Log24

Monday, September 13, 2021

Cube Space Revisited

Filed under: General — Tags: , , , , — m759 @ 3:02 PM

The above Quanta  article mentions

"Maryna Viazovska’s 2016 discovery of the most efficient
ways of packing spheres in dimensions eight and 24."

From a course to be taught by Viazovska next spring:

The Lovasz reference suggests a review of my own webpage
Cube Space, 1984-2003.

See as well a review of Log24 posts on Packing.

Saturday, August 28, 2021

Solomon’s Super*Cube

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

Geometry for Jews  continues.
 

210828-Golomb-2x2x2-Super_Cube.jpg (500×373)

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 .

Sunday, February 21, 2021

Cube Woo

Filed under: General — Tags: — m759 @ 7:01 PM

“Before time began, there was the Cube.”
— Hassenfeld Brothers merchandising slogan

Saturday, September 19, 2020

Cube School

Filed under: General — Tags: — m759 @ 3:16 AM

The new domain http://cube.school
points to posts tagged Cube School here.

Sunday, July 5, 2020

The Enigma Cube

Filed under: General — Tags: , — m759 @ 5:03 AM

Promotional material —

“Did you buckle up?” —  Harlan Kane

The publication date of The Enigma Cube  reported above was February 13, 2020.

Related material — Log24 posts around that date now tagged The Reality Bond.

Friday, February 28, 2020

Up the Trinity Staircase

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

Or:  The Newman Prize  Continues.

Freeman Dyson reportedly died today.  In memoriam ,
some remarks by Dyson from Hiroshima Day 1979 —

(Click to enlarge.)

Monday, February 24, 2020

For “Time Cube” Fans

Filed under: General — Tags: , — m759 @ 3:53 AM

See also Time Cube elsewhere in this  journal.

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.

Friday, June 21, 2019

Cube Tales for Solstice Day

Filed under: General — Tags: , , — m759 @ 3:45 PM

See also "Six-Set" in this journal
and "Cube Geometry Continues."

 
 

Cubehenge

Filed under: General — Tags: — m759 @ 1:15 AM

Tuesday, May 21, 2019

Inside the White Cube

Filed under: General — Tags: , , , — m759 @ 11:01 AM

(Continued)

Monday, May 13, 2019

Star Cube

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

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

Saturday, May 4, 2019

Inside the White Cube

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

Structure of the eightfold cube

See also Espacement  and The Thing and I.

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 . . .

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.

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.

Tuesday, September 25, 2018

Trinity

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

See some posts related to three names
associated with Trinity College, Cambridge —

Atiyah + Shaw + Eddington .

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

Thursday, June 28, 2018

Trinity Meditation

Filed under: General — Tags: — m759 @ 2:45 AM

See Interpenetration and Trinity Cube.

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)

Thursday, March 22, 2018

The Diamond Cube

Filed under: General,Geometry — Tags: , — m759 @ 11:32 AM

The Java applets at the webpage "Diamonds and Whirls"
that illustrate Cullinane cubes may be difficult to display.

Here instead is an animated GIF that shows the basic unit
for the "design cube" pages at finitegeometry.org.

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

Sunday, June 4, 2017

In Memory of the Time Cube Page*

Filed under: General,Geometry — Tags: , — m759 @ 10:00 PM

From this journal on August 18, 2015, "A Wrinkle in Terms" —

For two misuses by John Baez of the phrase “permutation group”
at the n-Category Café, see “A Wrinkle in the Mathematical Universe
and “Re: A Wrinkle…” —

“There is  such a thing as a permutation group.”
— Adapted from A Wrinkle in Time , by Madeleine L’Engle

* See RIP, Time Cube at gizmodo.com (September 1, 2015).

Tuesday, April 4, 2017

White Cube

Filed under: General — Tags: , , — m759 @ 12:21 PM

Inside the White Cube” —

“We have now reached
a point where we see
not the art but the space first….
An image comes to mind
of a white, ideal space
that, more than any single picture,
may be the archetypal image
of 20th-century art.”

http://www.log24.com/log/pix09/090205-cube2x2x2.gif

“Space: what you
damn well have to see.”

— James Joyce, Ulysses  

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.

