Monday, January 1, 2018

Diamond Theory 1976

Filed under: Geometry — m759 @ 8:26 PM

The first 12 pages of my 1976 preprint "Diamond Theory" are 
now scanned and uploaded.  See a slideshow.

For downloading, all 12 pages are combined in a PDF.

Saturday, December 23, 2017

The Right Stuff

Filed under: Geometry — Tags: — m759 @ 1:12 PM

A figure related to the general connecting theorem  of Koen Thas —

Anticommuting Dirac matrices as spreads of projective lines

Ron Shaw on the 15 lines of the classical generalized quadrangle W(2), a general linear complex in PG(3,2)

See also posts tagged Dirac and Geometry in this  journal.

Those who prefer narrative to mathematics may, if they so fancy, call
the above Thas connecting theorem a "quantum tesseract theorem ."

Friday, December 22, 2017


Filed under: Geometry — Tags: — m759 @ 4:08 PM

Movie marquee on Camazotz, from the 2003 film of 'A Wrinkle in Time'

From a Log24 post of October 10, 2017

Koen Thas, 'Unextendible Mututally Unbiased Bases' (Sept. 2016)

Related material from May 25, 2016 —

Thursday, December 21, 2017

For Winter Solstice 2017

Filed under: Geometry — m759 @ 10:30 AM

A review —

Some context —

Webpage demonstrating symmetries of 'Solomon's Cube'

Wednesday, December 20, 2017

A Snow Ball for Clifford Irving (1930-2017)

Filed under: Geometry — m759 @ 11:45 PM

William Grimes in The New York Times  this evening —

"Clifford Irving, who perpetrated one of the biggest literary hoaxes
of the 20th century in the early 1970s when he concocted a
supposedly authorized autobiography of the billionaire Howard Hughes
based on meetings and interviews that never took place, died on Tuesday
at a hospice facility near his home in Sarasota, Fla. He was 87."

A figure reproduced here on Tuesday

A related figure —

See too the 1973 Orson Welles film "F for Fake."

Some background on the second figure above —
posts tagged April 8-11, 2016.

Some background on the first figure above —
today's previous post, January 2018 AMS Notices.

Devil’s Claws, Etc., Etc.

Filed under: Geometry — m759 @ 1:04 PM

"And Cabots speak only to God."

Tuesday, December 19, 2017

A Mythologem for Meletinsky

Filed under: Geometry — m759 @ 11:00 AM

The word "mythologem" on page 55 of The Burning Fountain 
by Philip Wheelwright, revised edition of 1968 (p. 91 in the 1954
edition), suggests a Web search for that word. It was notably often
used in the 1998 English translation of a book by Eleazar Meletinsky
first published in Russian in 1976 —

Meletinsky reportedly died on December 17, 2005.

In his memory, Log24 posts from that date are now tagged Mythologem Day.

"And we may see the meadow in December,
icy white and crystalline" — Johnny Mercer

Monday, December 18, 2017

Mathematics and Art

Filed under: Geometry — m759 @ 5:09 PM

From the American Mathematical Society homepage today —

From concinnitasproject.org

"Concinnitas  is the title of a portfolio of fine art prints. . . .
The portfolio draws its name from a word famously used
by the Renaissance scholar, artist, architect, and philosopher
Leon Battista Alberti (1404-1472) to connote the balance of
number, outline, and position (in essence, number, geometry,
and topology) that he believed characterize a beautiful work of art."

The favicon of the Concinnitas Project —

The structure of the Concinnitas favicon —

This structure is from page 15 of
"Diamond Theory," a 1976 preprint —


Wheelwright and the Dance

Filed under: Geometry — m759 @ 1:00 PM

The page preceding that of yesterday's post  Wheelwright and the Wheel —

See also a Log24 search for 
"Four Quartets" + "Four Elements".

