown below is an illustration from "The Puzzle Layout Problem" —

Exercise:  Using the above numerals 1 through 24
(with 23 as 0 and 24 as ∞) to represent the points 
, 0, 1, 2, 3 … 22  of the projective line over GF(23),
reposition the labels 1 through 24 in the above illustration
so that they appropriately* illustrate the cube-parts discussed
by Iain Aitchison in his March 2018 Hiroshima slides on 
cube-part permutations by the Mathieu group M24

A note for Northrop Frye —

Interpenetration in the eightfold cube — the three midplanes —

IMAGE- The Trinity Cube (three interpenetrating planes that split the eightfold cube into its eight subcubes)

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.

* "Appropriately" — I.e. , so that the Aitchison cube octads correspond
exactly, via the projective-point labels, to the Curtis MOG octads.

Saturday, May 23, 2020

Eightfold Geometry: A Surface Code “Unit Cell”

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

A unit cell in 'a lattice geometry for a surface code'

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.

Saturday, May 4, 2019

Inside the White Cube

Structure of the eightfold cube

See also Espacement  and The Thing and I.

Thursday, December 6, 2018

The Mathieu Cube of Iain Aitchison

This journal ten years ago today —

Surprise Package

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

The Mathieu group cube of Iain Aitchison (2018, Hiroshima)

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 . . .

The Eightfold Cube: The Beauty of Klein's Simple Group

   Image from Christmas Day 2005.

Thursday, November 29, 2018

The White Cube

Filed under: G-Notes,General,Geometry — m759 @ 9:57 am

Clicking on Zong in the above post leads to a 2005 article
in the Bulletin of the American Mathematical Society .

See also the eightfold  cube and interality .

Thursday, May 31, 2018

Eightfold Suffering:

Filed under: General,Geometry — Tags: — m759 @ 9:23 pm

A New, Improved Version of  Quantum Suffering !

Background for group actions on the eightfold cube

See also other posts now tagged Quantum Suffering 
as well as — related to the image above of the Great Wall

Myspace China.

Wednesday, January 17, 2018

“Before Time Began, There Was the Cube”

Filed under: General,Geometry — m759 @ 8:00 am

See Eightfold Froebel.

The Paradise of Childhood'-- Froebel's Third Gift

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

James Propp on the eightfold cube

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

Tuesday, August 8, 2017

Cube Quaternions

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

See posts now tagged with the above title.

IMAGE- Quaternion group acting on an 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 —

The Eightfold Cube

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

See also the recent post Cube Bricks 1984

An Approach to Symmetric Generation of the Simple Group of Order 168

Related remark from the literature —

http://www.log24.com/log/pix11B/110918-Felsner.jpg

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.)

Friday, December 30, 2011

Quaternions on a Cube

The following picture provides a new visual approach to
the order-8 quaternion  group's automorphisms.

IMAGE- Quaternion group acting on an eightfold cube

Click the above image for some context.

Here the cube is called "eightfold" because the eight vertices,
like the eight subcubes of a 2×2×2 cube,* are thought of as
independently movable. See The Eightfold Cube.

See also…

Related material: Robin Chapman and Karen E. Smith
on the quaternion group's automorphisms.

* See Margaret Wertheim's Christmas Eve remarks on mathematics
and the following eightfold cube from an institute she co-founded—

Froebel's third gift, the eightfold cube
© 2005 The Institute for Figuring

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

Sunday, September 18, 2011

Anatomy of a Cube

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

R.D. Carmichael’s seminal 1931 paper on tactical configurations suggests
a search for later material relating such configurations to block designs.
Such a search yields the following

“… it seems that the relationship between
BIB [balanced incomplete block ] designs
and tactical configurations, and in particular,
the Steiner system, has been overlooked.”
— D. A. Sprott, U. of Toronto, 1955

http://www.log24.com/log/pix11B/110918-SprottAndCube.jpg

The figure by Cullinane included above shows a way to visualize Sprott’s remarks.