Tuesday, April 5, 2016

“Puzzle Cube of a Novel”

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

"To know the mind of the creator"

Or that of Orson Welles

Related material — Cube Coloring.

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.)

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, March 4, 2016

Cube Bricks 1984

Filed under: General,Geometry — Tags: , — m759 @ 12:06 PM

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

Related aesthetics —

"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

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.

Sunday, December 28, 2014

Cube of Ultron

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

The Blacklist “Pilot” Review

"There is an element of camp to this series though. Spader is
quite gleefully channeling Anthony Hopkins, complete with being
a well educated, elegant man locked away in a super-cell.
Speaking of that super-cell, it’s kind of ridiculous. They’ve got him
locked up in an abandoned post office warehouse on a little
platform with a chair inside  a giant metal cube that looks like
it could have been built by Tony Stark. And as Liz approaches
to talk to him, the entire front of the cube  opens and the whole
thing slides back to leave just the platform and chair. Really? 
FUCKING REALLY ? "

Kate Reilly at Geekenstein.com (Sept. 27, 2013)

Monday, May 19, 2014

Cube Space

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

A sequel to this afternoon’s Rubik Quote:

“The Cube was born in 1974 as a teaching tool
to help me and my students better understand
space and 3D. The Cube challenged us to find
order in chaos.”

— Professor Ernő Rubik at Chrome Cube Lab

IMAGE- Weyl on symmetry

(Click image below to enlarge.)

Thursday, January 24, 2013

Cube Space

Filed under: General — Tags: , , — m759 @ 12:24 PM

For the late Cardinal Glemp of Poland,
who died yesterday, some links:

Friday, December 28, 2012

Cube Koan

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

From Don DeLillo's novel Point Omega —

I knew what he was, or what he was supposed to be, a defense intellectual, without the usual credentials, and when I used the term it made him tense his jaw with a proud longing for the early weeks and months, before he began to understand that he was occupying an empty seat. "There were times when no map existed to match the reality we were trying to create."

"What reality?"

"This is something we do with every eyeblink. Human perception is a saga of created reality. But we were devising entities beyond the agreed-upon limits of recognition or interpretation. Lying is necessary. The state has to lie. There is no lie in war or in preparation for war that can't be defended. We went beyond this. We tried to create new realities overnight, careful sets of words that resemble advertising slogans in memorability and repeatability. These were words that would yield pictures eventually and then become three-dimensional. The reality stands, it walks, it squats. Except when it doesn't."

He didn't smoke but his voice had a sandlike texture, maybe just raspy with age, sometimes slipping inward, becoming nearly inaudible. We sat for some time. He was slouched in the middle of the sofa, looking off toward some point in a high corner of the room. He had scotch and water in a coffee mug secured to his midsection. Finally he said, "Haiku."

I nodded thoughtfully, idiotically, a slow series of gestures meant to indicate that I understood completely.

"Haiku means nothing beyond what it is. A pond in summer, a leaf in the wind. It's human consciousness located in nature. It's the answer to everything in a set number of lines, a prescribed syllable count. I wanted a haiku war," he said. "I wanted a war in three lines. This was not a matter of force levels or logistics. What I wanted was a set of ideas linked to transient things. This is the soul of haiku. Bare everything to plain sight. See what's there. Things in war are transient. See what's there and then be prepared to watch it disappear."

What's there—

This view of a die's faces 3, 6, and 5, in counter-
clockwise order (see previous post) suggests a way
of labeling the eight corners  of a die (or cube):

123, 135, 142, 154, 246, 263, 365, 456.

Here opposite faces of the die sum to 7, and the
three faces meeting at each corner are listed
in counter-clockwise order. (This corresponds
to a labeling of one of MacMahon's* 30 colored cubes.)
A similar vertex-labeling may be used in describing 
the automorphisms of the order-8 quaternion group.

For a more literary approach to quaternions, see
Pynchon's novel Against the Day .

* From Peter J. Cameron's weblog:

  "The big name associated with this is Major MacMahon,
   an associate of Hardy, Littlewood and Ramanujan,
   of whom Robert Kanigel said,

His expertise lay in combinatorics, a sort of
glorified dice-throwing, and in it he had made
contributions original enough to be named
a Fellow of the Royal Society.