A graphic approach to this concept:

"The Bounded Space" —

'Space Cross' from the Cullinane diamond theorem

"The Fire, Air, Earth, and Water" —

Logo for 'Elements of Finite Geometry'

Saturday, December 16, 2017

Dagger Definitions (Review)

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

The previous post suggests a review of
the philosophical concept of universals —

A part of the above-mentioned 2011 "Saturday evening's post" that is
relevant to the illustration at the end of today's previous post —


Note the whatness of Singer's  dagger definitions —


Filed under: Geometry — Tags: — m759 @ 12:24 AM

Two readings by James Parker —

From next year's first Atlantic  issue

From last month's Atlantic  issue

"Let’s return to that hillside where Clayton exited his Mercedes.
In the gray light, he climbs the pasture. Halfway up the slope,
three horses are standing: sculpturally still, casually composed
in a perfect triptych of horsitude."

James Parker in The Atlantic , Nov. 2017 issue

Logos-related material 

Monday, December 11, 2017

The Diamond Theorem at SASTRA

Filed under: Geometry — m759 @ 12:35 PM

The following IEEE paper is behind a paywall,
but the first page is now available for free
at deepdyve.com

For further details on the diamond theorem, see
finitegeometry.org/sc/ or the archived version at . . .


Sunday, December 10, 2017


Filed under: Geometry — Tags: , — m759 @ 8:45 PM

Google search result for Plato + Statesman + interlacing + interweaving

See also Symplectic in this journal.

From Gotay and Isenberg, “The Symplectization of Science,”
Gazette des Mathématiciens  54, 59-79 (1992):

“… what is the origin of the unusual name ‘symplectic’? ….
Its mathematical usage is due to Hermann Weyl who,
in an effort to avoid a certain semantic confusion, renamed
the then obscure ‘line complex group’ the ‘symplectic group.’
… the adjective ‘symplectic’ means ‘plaited together’ or ‘woven.’
This is wonderfully apt….”

IMAGE- A symplectic structure -- i.e. a structure that is symplectic (meaning plaited or woven)

The above symplectic  figure appears in remarks on
the diamond-theorem correlation in the webpage
Rosenhain and Göpel Tetrads in PG(3,2). See also
related remarks on the notion of  linear  (or line ) complex
in the finite projective space PG(3,2) —

Anticommuting Dirac matrices as spreads of projective lines

Ron Shaw on the 15 lines of the classical generalized quadrangle W(2), a general linear complex in PG(3,2)

Friday, December 8, 2017

Mythos and Logos

Filed under: Geometry — m759 @ 9:48 PM

Part I:  Black Magician

"Schools of criticism create their own canons, elevating certain texts,
discarding others. Yet some works – Malcolm Lowry’s Under the Volcano
is one of them – lend themselves readily to all critical approaches."

— Joan Givner, review of 
A Darkness That Murmured: Essays on Malcolm Lowry and the Twentieth Century
by Frederick Asals and Paul Tiessen, eds.

The Asals-Tiessen book (U. of Toronto Press, 2000) was cited today
by Margaret Soltan (in the link below) as the source of this quotation —

"When one thinks of the general sort of snacky
under-earnest writers whose works like wind-chimes
rattle in our heads now, it is easier to forgive Lowry
his pretentious seriousness, his old-fashioned ambitions,
his Proustian plans, [his efforts] to replace the reader’s
consciousness wholly with a black magician’s."

A possible source, Perle Epstein, for the view of Lowry as black magician —

Part II:  Mythos  and Logos

Part I above suggests a review of Adam Gopnik as black magician
(a figure from Mythos ) —

Tuesday, November 7, 2017

Polarities and Correlation

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

— and of an opposing figure from Logos
     Paul B. Yale, in the references below:

Friday, December 1, 2017

The Architect and the Matrix

Filed under: Geometry — m759 @ 1:00 PM

In memory of Yale art historian Vincent Scully, who reportedly
died at 97 last night at his home in Lynchburg, Va., some remarks
from the firm of architect John Outram and from Scully —

Update from the morning of December 2 —

The above 3×3 figure is of course not unrelated to
the 4×4 figure in The Matrix for Quantum Mystics:


See as well Tsimtsum in this journal.