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

Update of 7:42 PM Sept. 18, 2011—

From a Summer 2011 course on discrete structures at a Berlin website—

A different illustration of the eightfold cube as the Steiner system S(3, 4, 8)—

http://www.log24.com/log/pix11B/110918-Felsner.jpg

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

Thursday, May 26, 2011

Prime Cubes

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

The title refers not to numbers  of the form p 3, p  prime, but to geometric  cubes with p 3 subcubes.

Such cubes are natural models for the finite vector spaces acted upon by general linear groups viewed as permutation  groups of degree  (not order ) p 3.

IMAGE- From preface to Larry C. Grove, 'Classical Groups and Geometric Algebra

For the case p =2, see The Eightfold Cube.

For the case p =3, see the "External links" section of the Nov. 30, 2009, version of Wikipedia article "General Linear Group." (That is the version just prior to the Dec. 14, 2009, revision by anonymous user "Greenfernglade.")

For symmetries of group actions for larger primes, see the related 1985 remark* on two -dimensional linear groups—

"Actions of GL(2,p )  on a p ×p  coordinate-array
have the same sorts of symmetries,
where p  is any odd prime."

* Group Actions, 1984-2009

Monday, June 21, 2010

Cube Spaces

Cubic models of finite geometries
display an interplay between
Euclidean and Galois geometry.

 

Example 1— The 2×2×2 Cube—

also known as the eightfold  cube

2x2x2 cube

Group actions on the eightfold cube, 1984—

http://www.log24.com/log/pix10A/100621-diandwh-detail.GIF

Version by Laszlo Lovasz et al., 2003—

http://www.log24.com/log/pix10A/100621-LovaszCubeSpace.gif

Lovasz et al. go on to describe the same group actions
as in the 1984 note, without attribution.

Example 2— The 3×3×3 Cube

A note from 1985 describing group actions on a 3×3 plane array—

http://www.log24.com/log/pix10A/100621-VisualizingDetail.gif

Undated software by Ed Pegg Jr. displays
group actions on a 3×3×3 cube that extend the
3×3 group actions from 1985 described above—

Ed Pegg Jr.'s program at Wolfram demonstrating concepts of a 1985 note by Cullinane

Pegg gives no reference to the 1985 work on group actions.

Example 3— The 4×4×4 Cube

A note from 27 years ago today—

http://www.log24.com/log/pix10A/100621-Cube830621.gif

As far as I know, this version of the
group-actions theorem has not yet been ripped off.

Tuesday, March 30, 2010

Eightfold Symmetries

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

Harvard Crimson headline today–
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” —

Dharma Wheel from Wikipedia

Adapted detail–

Adapted Dharma Wheel detail

See also, from
St. Joseph’s Day

Weyl's 'Symmetry,' the triquetrum, and the eightfold cube

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.

Tuesday, February 20, 2024

Backlight

Filed under: General — m759 @ 12:09 am

The epigraph of the previous post

"To Phaedrus, this backlight from the conflict between
the Sophists and the Cosmologists adds an entirely
new dimension to the Dialogues of Plato." — Robert M. Pirsig

Related reading and art for academic nihilists — See . . .

Reading and art I prefer —

Love in the Ruins , by Walker Percy, and . . .

Van Gogh  (by Ed Arno) and an image and
a passage from The Paradise of Childhood
(by Edward Wiebé):

'Dear Theo' cartoon of van Gogh by Ed Arno, adapted to illustrate the eightfold cube

Tuesday, October 24, 2023

A Bond with Reality:  The Geometry of Cuts

Filed under: General — Tags: , , — m759 @ 12:12 pm


Illustrations of object and gestures
from finitegeometry.org/sc/ —

Object

Gestures

An earlier presentation of the above
seven partitions of the eightfold cube:

Seven partitions of the 2x2x2 cube in a book from 1906

Related mathematics:

The use  of binary coordinate systems
as a conceptual tool

Natural physical  transformations of square or cubical arrays
of actual physical cubes (i.e., building blocks) correspond to
natural algebraic  transformations of vector spaces over GF(2).
This was apparently not previously known.

See "The Thing and I."

and . . .

Galois.space .

 

Related entertainment:

Or Matt Helm by way of a Jedi cube.

