Log24

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 — m759 @ 2:00 AM

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

Sunday, December 22, 2019

M24 from the Eightfold Cube

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

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

Click image below to download a Guitart PowerPoint presentation.

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

Friday, May 24, 2019

Hidden Eightfold Patterns

Filed under: General — Tags: — m759 @ 5:23 PM

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 — 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

Filed under: General,Geometry — Tags: — m759 @ 9:10 PM

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, 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.

Tuesday, January 10, 2017

Eightfold Epiphany

Filed under: General — m759 @ 1:00 PM

The reported death today at 105 of an admirable war correspondent,
"a perennial fixture at the Foreign Correspondents’ Club in Hong Kong," 
suggested a search in this journal for that city.

The search recalled to mind a notable quotation from
a Montreal philosopher —

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

Charles Taylor, "Epiphanies of Modernism,"
Chapter 24 of Sources of the Self
(Cambridge U. Press, 1989, p. 477)

For some context, see St. Lucia's Day, 2012.
See also Epiphany 2017 —

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:

Wednesday, November 30, 2016

Eightfold Roman

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

"Frye's largely imaginary eightfold roman 
may have provided him a personal substitute—
or alternative— for both ideology and myth."

— P. 63 of James C. Nohrnberg, "The Master of
the Myth of Literature: An Interpenetrative Ogdoad
for Northrop Frye," Comparative Literature  Vol. 53,
No. 1 (Winter, 2001), pp. 58-82

See also today's earlier post In Nuce .

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 — 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.

Thursday, May 19, 2011

The Aleph, the Lottery, and the Eightfold Way

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

Three links with a Borges flavor—

Related material

The 236 in yesterday evening's NY lottery may be
 viewed as the 236 in March 18's Defining Configurations.
For some background, see Configurations and Squares.

A new illustration for that topic—

http://www.log24.com/log/pix11A/110519-8-3-Configuration.jpg

This shows a reconcilation of the triples described by Sloane
 in Defining Configurations with the square geometric
arrangement described by Coxeter in the Aleph link above.

Note that  the 56 from yesterday's midday NY lottery
describes the triples that appear both in the Eightfold Way
link above and also in a possible source for
the eight triples of  Sloane's 83 configuration—

http://www.log24.com/log/pix11A/110519-SloaneDesign.jpg

The geometric square arrangement discussed in the Aleph link
above appears in a different, but still rather Borgesian, context
in yesterday morning's Minimalist Icon.

Wednesday, April 28, 2010

Eightfold Geometry

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

Image-- The 35 partitions of an 8-set into two 4-sets

Image-- Analysis of structure of the 35 partitions of an 8-set into two 4-sets

Image-- Miracle Octad Generator of R.T. Curtis

Related web pages:

Miracle Octad Generator,
Generating the Octad Generator,
Geometry of the 4×4 Square

Related folklore:

"It is commonly known that there is a bijection between the 35 unordered triples of a 7-set [i.e., the 35 partitions of an 8-set into two 4-sets] and the 35 lines of PG(3,2) such that lines intersect if and only if the corresponding triples have exactly one element in common." –"Generalized Polygons and Semipartial Geometries," by F. De Clerck, J. A. Thas, and H. Van Maldeghem, April 1996 minicourse, example 5 on page 6

The Miracle Octad Generator may be regarded as illustrating the folklore.

Update of August 20, 2010–

For facts rather than folklore about the above bijection, see The Moore Correspondence.

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.

Sunday, May 17, 2020

“The Ultimate Epistemological Fact”

Filed under: General — 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 — 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.

 

Sunday, February 2, 2020

Tetrads for McLuhan, or “Blame It on Video”

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

"I like to put people on myself by skipping logical steps
in the conversation until they're dizzy." — Jemima Brown
in The Eiger Sanction

Related posts — See "McLuhan Tetrad" in this journal.

Related theology — See  "The Meaning of Perichoresis."
Background — The New Yorker , "On Religion:
Richard Rohr Reorders the Universe," by Eliza Griswold
on February 2, 2020, and a different reordering in posts
tagged Eightfold Metaphysics.

Sunday, January 26, 2020

Harmonic-Analysis Building Blocks

Filed under: General — Tags: — m759 @ 1:14 PM

See also The Eightfold Cube.

Monday, January 20, 2020

Dyadic Harmonic Analysis:

Filed under: General — Tags: — m759 @ 8:26 PM

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

Wednesday, January 1, 2020

Le Mot Juste

Filed under: General — Tags: — m759 @ 3:23 PM

Related art

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

Wednesday, December 11, 2019

Miracle Octad Generator Structure

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

Miracle Octad Generator — Analysis of Structure

(Adapted from Eightfold Geometry, a note of April 28, 2010.
  See also the recent post Geometry of 6 and 8.)