Harold Bloom on tsimtsum as sublimation

Thursday, November 30, 2017

The Matrix for Quantum Mystics

Filed under: Geometry — Tags: — m759 @ 10:29 PM

Scholia on the title — See Quantum + Mystic in this journal.

The Matrix of Lévi-Strauss

"In Vol. I of Structural Anthropology , p. 209, I have shown that
this analysis alone can account for the double aspect of time
representation in all mythical systems: the narrative is both
'in time' (it consists of a succession of events) and 'beyond'
(its value is permanent)." — Claude Lévi-Strauss, 1976

I prefer the earlier, better-known, remarks on time by T. S. Eliot
in Four Quartets , and the following four quartets (from
The Matrix Meets the Grid) —


From a Log24 post of June 26-27, 2017:

A work of Eddington cited in 1974 by von Franz

See also Dirac and Geometry and Kummer in this journal.

Ron Shaw on Eddington's triads "associated in conjugate pairs" —

For more about hyperbolic  and isotropic  lines in PG(3,2),
see posts tagged Diamond Theorem Correlation.

For Shaw, in memoriam — See Contrapuntal Interweaving and The Fugue.

Wednesday, November 29, 2017


Filed under: Geometry — m759 @ 11:25 PM

See also Inscape in this journal and, for a related Chapel Hill thesis,
the post Kummer and Dirac.

Monday, November 27, 2017

The Golay Code via Witt’s Construction

Filed under: Geometry — Tags: — m759 @ 2:08 PM


Hansen, Robert Peter, "Construction and Simplicity of
the Large Mathieu Groups" (2011). Master's Theses. 4053. 

See also The Matrix Meets the Grid (Log24, Nov. 24).
More generally, see SPLAG in this journal.

Friday, November 24, 2017

The Matrix Meets the Grid

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

The Matrix —

  The Grid —

  Picturing the Witt Construction

     "Read something that means something." — New Yorker  ad

Wednesday, November 22, 2017

The Prize

Filed under: Geometry — m759 @ 2:45 PM

Bernd Sturmfels to Receive 2018
George David Birkhoff Prize in Applied Mathematics

— American Mathematical Society on
     Monday, November 20th, 2017

See also Sturmfels and Birkhoff + Geometry in this  journal.

Sunday, November 19, 2017

Galois Space

Filed under: Geometry — m759 @ 8:00 PM

This is a sequel to yesterday's post Cube Space Continued.

Saturday, November 18, 2017

Cube Space Continued

Filed under: Geometry — m759 @ 4:44 AM

James Propp in the current Math Horizons  on the eightfold cube

James Propp on the eighfold cube

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

Thursday, November 16, 2017

A Line at Infinity

Filed under: Geometry — m759 @ 12:00 PM

Lost Horizon

Filed under: Geometry — m759 @ 11:29 AM

Related material —

The following image in this journal


Sunday, November 12, 2017


Filed under: Geometry — m759 @ 7:04 PM

Sunday, October 29, 2017

File System… Unlocked

Filed under: Uncategorized — m759 @ 2:16 PM 

. . . .

"Wikipedia's first true logo . . . . included a quote from the preface
of Euclid and His Modern Rivals  by Lewis Carroll . . . ." 

. . . .

Related dialogue from the new film "Unlocked" —

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

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

See as well Chloë Grace Moretz portraying  a schoolgirl problem.

Friday, November 10, 2017

Annals of Rarefied Scholarship

Filed under: Geometry — m759 @ 3:00 PM

From Cambridge Core, suggested by a reference to
that website in the previous post and by the following
bibliographic data . . .


Downloaded from https://www.cambridge.org/core
on 10 Nov 2017 at 19:06:19 

See Conwell + Princeton in this journal.