Saturday, July 1, 2023

Mechanical Plaything (Hinged) vs. Conceptual Art (Unhinged)

Filed under: General — Tags: , , — m759 @ 2:44 pm

"Infinity Cube" … hinged plaything, for sale —

"Eightfold Cube" … un hinged concept, not for sale—

See as well yesterday's Trickster Fuge ,
and a 1906 discussion of the eightfold cube:

Page from 'The Paradise of Childhood,' 1906 edition

Wednesday, June 21, 2023

Annals of Magical Thinking: The Rushmore Embedding

Filed under: General — Tags: , — m759 @ 11:53 am

http://www.log24.com/log/pix18/180830-Harry_Potter-Phil_Stone-Bloomsbury-2004-p168.jpg

Less magical: "What are vector embeddings?"

Saturday, June 10, 2023

Green, Orange, Black

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

https://www.dailymail.co.uk/tvshowbiz/article-12179599/
Emma-Watson-stuns-revealing-black-bandeau-Prada.html

The colors surrounding Watson's body in the above
"bandeau" photo suggest a review.  A search in this  journal
for Green+Orange+Black  yields . . .

In the above image, the "hard core of objectivity" is represented
by the green-and-white eightfold cube.  The orange and black are,
of course, the Princeton colors.

Friday, June 2, 2023

Reichenbach’s Fell Swoop

Filed under: General — Tags: , — m759 @ 12:18 pm

See The Eightfold Cube  and . . .

Truth, Beauty, and The Good

Art is magic delivered from
the lie of being truth.
 — Theodor Adorno, Minima moralia,
London, New Left Books, 1974, p. 222
(First published in German in 1951.)

The director, Carol Reed, makes…
 impeccable use of the beauty of black….
— V. B. Daniel on The Third Man 

I see your ironical smile.
— Hans Reichenbach 

Adorno, The Third Man, and Reichenbach
are illustrated below (l. to r.) above the names of
cities with which they are associated. 

 

Friday, January 27, 2023

The Stone

Filed under: General — Tags: , , — m759 @ 11:00 pm

Here stands the mean, uncomely stone,
’Tis very cheap in price!
The more it is despised by fools,
The more loved by the wise.

— https://jungcurrents.com/
the-story-of-the-stone-at-bollingen

Not so cheap:

Identical copies of the above image are being offered for sale
on three websites as representing a Masonic "cubic stone."

None of the three sites say where, exactly, the image originated.
Image searches for "Masonic stone," "Masonic cube," etc.,
fail to yield any other  pictures that look like the above image —
that of a 2x2x2 array of eight identical subcubes.

For purely mathematical — not  Masonic — properties of such
an array, see "eightfold cube" in this journal.

The websites offering to sell the questionable image —

Getty —

https://www.gettyimages.co.nz/detail/news-photo/
freemasonry-cubic-stone-masonic-symbol-news-photo/535802541

Alamy —

https://www.alamy.com/
stock-photo-cubic-stone-masonic-symbol-49942969.html

Photo12 —

https://www.photo12.com/en/image/
hac03239_2002_p1800264

No price quoted on public page:

Saturday, January 14, 2023

Châtelet on Weil — A “Space of Gestures”

Filed under: General — Tags: , , , — m759 @ 2:21 pm
 

From Gilles Châtelet, Introduction to Figuring Space
(Springer, 1999) —

Metaphysics does have a catalytic effect, which has been described in a very beautiful text by the mathematician André Weil:

Nothing is more fertile, all mathematicians know, than these obscure analogies, these murky reflections of one theory in another, these furtive caresses, these inexplicable tiffs; also nothing gives as much pleasure to the researcher. A day comes when the illusion vanishes: presentiment turns into certainty … Luckily for researchers, as the fogs clear at one point, they form again at another.4

André Weil cuts to the quick here: he conjures these 'murky reflections', these 'furtive caresses', the 'theory of Galois that Lagrange touches … with his finger through a screen that he does not manage to pierce.' He is a connoisseur of these metaphysical 'fogs' whose dissipation at one point heralds their reforming at another. It would be better to talk here of a horizon that tilts thereby revealing a new space of gestures which has not as yet been elucidated and cut out as structure.

4 A. Weil, 'De la métaphysique aux mathématiques', (Oeuvres, vol. II, p. 408.)

For gestures as fogs, see the oeuvre of  Guerino Mazzola.