   Glorified dice-throwing, indeed…"

Sunday, August 5, 2012

Cube Partitions

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

The second Logos  figure in the previous post
summarized affine group actions on partitions
that generate a group of about 1.3 trillion
permutations of a 4x4x4 cube (shown below)—

IMAGE by Cullinane- 'Solomon's Cube' with 64 identical, but variously oriented, subcubes, and six partitions of these 64 subcubes

Click for further details.

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.

Sunday, February 5, 2012

Cuber

Filed under: General,Geometry — Tags: , — m759 @ 5:15 PM

(Continued from January 11, 2012)

Wednesday, January 11, 2012

Cuber

“Examples galore of this feeling must have arisen in the minds of the people who extended the Magic Cube concept to other polyhedra, other dimensions, other ways of slicing.  And once you have made or acquired a new ‘cube’… you will want to know how to export a known algorithm , broken up into its fundamental operators , from a familiar cube.  What is the essence of each operator?  One senses a deep invariant lying somehow ‘down underneath’ it all, something that one can’t quite verbalize but that one recognizes so clearly and unmistakably in each new example, even though that example might violate some feature one had thought necessary up to that very moment.  In fact, sometimes that violation is what makes you sure you’re seeing the same thing , because it reveals slippabilities you hadn’t sensed up till that time….

… example: There is clearly only one sensible 4 × 4 × 4 Magic Cube.  It is the  answer; it simply has the right spirit .”

— Douglas R. Hofstadter, 1985, Metamagical Themas: Questing for the Essence of Mind and Pattern  (Kindle edition, locations 11557-11572)

See also Many Dimensions in this journal and Solomon’s Cube.

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.”

Saturday, August 27, 2011

Cosmic Cube*

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

IMAGE- Anthony Hopkins exorcises a Rubik cube

Prequel (Click to enlarge)

IMAGE- Galois vs. Rubik: Posters for Abel Prize, Oslo, 2008

Background —

IMAGE- 'Group Theory' Wikipedia article with Rubik's cube as main illustration and argument by a cuber for the image's use

See also Rubik in this journal.

* For the title, see Groups Acting.

Friday, June 24, 2011

The Cube

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

IMAGE- 'The Stars My Destination' (with cover slightly changed)

Click the above image for some background.

Related material:
Skateboard legend Andy Kessler,
this morning's The Gleaming,
and But Sometimes I Hit London.

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

Filed under: General,Geometry — Tags: , , — m759 @ 11:30 AM

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.

Thursday, October 22, 2009

Chinese Cubes

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

From the Bulletin of the American Mathematical Society, Jan. 26, 2005:

What is known about unit cubes
by Chuanming Zong, Peking University

Abstract: Unit cubes, from any point of view, are among the simplest and the most important objects in n-dimensional Euclidean space. In fact, as one will see from this survey, they are not simple at all….

From Log24, now:

What is known about the 4×4×4 cube
by Steven H. Cullinane, unaffiliated

Abstract: The 4×4×4 cube, from one point of view, is among the simplest and the most important objects in n-dimensional binary space. In fact, as one will see from the links below, it is not simple at all.

Solomon’s Cube

The Klein Correspondence, Penrose Space-Time, and a Finite Model

Non-Euclidean Blocks

Geometry of the I Ching

Related material:

Monday’s entry Just Say NO and a poem by Stevens,

The Well Dressed Man with a Beard.”

Sunday, August 6, 2017

Dark Tower Theology

Filed under: General — m759 @ 9:00 PM

In memory of a TV gunslinger who reportedly died Thursday, August 3, 2017 . . .

From this journal on that day (posts now tagged Dark Tower Theology) —

"The concept under review is that of the Holy Trinity.
  See also, in this  journal, Cube Trinity.
  For a simpler Trinity model, see the three-point line  "

"Would that it were so simple."

Thursday, August 3, 2017

Poetic Theology at the New York Times

Filed under: General — Tags: — m759 @ 12:19 PM

Or:  Trinity Test Site

From the New York Times Book Review  of
next Sunday, August 6, 2017 —

"In a more conventional narrative sequence,
even a sequence of poems,
this interpenetration would acquire
sequence and evolution." [Link added.]

The concept under review is that of the Holy Trinity.

See also, in this  journal, Cube Trinity.