Sunday, December 8, 2019

Geometry of 6 and 8

Filed under: General — Tags: , — m759 @ 4:03 AM

Just as
the finite space PG(3,2) is
the geometry of the 6-set, so is
the finite space PG(5,2)
the geometry of the 8-set.*

Selah.

* Consider, for the 6-set, the 32
(16, modulo complementation)
0-, 2-, 4-, and 6-subsets,
and, for the 8-set, the 128
(64, modulo complementation)
0-, 2-, 4-, 6-, and 8-subsets.

Update of 11:02 AM ET the same day:

See also Eightfold Geometry, a note from 2010.

Sunday, October 27, 2019

Friday Night Lights

Filed under: General — m759 @ 2:05 PM

Entertainment from NBC on Friday night —

The above question, and Saturday morning's post on a film director
from Melbourne, suggest an image from December's Melbourne Noir

 (March 8, 2018, was the date of death for Melbourne author Peter Temple.)

Saturday, October 26, 2019

Director’s Cut

Filed under: General — m759 @ 6:14 AM

The title was suggested by the previous post and by
the title illustration in the weblog of the director,
Leigh Whannell, of the 2018 film "Upgrade."

Related visual details —

For the Church of Synchronology

Related remarks:  "The Thing and I."

Wednesday, October 9, 2019

Philosophical Infanticide

Filed under: General — Tags: — m759 @ 5:51 AM

From Wallace Stevens —

"Reality is the beginning not the end,
Naked Alpha, not the hierophant Omega,
Of dense investiture, with luminous vassals."

— “An Ordinary Evening in New Haven” VI

From The Point  magazine yesterday, October 8, 2019
Parricide:  On Irad Kimhi's Thinking and Being .
Book review by Steven Methven.

The conclusion:

"Parricide is nothing that the philosopher need fear . . . .
What sustains can be no threat. Perhaps what the
unique genesis of this extraordinary work suggests is that
the true threat to philosophy is infanticide."

This remark suggests revisiting a post from Monday

Monday, October 7, 2019

Berlekamp Garden vs. Kinder Garten

Filed under: General — 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.

. . . .

Monday, October 7, 2019

Berlekamp Garden vs. Kinder Garten

Filed under: General — 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

Tuesday, October 1, 2019

Hijacking the Vatican

Filed under: General — Tags: , — m759 @ 12:42 PM

Rothstein's 'Emblems of Mind,' 1995, cover illustrations by Pinturicchio from Vatican

Cover illustration— Arithmetic and Music,
Borgia Apartments, the Vatican.

See also Rothstein in this journal.

Related posts: The Eightfold Hijacking.

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.

Thursday, August 8, 2019

The Embedding

Filed under: General — m759 @ 1:33 PM

(Continued from Nov. 28, 2010)

Saturday, August 3, 2019

Butterfield’s Eight

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

What is going on in this picture?

Tuesday, July 16, 2019

Morf Vandewalt, Social Prism

Filed under: General — Tags: , — m759 @ 8:38 AM

From the 2019 film "Velvet Buzzsaw" —

What is going on in this picture?
 

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

Tuesday, June 18, 2019

Paris Review

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

"The loveliness of Paris seems somehow sadly gay." — Song lyric

Stewart also starred in "Equals" (2016). From a synopsis —

"Stewart plays Nia, a writer who works at a company that extols
the virtues of space exploration in a post-apocalyptic society.
She falls in love with the film's main character, Silas (Nicholas Hoult),
an illustrator . . . ."

Space art in The Paris Review

For a different sort of space exploration, see Eightfold 1984.

Thursday, June 13, 2019

The Reality Blocks

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

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

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

See Gell-Mann in this journal and May 24, 2003.

Seeing the Seing

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

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

" 'experimental metaphysics' is a term coined by Abner Shimony …."

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

Sunday, May 26, 2019

Nine-Dot Patterns

Filed under: General — 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

Burning Bright

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

Gell-Mann's 'eightfold way' as 'a mosaic of simple triangular building blocks' — George Johnson, 1999

Compare and contrast with . . .

The Brightburn Logo:

Related material from the May 12 post

"The Collective Unconscious
in a Cartoon Graveyard
" —

"When they all finally reach their destination —
a deserted field in the Florida Panhandle…." 