For some clearer remarks, see . . .


Illustrations of object and gestures
from finitegeometry.org/sc/ —

 

Object

 

Gestures

An earlier presentation
of the above seven partitions
of the eightfold cube:

Seven partitions of the 2x2x2 cube in a book from 1906

Related material: Galois.space .

Monday, December 26, 2022

Super-8 Box

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

For the title, see other posts tagged Super-8.

Box containing Froebel's Third Gift-- The Eightfold Cube

Click image for some background.

Related material —

Friday, December 23, 2022

Was ist Raum?” — Bauhaus Founder Walter Gropius

Filed under: General — Tags: — m759 @ 10:43 am

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

Walter Gropius,

The Theory and
Organization of the
Bauhaus
  (1923)

A relevant illustration:

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

Is there a geometric realization of the Quaternion group?” —

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).

These references will not appeal to those who enjoy modernism as a religion.
(For such a view, see Rosalind Krauss on grids and another writer's remarks
on the religion's 100th anniversary this year.)

Some related nihilist philosophy from Cormac McCarthy —

"Forms turning in a nameless void."

Tuesday, November 1, 2022

From “Goethe on All Souls’ Day”

Filed under: General — Tags: — m759 @ 3:04 am

The above title is that of a Log24 post on St. Cecilia's Day in 2017
that quoted some earlier All Souls' Day remarks from Berlin.

From that post —

Exercise:  Explain why the lead article in the November issue of
Notices of the American Mathematical Society  misquotes Weyl
and gives the misleading impression that the example above,
the eightfold cube ,  might be part of the mathematical pursuit
known as geometric group theory .

    Background:  Earlier instances here  of the phrase "geometric group theory." 

Saturday, September 3, 2022

1984 Revisited

Filed under: General — m759 @ 2:46 pm

Cube Bricks 1984 —

An Approach to Symmetric Generation of the Simple Group of Order 168

Related material

Note the three quadruplets of parallel edges  in the 1984 figure above.

Further Reading

The above Gates article appeared earlier, in the June 2010 issue of
Physics World , with bigger illustrations. For instance —

Exercise: Describe, without seeing the rest of the article,
the rule used for connecting the balls above.

Wikipedia offers a much clearer picture of a (non-adinkra) tesseract —

      And then, more simply, there is the Galois tesseract

For parts of my own  world in June 2010, see this journal for that month.

The above Galois tesseract appears there as follows:

Image-- The Dream of the Expanded Field

See also the Klein correspondence in a paper from 1968
in yesterday's 2:54 PM ET post

Monday, August 1, 2022

Enowning

Filed under: General — Tags: — m759 @ 3:26 pm

Related material — The Eightfold Cube.

See also . . .

"… 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."

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

Tuesday, April 12, 2022

Dealing with Cubism

Filed under: General — Tags: , — m759 @ 10:45 am

"It’s important, as art historian Reinhard Spieler has noted,
that after a brief, unproductive stay in Paris, circa 1907,
Kandinsky chose to paint in Munich. That’s where he formed
the Expressionist art group Der Blaue Reiter  (The Blue Rider) —
and where he avoided having to deal with cubism."

— David Carrier, 

Images from an earlier Christmas Day, in 2005 —

The Eightfold Cube

The image “http://www.log24.com/theory/images/EightfoldWayCover.jpg” cannot be displayed, because it contains errors.

Saturday, February 12, 2022

Das Geheimnis der Einheit

Filed under: General — Tags: , , — m759 @ 11:13 pm

Thomas Mann on "the mystery of the unity"

Mann on Schopenhauer: Psychoanalysis and 'The Will'

"Denn um zu wiederholen, was ich anfangs sagte:
in dem Geheimnis der Einheit von Ich und Welt,
Sein und Geschehen, in der Durchschauung des
scheinbar Objectiven und Akzidentellen als
Veranstaltung der Seele glaube ich den innersten Kern
der analytischen Lehre zu erkennen." (GW IX 488)

An Einheit-Geheimnis  that is perhaps* more closely related
to pure mathematics** —

"What is the nature of the original unity
that throws itself apart in this separation,
and in what sense are the separated ones
here as the essence of the abyss? 