For a simpler Trinity model, see the three-point line  

Friday, March 11, 2016

Spacey

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

"You know that in space you can move in three ways…."

See also Cube Trinity and Many Dimensions.

Thursday, January 8, 2015

ABC Verlag, Zurich

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

IMAGE- Dust jacket, 'Conceptions of International Exhibitions,' by Hans Neuburg, ABC Verlag, Zurich, 1969

"The motive for metaphor, shrinking from
The weight of primary noon,
The A B C of being…." — Wallace Stevens

See also Cube Trinity in this journal.

Saturday, February 25, 2012

The Rock

Filed under: General — Tags: — m759 @ 9:26 PM

(Continued. See previous post and Red and Gray in this journal.)

“Give faith a fighting chance.” —Country song

From a post of June 3, 2007—

Related illustration relevant to theology—

http://www.log24.com/log/pix11A/110625-CubeHypostases.gif

For some background, see Cube Trinity in this journal.

For greater depth, see Levering’s Scripture and Metaphysics:
Aquinas and the Renewal of Trinitarian Theology 
,
Blackwell, 2004, page 150.

Sunday, August 28, 2011

The Cosmic Part

Filed under: General,Geometry — Tags: , , , — m759 @ 6:29 PM

Yesterday’s midday post, borrowing a phrase from the theology of Marvel Comics,
offered Rubik’s mechanical contrivance as a rather absurd “Cosmic Cube.”

A simpler candidate for the “Cube” part of that phrase:

http://www.log24.com/log/pix10/100214-Cube2x2x2.gif

The Eightfold Cube

As noted elsewhere, a simple reflection group* of order 168 acts naturally on this structure.

“Because of their truly fundamental role in mathematics,
even the simplest diagrams concerning finite reflection groups
(or finite mirror systems, or root systems—
the languages are equivalent) have interpretations
of cosmological proportions.”

Alexandre V. Borovik in “Coxeter Theory: The Cognitive Aspects

Borovik has a such a diagram—

http://www.log24.com/log/pix11B/110828-BorovikM.jpg

The planes in Borovik’s figure are those separating the parts of the eightfold cube above.

In Coxeter theory, these are Euclidean hyperplanes. In the eightfold cube, they represent three of seven projective points that are permuted by the above group of order 168.

In light of Borovik’s remarks, the eightfold cube might serve to illustrate the “Cosmic” part of the Marvel Comics phrase.

For some related theological remarks, see Cube Trinity in this journal.

Happy St. Augustine’s Day.

* I.e., one generated by reflections : group actions that fix a hyperplane pointwise. In the eightfold cube, viewed as a vector space of 3 dimensions over the 2-element Galois field, these hyperplanes are certain sets of four subcubes.

Monday, September 13, 2021

“What a Swell Party This Is” — Cole Porter

Filed under: General — Tags: — m759 @ 4:59 PM

Using the Dreidel*

Filed under: General — Tags: , — m759 @ 3:53 PM

The date of the Rubber Ducky article in the previous post was . . .

November 11, 2019.

Synchronology check:

* A phrase by Woody Allen (NY Times , May 5, 2011).

Duck Sup

Filed under: General — Tags: , — m759 @ 3:27 PM

Tuesday, August 31, 2021

Annals of Geometry

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

A passage by Max Jammer quoted in yesterday's post
A Brief Introduction to Ideas suggests further remarks:

There are geometries in which lengths are not invariant 
because they are not  relevant — for instance, projective 
geometry,  finite  geometry, and of course finite projective 
geometry.

See the annus mirabilis  introduction to that subject 
cited by Jammer in yesterday's Brief Introduction —

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 — 

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.”

Monday, March 29, 2021

Graduate School

Filed under: General — Tags: , — m759 @ 5:37 PM

Also on 18 November 2010 —

Sunday, March 21, 2021

Mathematics and Narrative: The Unity

Filed under: General — Tags: , — m759 @ 10:25 PM

“To conquer, three boxes* have to synchronize and join together into the Unity.”

―Wonder Woman in Zack Snyder’s Justice League

See also The Unity of Combinatorics  and The Miracle Octad Generator.

* Cf.  Aitchison’s Octads

Mind the Gaps…

Filed under: General — Tags: , , — m759 @ 1:18 PM

Continues from March 17.