Related art —

Saturday, November 4, 2017

Seven-Cycles in an Octad

Filed under: Geometry — m759 @ 8:00 PM

Figures from a search in this journal for Springer Knight
and from the All Souls' Day post The Trojan Pony

     Binary coordinates for a 4x2 array  Chess knight formed by a Singer 7-cycle

For those who prefer pure abstraction to the quasi-figurative
1985 seven-cycle above, a different 7-cycle for M24 , from 1998 —

Compare and contrast with my own "knight" labeling
of a 4-row 2-column array (an M24 octad, in the system
of R. T. Curtis)  by the 8 points of the projective line
over GF(7),  from 2008 —

'Knight' octad labeling by the 8 points of the projective line over GF(7)

Thursday, November 2, 2017

The Trojan Pony

Filed under: Geometry — m759 @ 7:31 PM

From a search in this journal for Springer Knight

     Binary coordinates for a 4x2 array  Chess knight formed by a Singer 7-cycle

Related material from Academia —

Nash and Needleman, 'On Magic Finite Projective Space,' Dec. 4, 2014

See also Log24 posts from the above "magic" date,
December 4, 2014, now tagged The Pony Argument.

Wednesday, November 1, 2017

Many Years

Filed under: Geometry — m759 @ 9:00 PM

From Tony Phillips's American Mathematical Society column
for November 2017 —

" It is significant that the authors chose to place this announcement
of their results not in a mathematics journal but in one aimed at a
much larger scientific audience; their writing is appropriately
expository, especially in the introduction. Nature  itself ran an 
assessment of the paper in their 'News and Views' section,
October 4: 'Mathematics: A pariah finds a home,' by Terry Gannon.
Gannon sets the stage, again in terms suitable for wide consumption,
and sketches out the story. He ends 'It is always difficult to gauge
the importance of a mathematical result without the hindsight
that many years brings.
Nevertheless, Duncan et al.  have shown us
a door. Whether it is to a new closet, house or world, we cannot yet say,
but the results are certainly unexpected, and no one will think of
the pariahs in the same way again.' "

See as well Log24 on the above date — Text and Context.

Monday, October 30, 2017


Filed under: Geometry — m759 @ 10:28 PM

Plato and His Modern Rivals

Filed under: Geometry — m759 @ 1:00 PM

The previous post's Lewis Carroll cover,
modified to illustrate Plato's diamond

Book cover modified to illustrate 'Plato and His Modern Rivals'

See also "To Forge
a Head
" (Oct. 27).

Sunday, October 29, 2017


Filed under: Geometry — m759 @ 11:00 PM

The passage from Lewis Carroll's Euclid and His Modern Rivals 
in the previous post suggests two illustrations —

Click the Trudeau book for related Log24 posts.

File System… Unlocked

Filed under: 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"

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

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

Saturday, October 28, 2017

Lowell Brown at Vanity Fair

Filed under: Geometry — Tags: — m759 @ 8:18 PM

A sequel to the post  CP  is for Consolation Prize  (Sept. 3, 2016)

An image from Log24 on this date last year:

A recent comment on a discussion of CP symmetry

Just the Facts

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

The New Yorker  on the recent film "The Square"

"It’s an aesthetic that presents,
so to speak, just the facts, 
as if the facts themselves weren’t
deeply layered with living history
and crisscrossed with vectors
of divergent ideas and ideals."

— Richard Brody, Thursday, Oct. 26, 2017

For other images deeply layered  and crisscrossed ,
see Geometry of the I Ching.

Friday, October 27, 2017

To Forge a Head

Filed under: Geometry — m759 @ 12:25 PM

The title was suggested by a 2014 Vanity Fair  piece
by James Toback (Harvard '66).

"He squinted at this vision of a Qualityless world for a while,
conjured up more details, thought about it, and then squinted
some more and thought some more and then finally circled
back to where he was before.


That's the look. That sums it. Squareness. When you subtract
quality you get squareness. Absence of Quality is the essence
of squareness."

— Robert M. Pirsig, Zen and the Art of Motorcycle Maintenance

And when you add  quality?