" When asked about the film's similarities to the 2015 Disney movie Tomorrowland , which also posits a futuristic world that exists in an alternative dimension, Nichols sighed. 'I was a little bummed, I guess,' he said of when he first learned about the project. . . . 'Our die was cast. Sometimes this kind of collective unconscious that we're all dabbling in, sometimes you're not the first one out of the gate.' "

Sunday, May 19, 2019

The Building Blocks of Geometry

Filed under: General — Tags: , — m759 @ 10:21 PM

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

Filed under: General — Tags: , , , — m759 @ 1:17 PM

Building blocks?

From a post of May 4

Structure of the eightfold cube

See also Espacement  and The Thing and I.

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.

Thursday, April 18, 2019

Expanding the Unfolding*

Filed under: General — m759 @ 8:06 PM

From a New York Times  book review of a new novel about
Timothy Leary that was in the Times online on April 10 —

"Most of the novel resides in the perspective
of Fitzhugh Loney, one of Leary’s graduate students."

"A version of this article appears in print on ,
on Page 10 of the Sunday Book Review with the headline
Strange Days." 

For material about one of Leary's non -fictional grad students,
Ralph Metzner, see posts now tagged Metzner's Pi Day.

Related material —

The reported publication date of Searching for the Philosophers' Stone
was January 1, 2019.  

A related search published here  on that date:

* Title suggested by two of Ralph Metzner's titles,
   The Expansion of Consciousness  and The Unfolding Self .

Monday, April 8, 2019

Misère Play

Filed under: General — Tags: , — m759 @ 5:21 PM

Facebook on Bloomsday 2017 —

Also on that Bloomsday —

Chalkroom Jungle Revisited —

Sunday, April 7, 2019

Chess King

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

Meanwhile . . .

Front page top center, online NY Times: Bobby Fischer Dead at 64

Wednesday, April 3, 2019

Nocturnal Object of Beauty

Filed under: General — Tags: — m759 @ 2:41 AM

http://www.log24.com/log/pix11B/110712-ObjectOfBeauty.jpg
 

 

What is going on in this picture?
 

Monday, March 25, 2019

Espacement

Filed under: General — Tags: , , , , — m759 @ 1:46 PM

(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 — m759 @ 4:53 PM

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

Tuesday, January 1, 2019

Child’s Play Continues — La Despedida

Filed under: General — m759 @ 1:03 PM

This post was suggested by the phrase "Froebel Decade" from
the search results below.

This journal a decade ago had a post on the late Donald Westlake,
an author who reportedly died of a heart attack in Mexico on Dec. 31,
2008, while on his way to a New Year's Eve dinner.

One of Westlake's books —

Related material —

"La Despedida " and "Finality indeed, and cleavage!"

Sunday, December 9, 2018

Quaternions in a Small Space

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

The previous post, on the 3×3 square in ancient China,
suggests a review of group actions on that square
that include the quaternion group.

Click to enlarge

Three links from the above finitegeometry.org webpage on the
quaternion group —

Related material —

Iain Aitchison on the 'symmetric generation' of R. T. Curtis

See as well the two Log24 posts of December 1st, 2018 —

Character and In Memoriam.

Thursday, December 6, 2018

The Mathieu Cube of Iain Aitchison

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

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.

Tuesday, December 4, 2018

Melbourne Noir

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

 March 8, 2018, was the date of death for Melbourne author Peter Temple.

Sunday, December 2, 2018

Symmetry at Hiroshima

Filed under: G-Notes,General,Geometry — Tags: , — m759 @ 6:43 AM

A search this morning for articles mentioning the Miracle Octad Generator
of R. T. Curtis within the last year yielded an abstract for two talks given
at Hiroshima on March 8 and 9, 2018

http://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 mathematical objects have played a special role, underpinning new mathematics as understanding deepened. Perhaps archetypal are the Platonic polyhedra, subsequently related to Platonic idealism, and the contentious notion of existence of mathematical reality independent of human consciousness.

Exceptional or unique objects are often associated with symmetry – manifest or hidden. In topology and geometry, we have natural base points for the moduli spaces of closed genus 2 and 3 surfaces (arising from the 2-fold branched cover of the sphere over the 6 vertices of the octahedron, and Klein’s quartic curve, respectively), and Bring’s genus 4 curve arises in Klein’s description of the solution of polynomial equations of degree greater than 4, as well as in the construction of the Horrocks-Mumford bundle. Poincare’s homology 3-sphere, and Kummer’s surface in real dimension 4 also play special roles.