Here it cannot be a question of any kind of 'dialectic,' 
but only of the essence of the ground
(that is, of truth) itself." [Tr. by Google]

" Welcher Art ist die ursprüngliche Einheit,
daß sie sich in diese Scheidung auseinanderwirft,
und in welchem Sinn sind die Geschiedenen
hier als Wesung der Ab-gründigkeit gerade einig?
Hier kann es sich nicht um irgend eine »Dialektik«
handeln, sondern nur um die Wesung des Grundes
(der Wahrheit also) selbst."

Heidegger 

* Or perhaps not .

** For a relevant Scheidung , see Eightfold Cube.

Friday, February 11, 2022

For Space Groupies

Filed under: General — Tags: , — m759 @ 5:31 pm

A followup to Wednesday's post Deep Space

Related material from this journal on July 9, 2019

Cube Bricks 1984 —

An Approach to Symmetric Generation of the Simple Group of Order 168

From "Tomorrowland" (2015) —

From other posts tagged 1984 Cubes

Friday, December 31, 2021

Aesthetics in Academia

Filed under: General — Tags: — m759 @ 9:33 am

Related art — The non-Rubik 3x3x3 cube —

The above structure illustrates the affine space of three dimensions
over the three-element finite (i.e., Galois) field, GF(3). Enthusiasts
of Judith Brown's nihilistic philosophy may note the "radiance" of the
13 axes of symmetry within the "central, structuring" subcube.

I prefer the radiance  (in the sense of Aquinas) of the central, structuring 
eightfold cube at the center of the affine space of six dimensions over
the two-element field GF(2).

Thursday, December 30, 2021

Antidote to Chaos?

Filed under: General — Tags: , , , — m759 @ 3:57 pm

Some formal symmetry —

"… 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 ."

— Steven H. Cullinane on April 19, 2016 — The Folding.

Related art-historical remarks:

The Shape of Time  (Kubler, Yale U.P., 1962).

See yesterday's post The Thing 

Tuesday, October 5, 2021

The Tidier

Filed under: General — m759 @ 2:00 pm


 

"A Little Tidier" —
 

Cover of 'The Eightfold Way: The Beauty of Klein's Quartic Curve' Versus The Eightfold Cube: The Beauty of Klein's Simple Group

Wednesday, September 15, 2021

The Razor and the Touchstone

Filed under: General — m759 @ 10:28 pm

It is often good to remember that writers of headlines (and subheadlines)
are usually not the same people as the authors of the following texts.

In particular, in the above example, neither the word "touchstone" nor
the use of "enquires" to mean "enquiries" appears in the text proper.

Still, the mixed metaphor of "razor" as "touchstone" is not without interest.

See The Eightfold Cube and Modernist Cuts.

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"

Two years ago on August 3 . . .

The Eightfold Cube

What is going on in this picture?

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

IMAGE- The Trinity Cube (three interpenetrating planes that split the eightfold cube into its eight subcubes)

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:

The Eightfold Cube: The Beauty of Klein's Simple Group
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

Cover of 'The Eightfold Way: The Beauty of Klein's Quartic Curve'

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.

The arXiv versions

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

See also "Sprechen Sie Neutsch?".
 

Update of December 29, 2022 —

 

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 —

IMAGE- The Trinity Cube (three interpenetrating planes that split the eightfold cube into its eight subcubes)

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 

Order-8 group generated by reflections in midplanes of cube parallel to faces

… and in some Galois spaces —

Generating permutations for the Klein simple group of order 168 acting on the eightfold cube .

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
special about this room other than the coincidence that
its previous occupant was Freeman Dyson.”

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

See also other posts now tagged Trinity Staircase.

Illuminati enthusiasts  may enjoy the following image:

'Ex Fano Apollinis'- Fano plane, eightfold cube, and the two combined.

Saturday, February 29, 2020

Template

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

Roberta Smith on Donald Judd’s
ARTnews Writings:
‘A Great Template for Criticism’ 

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 —

'Ex Fano Apollinis'- Fano plane, eightfold cube, and the two combined.