A related Zen joke from Final Club (June 19, 2017) —


Finite Geometry at Zenodo

Filed under: Geometry — m759 @ 11:32 AM


Thursday, October 26, 2017

A Center

Filed under: Geometry — m759 @ 8:40 PM

This post was suggested by a New York Times  obituary this evening —

"Tom Mathews, Promoter of Liberal Causes and Candidates, Dies at 96."

Mathews reportedly died on October 14, 2017.

"Mr. Mathews and his business partner Roger Craver 'dreamed for years
of finding the perfect citizen-candidate,' the authors wrote, 'a man or
woman of the center-left with a feel for issues, a history of independence,
a winning television manner and, most important of all, a center — a core
of beliefs more important to him or her than getting elected.'

Dream on.

From the date of Mathews's death:

Posts now tagged A Center for Krauss

"Let no one ignorant of geometry enter"

SourceForge Finite Geometry Download

Filed under: Geometry — m759 @ 2:22 PM

Wednesday, October 25, 2017

The Palo Alto Edge

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

From Stanford — The death on October 9, 2017, of a man who
"always wanted to be at the most cutting of cutting-edge technology."

Related material from Log24 on April 26, 2017

A sketch, adapted from Girl Scouts of Palo Alto —

Click the sketch for further details.

Tuesday, October 24, 2017

Visual Insight

Filed under: Geometry — m759 @ 1:00 PM

The most recent post in the "Visual Insight" blog of the
American Mathematical Society was by John Baez on Jan. 1, 2017

A visually  related concept — See Solomon's Cube in this  journal.
Chronologically  related — Posts now tagged New Year's Day 2017.
Solomon's cube is the 4x4x4 case of the diamond theorem — 

Monday, October 23, 2017

Plan 9 Continues

Filed under: Geometry — m759 @ 9:00 PM

Click for some background

Another approach, for Dan Brown fans —

In the following passage, Brown claims that an eight-ray star
with arrowheads at the rays' ends is "the mathematical symbol for
entropy."  Brown may have first encountered this symbol at a 
questionable "Sacred Science" website.  Wikipedia discusses
some even less  respectable uses of the symbol.

Saturday, October 21, 2017

Crimson Algebra

Filed under: Geometry — m759 @ 7:23 PM

"Category theory has become the central gateway
through which to learn pure mathematics."

— David Spivak, Harvard Math Table, Oct. 24, 2017

The New Yorker , issue of October 23, 2017

See as well posts tagged Death Warmed Over.

Thursday, October 19, 2017

Design Grammar***

Filed under: Geometry — m759 @ 10:22 PM

The elementary shapes at the top of the figure below mirror
the looking-glass property  of the classical Lo Shu square.

The nine shapes at top left* and their looking-glass reflection
illustrate the looking-glass reflection relating two orthogonal
Latin squares over the three digits of modulo-three arithmetic.

Combining these two orthogonal Latin squares,** we have a
representation in base three of the numbers from 0 to 8.

Adding 1 to each of these numbers yields the Lo Shu square.

* The array at top left is from the cover of
Wonder Years:
Werkplaats Typografie 1998-2008

** A well-known construction.

*** For other instances of what might be
called "design grammar" in combinatorics,
see a slide presentation by Robin Wilson.
No reference to the work of Chomsky is

Graphic Design: Fast Forward

Filed under: Geometry — m759 @ 8:18 PM

Typographical: » 

Eightfold Cube:


Wednesday, October 18, 2017

Three Small Grids

Filed under: Geometry — m759 @ 8:48 PM

An earlier post today, now tagged "Three Small Magic Squares,"
suggests a review of a post from October 25 three years ago
that contains the following figure —

Fans of the October Revolution may enjoy a passage
by Rosalind Krauss on grids:

Dürer for St. Luke’s Day

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

Structure of the Dürer magic square 

16   3   2  13
 5  10  11   8   decreased by 1 is …
 9   6   7  12
 4  15  14   1

15   2   1  12
 4   9  10   7
 8   5   6  11
 3  14  13   0 .