In other areas: we have the exceptional Lie algebras such as E8; the sporadic finite simple groups; the division algebras: Golay’s binary and ternary codes; the Steiner triple systems S(5,6,12) and S(5,8,24); the Leech lattice; the outer automorphisms of the symmetric group S6; the triality map in dimension 8; and so on. We also note such as: the 27 lines on a cubic, the 28 bitangents of a quartic curve, the 120 tritangents of a sextic curve, and so on, related to Galois’ exceptional finite groups PSL2(p) (for p= 5,7,11), and various other so-called `Arnol’d Trinities’.

Motivated originally by the `Eightfold Way’ sculpture at MSRI in Berkeley, we discuss inter-relationships between a selection of these objects, illustrating connections arising via highly symmetric Riemann surface patterns. These are constructed starting with a labeled polygon and an involution on its label set.

Necessarily, in two lectures, we will neither delve deeply into, nor describe in full, contexts within which exceptional objects arise. We will, however, give sufficient definition and detail to illustrate essential inter-connectedness of those exceptional objects considered.

Our starting point will be simplistic, arising from ancient Greek ideas underlying atomism, and Plato’s concepts of space. There will be some overlap with a previous talk on this material, but we will illustrate with some different examples, and from a different philosophical perspective.

Some new results arising from this work will also be given, such as an alternative graphic-illustrated MOG (Miracle Octad Generator) for the Steiner system S(5,8,24), and an alternative to Singerman – Jones’ genus 70 Riemann surface previously proposed as a completion of an Arnol’d Trinity. Our alternative candidate also completes a Trinity whose two other elements are Thurston’s highly symmetric 6- and 8-component links, the latter related by Thurston to Klein’s quartic curve.

See also yesterday morning’s post, “Character.”

Update: For a followup, see the next  Log24 post.

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 .

Wednesday, November 28, 2018

Geometry and Experience

Filed under: G-Notes,General,Geometry — 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 — 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, November 3, 2018

Foundations of Geometry

Filed under: G-Notes,General,Geometry — m759 @ 1:40 PM

"costruire (o, dirò meglio immaginare) un ente" — Fano, 1892

"o, dirò meglio, costruire" — Cullinane, 2018

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, September 14, 2018

Warburg at Cornell, Continued

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

Warburg at Cornell U. Press

See also Epiphany 2017 —

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 — 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 — 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 — 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.

Tuesday, July 17, 2018

Deep Learning for Jews

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

From The New York Times  on June 20, 2018 —

" In a widely read article published early this year on arXiv.org,
a site for scientific papers, Gary Marcus, a professor at
New York University, posed the question:

'Is deep learning approaching a wall?'

He wrote, 'As is so often the case, the patterns extracted
by deep learning are more superficial than they initially appear.' "

See as well an image from posts tagged Quantum Suffering  . . .

The time above, 10:06:48 PM July 16, is when  I saw

"What you mean 'we,' Milbank?"

Wednesday, July 11, 2018

Clarity and Precision

Filed under: General,Geometry — m759 @ 9:13 AM

"The whole meaning of the word is
looking into something with clarity and precision,
seeing each component as distinct,
and piercing all the way through
so as to perceive the most fundamental reality
of that thing."

For the word itself, try a Web search on 
noteworthy phrases above.

“. . . the utterly real thing in writing is 
the only thing that counts . . . ."

— Maxwell Perkins to Ernest Hemingway, Aug. 30, 1935

"168"

— Page number in a 2016 Scribner edition
of Stephen King's IT

Sunday, July 1, 2018

Deutsche Ordnung

Filed under: General,Geometry — Tags: , , — m759 @ 8:22 PM

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 — 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.

Thursday, June 7, 2018

For Dan Brown

Filed under: General,Geometry — Tags: , — m759 @ 1:09 PM

See also Eightfold Trinity in this  journal.

Symbologist Robert Langdon views a corner of Solomon's Cube

Wednesday, June 6, 2018

Geometry for Goyim

Filed under: General,Geometry — 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 — 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

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, 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

* 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

Saturday, November 18, 2017

Cube Space Continued

Filed under: General,Geometry — 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).

Sunday, October 29, 2017

File System… Unlocked

Filed under: General,Geometry — 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 — m759 @ 8:18 PM
 

Typographical: » 

Eightfold Cube:

 

Saturday, October 7, 2017

Byte Space

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

The Eightfold Cube

"Before time began,
there was the Cube."

Optimus Prime

Wednesday, September 13, 2017

Summer of 1984

Filed under: General,Geometry — Tags: , , — m759 @ 9:11 AM

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.

Tuesday, August 8, 2017

Cube Quaternions

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

See posts now tagged with the above title.

IMAGE- Quaternion group acting on an eightfold cube

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

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

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.