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 —

IMAGE- Eightfold cube with detail of triskelion structure

Other geometry for enthusiasts of the esoteric —

Monday, November 4, 2019

As Above, So Below*

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

Braucht´s noch Text?

       — Deutsche Schule Montevideo

* An "established rule of law
across occult writings.
"

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." 

Mondrian, 1936  [Links added.]

An image search today (click to enlarge) —

Image search for 'Eightfold Cube'

Wednesday, February 19, 2020

Aitchison’s Octads

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 —

Aitchison's  octads are instead related to symmetries of the cube.

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 .

The Eightfold Cube: The Beauty of Klein's Simple Group

   Image from Christmas Day 2005.

http://www.math.sci.hiroshima-u.ac.jp/branched/files/2018/
presentations/Aitchison-Hiroshima-2-2018.pdf
.
See also Aitchison in this journal.

 
 

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.

Structure of the eightfold cube

 

Sunday, January 26, 2020

Harmonic-Analysis Building Blocks

See also The Eightfold Cube.

Monday, January 20, 2020

Dyadic Harmonic Analysis:

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

Structure of the eightfold cube

Saturday, December 14, 2019

Colorful Tale

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

(Continued)

Four-color correspondence in an eightfold array (eightfold cube unfolded)

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
Miracle Octad Generator Structure.

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
" —

http://www.log24.com/log/pix11B/110808-DwarfsParade500w.jpg

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.

Froebel's Third Gift: A cube made up of eight subcubes  

Cube Bricks 1984 —

An Approach to Symmetric Generation of the Simple Group of Order 168

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 —

An Approach to Symmetric Generation of the Simple Group of Order 168

From "Tomorrowland" (2015) —

From John Baez (2018) —

See also this morning's post Perception of Space 
and yesterday's Exploring Schoolgirl Space.

Thursday, June 20, 2019

The Lively Hallows

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

Structure of the eightfold cube

Sunday, May 26, 2019

Nine-Dot Patterns

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

Some nine-dot patterns of greater interest:

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

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).


Restoring the severed link —

Structure of the eightfold cube

See also Espacement  and The Thing and I.
 

Related material —

 
 

Monday, May 6, 2019

One Stuff

Building blocks?

From a post of May 4

Structure of the eightfold cube

See also Espacement  and The Thing and I.

Monday, March 25, 2019

Espacement

(Continued from the previous post.)

In-Between "Spacing" and the "Chôra "
in Derrida: A Pre-Originary Medium?

By Louise Burchill

(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

See also, from the reported publication date of the above book
Intermedialities , the Log24 post Synchronicity.

Saturday, March 16, 2019

Grundlagen

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

See also eightfold cube.

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.

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

From https://blogs.scientificamerican.com/…

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."

Kurt Gödel as quoted by Gian-Carlo Rota

Some results from a Google search —

Eidetic reduction | philosophy | Britannica.com

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 …

Phenomenology Online » Eidetic Reduction

The eidetic reduction: eidos. Method: Bracket all incidental meaning and ask: what are some of the possible invariate aspects of this experience? The research …

Eidetic reduction – New World Encyclopedia

Sep 19, 2017 – Eidetic reduction is a technique in Husserlian phenomenology, used to identify the essential components of the given phenomenon or experience.

Terminology: Eidos

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:

http://www.log24.com/log/pix18/180830-Harry_Potter_Phil_Stone-wand-movements-quote.jpg

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

http://www.log24.com/log/pix18/180830-Harry_Potter-Phil_Stone-Bloomsbury-2004-p168.jpg

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

http://www.log24.com/log/pix18/180831-NYer-back-cover-ad-Lifespan_of_a_Fact.jpg

"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

http://www.log24.com/log/pix18/180830-Sandback-perception-of-space-500w.jpg

http://www.log24.com/log/pix18/180830-Harry_Potter_Phil_Stone-wand-movements-quote.jpg

http://www.log24.com/log/pix18/180830-Harry_Potter-Phil_Stone-Bloomsbury-2004-p168.jpg

* 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 —

http://www.log24.com/log/pix18/180825-Ballard-on-Modernism.gif

Related material —

Waugh + Orwell in this journal and

Cube Bricks 1984

An Approach to Symmetric Generation of the Simple Group of Order 168

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.