See as well some remarks on Chinese  perspective
in the Log24 post “Gate” of June 13, 2013.

Wednesday, March 17, 2021

Mind the Gaps

Filed under: General — Tags: , , — m759 @ 12:23 PM

Katherine Neville's 'The Eight,' edition with knight on cover, on her April 4 birthday

Page from 'The Paradise of Childhood,' 1906 edition

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.

Saturday, February 27, 2021

The Pencil Case

Filed under: General — Tags: , , — m759 @ 11:55 PM

Clue

Here is  a midrash on “desmic,” a term derived from the Greek desmé
( δέσμη: bundle, sheaf , or, in the mathematical sense, pencil —
French faisceau ), which is related to the term desmos , bond …

(The term “desmic,” as noted earlier, is relevant to the structure of
Heidegger’s Sternwürfel .)

“Gadzooks, I’ve done it again!” — Sherlock Hemlock

Thursday, February 25, 2021

Compare and Contrast

Filed under: General — Tags: — m759 @ 12:31 PM

“… What is your dream—your ideal?  What is your News from Nowhere,
or, rather, What is the result of the little shake your hand has given to
the old pasteboard toy with a dozen bits of colored glass for contents?
And, most important of all, can you present it in a narrative or romance
which will enable me to pass an idle hour not disagreeably? How, for instance,
does it compare in this respect with other prophetic books on the shelf?”

— Hudson, W. H.. A Crystal Age  (p. 2). Open Road Media. Kindle Edition.

See as well . . .

The lexicographic Golay code
contains, embedded within it,
the Miracle Octad Generator.

Wednesday, February 24, 2021

Annals of Dim Antiquity

Filed under: General — Tags: , — m759 @ 2:20 PM

“Twenty-four glyphs, each one representing not a letter, not a word,
but a concept, arranged into four groups, written in Boris’s own hand,
an artifact that seemed to have resurrected him from the dead. It was
as if he were sitting across from Bourne now, in the dim antiquity of
the museum library.

This was what Bourne was staring at now, written on the unfolded
bit of onionskin.”

The Bourne Enigma , published on June 21, 2016

Passing, on June 21, 2016, into a higher dimension —

For those who prefer Borges to Bourne —

Monday, February 15, 2021

Raiders of the Lost Building Blocks

Filed under: General — Tags: — m759 @ 11:22 AM

In memory of a Dead Sea Scrolls scholar who
reportedly died on December 29, 2020, here are
links to two Log24 posts from that date:
I Ching  Geometry  and Raiders of the Lost Coordinates.

Tuesday, December 29, 2020

I Ching  Geometry

Filed under: General — Tags: — m759 @ 11:04 AM

“Before time began, there was the Cube.”
Hassenfeld Brothers cinematic merchandising slogan

Sunday, December 27, 2020

Knight Move for Trevanian

Filed under: General — Tags: — m759 @ 11:42 PM

Knight move” remark from The Eiger Sanction

“I like to put people on myself by skipping logical steps
in the conversation until they’re dizzy.”

The following logical step — a check of the date Nov. 18, 2017
was omitted in the post Futon Dream  on this year’s St. Stephen’s Day.

For further context, see James Propp in this journal.

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

Monday, November 16, 2020

The Garland Modelling

Filed under: General — Tags: , — m759 @ 11:37 AM

See also God’s Dice  and  Dead Poet Joke .

Wednesday, September 23, 2020

Geometry of Even Subsets

Filed under: General — Tags: , — m759 @ 12:06 AM

Various posts here on the geometry underlying the Mathieu group M24
are now tagged with the phrase “Geometry of Even Subsets.”

For example, a post with this diagram . . .

Monday, September 21, 2020

Zelig-Like?

Filed under: General — Tags: , — m759 @ 10:46 PM

“On their way to obscurity, the Simulmatics people
played minor parts in major events, appearing Zelig-like
at crucial moments of 1960s history.”

James Gleick reviewing a new book by Jill Lepore

Saturday, September 19, 2020

The Summerfield Prize

Filed under: General — Tags: — m759 @ 3:01 PM

“Like Coleridge” . . .

Related material:  Bloomsday 2006.

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 .