Wednesday, June 7, 2017

Three Things at Once

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

Rosalind Krauss in 1979

Nanavira Thera in 1959

Cambridge University Press in 1999 —

See also Cube Bricks.

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

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

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 — m759 @ 1:16 PM

From Wikipedia's Iceberg Theory

Related material: 

The Eightfold Cube and The Quantum Identity

See also the previous post.

Saturday, March 4, 2017

At 74

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

New York Times  headline about a death
on Friday, March 3, 2017 —

René Préval, President of Haiti
in 2010 Quake, Dies at 74

See also

This way to the egress.

Saturday, January 14, 2017

1984: A Space Odyssey

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

See Eightfold 1984 in this journal.

Related material —

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

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

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

"Perhaps every science must start with metaphor
and end with algebra; and perhaps without the metaphor
there would never have been any algebra."

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

Epiphany 2017 —

Click to enlarge:

Monday, January 9, 2017

Analogical Extension at Cornell

Filed under: General,Geometry — 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

 

Tuesday, December 27, 2016

Bright Star

Filed under: General,Geometry — m759 @ 8:15 AM

See instances of the title in this journal.

Material related to yesterday evening's post
"Bright and Dark at Christmas" —

The Buddha of Rochester:

See also the Gelman (i.e., Gell-Mann) Prize
in the film "Dark Matter" and the word "Eightfold"
in this journal.

" A fanciful mark is a mark which is invented
for the sole purpose of functioning as a trademark,
e.g., 'Kodak.' "

"… don't take my Kodachrome away." — Paul Simon

Wednesday, November 30, 2016

In Nuce

Filed under: General,Geometry — Tags: — m759 @ 3:48 AM
 

Excerpts from James C. Nohrnberg, "The Master of the Myth of Literature: An Interpenetrative Ogdoad for Northrop Frye," Comparative Literature  Vol. 53, No. 1 (Winter, 2001), pp. 58-82

From page 58 —
"… the posthumously revealed Notebooks. A major project of the latter was his 'Ogdoad': two groups of four books each. '[T]he second group of four […] were considered to be Blakean "emanations" or counterparts of the first four,' like 'the "double mirror" structure of The Great Code  and Words with Power : two inter-reflecting parts of four chapters apiece,' Michael Dolzani reports.* "

* P. 22 of Rereading Frye: The Published and Unpublished Works , ed. David Boyd and Imre Salusinszky, Frye Studies [series] (Toronto: University of Toronto Press, 1998). [Abbreviated as RF .]


From page 62 —
"Visionaries like Blake and dramatists like Wagner seem to be working from some larger, mythic blueprint present in nuce  from very early on."

From page 63 —
"Frye's hypothetical books and will-to-totality were obviously fruitful; if the beckoning star was illusory, it nonetheless settled on a real birthplace. The sought-for constructs substituted their scaffolding for a backbone-like confidence in pre-given beliefs; possession of the latter is why Tories like Dr. Johnson and T.S. Eliot could do quite nicely without the constructs. Frye's largely imaginary eightfold roman  may have provided him a personal substitute— or alternative— for both ideology and myth."

From page 69 —
"For Frye the chief element of imaginative or expressive form is the myth, which functions structurally in literature like geometric shapes in painting."

From page 71 —
"The metaphysical skyhook lifting the artist free from unreflective social commitment is often a latent or manifest archetype that his work renews or reworks."

From page 77 —
"Frye's treatises— so little annotated themselves— are the notes writ large; the notes in the Notebooks are treatises writ small. They interpenetrate. Denham quotes 'the masters of the T'ien-tai school of Mahayana Buddhism' as saying '[t]he whole world is contained in a mustard seed' (RF  158, 160), and Frye quotes Keats: 'Every point of thought is the center of an intellectual world' (Study  159; cf. Great Code  167-68 and AC  61). …. [Frye’s] complex books were all generated out of the monadic obiter dicta . His kingdom 'is like a grain of mustard seed, which a man took, and cast into his garden, and it grew' (Luke 13:18-19)."

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

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

(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 — 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

The Seven Seals

Filed under: General,Geometry — Tags: — m759 @ 7:23 AM

From Hermann Weyl's 1952 classic Symmetry —

"Galois' ideas, which for several decades remained
a book with seven seals  but later exerted a more
and more profound influence upon the whole
development of mathematics, are contained in
a farewell letter written to a friend on the eve of
his death, which he met in a silly duel at the age of
twenty-one. This letter, if judged by the novelty and
profundity of ideas it contains, is perhaps the most
substantial piece of writing in the whole literature
of mankind."

Some Galois geometry —

See the previous post for more narrative.

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 — 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 — 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.)

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