See also the materialization of 281 as a page number in
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

See also interality in the eightfold cube.

IMAGE- The Trinity Cube (three interpenetrating planes that split the eightfold cube into its eight subcubes)

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 Dr/ Yau’s own website

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

James Blish, 'Black Easter'

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

Steiner quadruple system in eightfold cube

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

See also, from an April 2013 philosophical conference:

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 —

Box containing Froebel's Third Gift-- The Eightfold Cube

Click image for some background.

See also Nicht Spielerei .

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 .

An Approach to Symmetric Generation of the Simple Group of Order 168

Some small Galois spaces (the Cullinane models)

Sunday, April 1, 2018

Logos

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

The Eightfold Cube

Quantum logo

Business logo

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) —
 

  

The Fano plane and the eightfold cube

Tuesday, March 27, 2018

Compare and Contrast

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

Weyl on symmetry, the eightfold cube, the Fano plane, and trigrams of the I Ching

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 

The Eightfold Cube.
 

Weyl on symmetry, the eightfold cube, the Fano plane, and trigrams of the I Ching


"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)

See also Interality  and the Eightfold Cube .

Friday, January 5, 2018

Seven Types of Interality*

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

'Paradise of CHildhood'— on Froebel's Third Gift

* 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

http://literaturwissenschaft-berlin.de/interview-with-david-wellbery/

as later republished in 

https://thepointmag.com/2017/dialogue/
irreducible-significance-david-wellbery-literature-goethe-cavell
 —

 

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

Weyl on symmetry, the eightfold cube, the Fano plane, and trigrams of the I Ching

Sunday, October 29, 2017

File System… Unlocked

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

Logo from the above webpage

See also the similar structure of  the eightfold cube,  and

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

The Eightfold Cube

"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 —

An Approach to Symmetric Generation of the Simple Group of Order 168

Another mathematical remark from 1984 —

For further details, see Triangles Are Square.

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:

Geometric Group theory at MSRI (pronounced 'Misery')

See also some writings of Gromov from 2015-16:

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 —

An Approach to Symmetric Generation of the Simple Group of Order 168

Related material — Posts now tagged Device Narratives.

Tuesday, May 2, 2017

Image Albums

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

Pinterest boards uploaded to the new m759.net/piwigo

Diamond Theorem 

Diamond Theorem Correlation

Miracle Octad Generator

The Eightfold Cube

Six-Set Geometry

Diamond Theory Cover

Update of May 2 —

Four-Color Decomposition

Binary Galois Spaces

The Galois Tesseract

Update of May 3 —

Desargues via Galois

The Tetrahedral Model

Solomon's Cube

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:

Is there a geometric realization of the Quaternion group?” —

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: 

The Eightfold Cube and The Quantum Identity

See also the previous post.

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."

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 —

The Eightfold Cube, core structure of the I Ching

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

See also the eightfold cube

The Eightfold Cube

 

Sunday, November 27, 2016

A Machine That Will Fit

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

Or:  Notes for the Metaphysical Club

Northrop Frye on Wallace Stevens:

"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:

See also the eightfold cube.

Thursday, November 3, 2016

Triple Cross

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

Cube Bricks 1984 —

An Approach to Symmetric Generation of the Simple Group of Order 168
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.

The Eightfold Cube

Platonic
solid

Jack in the Box, Natasha Wescoat, 2004
Natasha Wescoat, 2004

Shakespearean
Fool

Not to mention Euclid and Picasso.

 

The image “http://www.log24.com/theory/images/Pythagoras-I47.gif” cannot be displayed, because it contains errors.


The image “http://www.log24.com/log/pix06A/RobertFooteAnimation.gif” cannot be displayed, because it contains errors.

In the above pictures, Euclid is represented by 
Alexander Bogomolny, Picasso by Robert Foote.

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.

The Eightfold Cube

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.

See also the eightfold cube.

Thursday, September 15, 2016

Metaphysics at Notre Dame

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

Recommended reading —

"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:

See also the eightfold cube in the previous post,
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.

Older Posts »

Powered by WordPress