Thursday, September 10, 2020

Raiders of . . .

Filed under: General — Tags: , — m759 @ 11:34 PM

Wednesday, September 9, 2020

Portrait with Holocron

Filed under: General — Tags: , , — m759 @ 11:08 PM

Novus Ordo Seclorum — Harold Bloom and the Tetrahedral Model of PG(3,2)

Sith Holocron in 'Star Wars Rebels'

For a Jedi  holocron of sorts, see this  journal on the above YouTube date

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?“.

Thursday, July 9, 2020

The Enigma Glyphs

Filed under: General — Tags: , , — m759 @ 5:53 AM

IMAGE- The Diamond Theorem

For those who  prefer fiction —

“Twenty-four glyphs, each one representing not a letter, not a word,
but a concept, arranged into four groups, written in Boris’s own hand,
an artifact that seemed to have resurrected him from the dead. It was
as if he were sitting across from Bourne now, in the dim antiquity of
the museum library.

This was what Bourne was staring at now, written on the unfolded
bit of onionskin.”

— “Robert Ludlum’s”  The Bourne Enigma , published on June 21, 2016

Passing, on June 21, 2016, into a higher dimension —

Sunday, July 5, 2020

It’s Still the Same Old Story …

Filed under: General — Tags: , , — m759 @ 4:29 PM

“He recounted the story of Adam and Eve, who were banished
from paradise because of their curiosity. Their inability to resist
the temptation of the forbidden fruit. Which itself was a metaphorical
stand-in for knowledge and power. He urged us to find the restraint
needed to resist the temptation of the cube—the biblical apple
in modern garb. He urged us to remain in Eden until we were able
to work out the knowledge the apple offered, all by ourselves.”

— Richards, Douglas E.. The Enigma Cube  (Alien Artifact Book 1)
(pp. 160-161). Paragon Press, 2020. Kindle Edition.

The biblical apple also appears in the game, and film, Assassin’s Creed .

Related material —

See the cartoon version of Alfred North Whitehead in the previous post,
and some Whitehead-related projective geometry —

Enigma Variations

Filed under: General — Tags: , — m759 @ 2:24 PM

The previous post reported, perhaps inaccurately, a publication
date of February 13, 2020, for the novel The Enigma Cube .

A variant publication date, Jan. 21,  2020, is reported below.

This journal on that  date —

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 —

Friday, May 22, 2020

Surface Code News

Filed under: General — Tags: , — m759 @ 5:50 PM

From a paper cited in the above story:

“Fig. 4   A lattice geometry for a surface code.” —

The above figure suggests a search for “surface code” cube :

Related poetic remarks — “Illumination of a surface.”

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.

Friday, April 17, 2020

A Mechanism of Fission

Filed under: General — Tags: , — m759 @ 3:01 PM

The above title was suggested by the previous post, Explosive Remarks.

'On Froebel's Third Gift,' from 'Paradise of Childhood,' 1906

Explosive Remarks

Filed under: General — Tags: — m759 @ 2:15 PM

Here is the background to Wheeler’s explosive remarks.

John Archibald Wheeler, director of the Center
for Theoretical Physics at the University of Texas,
is one of the world’s top theoretical physicists.
In 1939 he and Niels Bohr published a paper on
‘The Mechanism of Nuclear Fission”
that laid the groundwork for atomic and hydrogen bombs.
Wheeler later played major roles in their development.”

For a rather different explosion of Wheeler’s views, see the previous post.

Introduction to Quantum Woo

Filed under: General — Tags: — m759 @ 1:07 AM

“In his big book, Gravity  [sic ], Wheeler puts our space
into what he calls superspace, and speculates on the
most basic physical laws which operate on superspace.
He comes to the (to me) surprising conclusion that the
rock-bottom laws are the laws of the propositional calculus!”

— Martin Gardner, letter to Donald E. Knuth, 8 January 1976,
on cover of Notices of the American Mathematical Society ,
March 2011 issue.

Fact check —

Related reading —

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.

Tuesday, March 17, 2020

Geometric Theology

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

“Before time began” — Optimus Prime

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

See also posts tagged Aitchison.

 

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."

Pythagorean Letter Meets Box of Chocolates

Filed under: General — Tags: , , , — m759 @ 10:30 AM

Friday, July 11, 2014

Spiegel-Spiel des Gevierts

Filed under: Uncategorized — m759 @ 12:00 PM

See Cube Symbology.

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

Da hats ein Eck 

Wednesday, March 4, 2020

Architect’s Elegy

Filed under: General — Tags: , — m759 @ 2:31 PM

On Boston's Hancock Tower:

"I reflect that all art, all beauty, is reflection."

Fictional character by John Updike (July 1976)

The architect of the tower reportedly died Monday.

See as well "Reflections: Disturbing the Universe I"
by the late Freeman Dyson in The New Yorker
issue dated August 6, 1979.

A reflection I prefer:

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

Potter’s Staircase

Filed under: General — Tags: — m759 @ 11:48 PM

See as well "Up the Trinity Staircase" (yesterday afternoon)
and "British Pottery" (Log24 , December 22, 2018).

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'

Friday, February 21, 2020

To and Fro, Back and …

Filed under: General — Tags: , , , — m759 @ 11:44 PM

Also on January 27, 2017 . . .

For other appearances of John Hurt here,
see 1984 Cubes.

Update of 12:45 AM Feb. 22 —

A check of later obituaries reveals that Hurt may well
have died on January 25, 2017, not January 27 as above.

Thus the following remarks may be more appropriate:

Not to mention what, why, who, and how.

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, November 11, 2019

Time and Chance

Filed under: General — Tags: — m759 @ 2:49 PM

http://www.log24.com/log/pix10B/101202-DreidelAndStone.jpg

The misleading image at right above is from the cover of
an edition of Charles Williams's classic 1931 novel 
Many Dimensions  published in 1993 by Wm. B. Eerdmans.

Compare and constrast —

Goedel Escher Bach cover

Cover of a book by Douglas Hofstadter

IMAGE- 'Solomon's Cube'

An Invariance of Symmetry

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.

Stage Direction: “Comments Off.”

Filed under: General — Tags: , , — m759 @ 11:29 AM

The previous post dealt with “magic” cubes, so called because of the
analogous “magic” squares. Douglas Hofstadter has written about a
different, physical , object, promoted as “the  Magic Cube,” that Hofstadter
felt embodied “a deep invariant”:

Monday, September 9, 2019

ART WARS at Harvard: The Wertham Professorship

Filed under: General — Tags: — m759 @ 8:38 PM

See as well an obituary for Mrs. Wertham from 1987.

Related art —

Friday, July 11, 2014

Spiegel-Spiel des Gevierts

Filed under: Uncategorized — m759 @ 12:00 PM 

See Cube Symbology.

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

Da hats ein Eck 

For further details, search the Web for "Wertham Professor" + Eck.

Monday, July 22, 2019

The Four Diamonds Meet the Five Red Herrings

Filed under: General — Tags: — m759 @ 11:34 AM

Lord Peter Wimsey (Balliol 1912) on the Balliol-Trinity rivalry at Oxford:

See also Balliol College in the post subtitled Spidey Goes to Church.

Signed, Sealed, Delivered

Filed under: General — Tags: — m759 @ 5:58 AM

Visual Languages

Filed under: General — Tags: — m759 @ 4:48 AM

Good question. See also this  journal on the above date —

September 15, 2018.

Space, Time, Form

Filed under: General — Tags: — m759 @ 3:33 AM

Magic cube and corresponding hexagram, or Star of David, with faces mapped to lines and edges mapped to points

Click the image for some remarks on a related novel.

Wednesday, July 10, 2019

Artifice* of Eternity …

Filed under: General — Tags: , , , — m759 @ 10:54 AM

… and Schoolgirl Space

"This poem contrasts the prosaic and sensual world of the here and now
with the transcendent and timeless world of beauty in art, and the first line,
'That is no country for old men,' refers to an artless world of impermanence
and sensual pleasure."

— "Yeats' 'Sailing to Byzantium' and McCarthy's No Country for Old Men :
Art and Artifice in the New Novel,"
Steven Frye in The Cormac McCarthy Journal ,
Vol. 5, No. 1 (Spring 2005), pp. 14-20.

See also Schoolgirl Space in this  journal.

* See, for instance, Lewis Hyde on the word "artifice" and . . .

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.

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