Base 4 —

33  02  01  30
10  21  22  13
20  11  12  23 
03  32  31  00 .

Two-part decomposition of base-4 array
as two (non-Latin) orthogonal arrays

3 0 0 3     3 2 1 0
1 2 2 1     0 1 2 3
2 1 1 2     0 1 2 3
0 3 3 0     3 2 1 0 .

Base 2 –

1111  0010  0001  1100
0100  1001  1010  0111
1000  0101  0110  1011
0011  1110  1101  0000 .

Four-part decomposition of base-2 array
as four affine hyperplanes over GF(2) —

1001  1001  1100  1010
0110  1001  0011  0101
1001  0110  0011  0101
0110  0110  1100  1010 .

— Steven H. Cullinane,
  October 18, 2017

See also recent related analyses of
noted 3×3 and 5×5 magic squares.

Tuesday, October 17, 2017

Plan 9 Continues

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

See also Holy Field in this journal.

Some related mathematics —

IMAGE- Herbert John Ryser, 'Combinatorial Mathematics' (1963), page 1

Analysis of the Lo Shu structure —

Structure of the 3×3 magic square:

4  9  2
3  5  7    decreased by 1 is
8  1  6

3  8  1
2  4  6
7  0  5

In base 3 —

10  22  01
02  11  20
21  00  12

As orthogonal Latin squares
(a well-known construction) —

1  2  0     0  2  1
0  1  2     2  1  0
2  0  1     1  0  2 .

— Steven H. Cullinane,
     October 17, 2017

The Movement of Analogy

Filed under: Geometry — m759 @ 4:20 PM

The title is a phrase by Octavio Paz from today's post
"Status Symbols."

Other phrases from a link target in Sunday's post 
The Strength at the Centre

                               … a single world
In which he is and as and is are one.

See also Four Dots in this journal.

Status Symbols

Filed under: Geometry — m759 @ 2:02 PM

"Status: Defunct"  

As is now its owner, who reportedly
died at 80 on Sunday, October 15, 2017.

In memoriam —

Excerpts from Log24 posts on Sunday night 
and yesterday evening


" … listen: there's a hell
of a good universe next door; let's go"

— e. e. cummings

Some literary background —

"At the point of convergence
the play of similarities and differences
cancels itself out in order that 
identity alone may shine forth
The illusion of motionlessness,
the play of mirrors of the one: 
identity is completely empty;
it is a crystallization and
in its transparent core
the movement of analogy 
begins all over once again."

— The Monkey Grammarian 

by Octavio Paz, translated by Helen Lane 

Monday, October 16, 2017

Highway 61 Revisited

Filed under: Geometry — Tags: , — m759 @ 10:13 AM

"God said to Abraham …." — Bob Dylan, "Highway 61 Revisited"

Related material — 

See as well Charles Small, Harvard '64, 
"Magic Squares over Fields" —

— and Conway-Norton-Ryba in this  journal.

Some remarks on an order-five  magic square over GF(52):

"Ultra Super Magic Square"

on the numbers 0 to 24:

22   5   18   1  14
  3  11  24   7  15
  9  17   0  13  21
10  23   6  19   2
16   4  12  20   8


42  10  33  01  24 
03  21  44  12  30 
14  32  00  23  41
20  43  11  34  02
31  04  22  40  13 

Regarding the above digits as representing
elements of the vector 2-space over GF(5)
(or the vector 1-space over GF(52)) 

All vector row sums = (0, 0)  (or 0, over GF(52)).
All vector column sums = same.

Above array as two
orthogonal Latin squares:
4 1 3 0 2     2 0 3 1 4
0 2 4 1 3     3 1 4 2 0 
1 3 0 2 4     4 2 0 3 1         
2 4 1 3 0     0 3 1 4 2
3 0 2 4 1     1 4 2 0 3

— Steven H. Cullinane,
      October 16, 2017

Sunday, October 15, 2017

Saturday Night Not-So-Live

Filed under: Geometry — m759 @ 11:59 PM

Hillel Italie at AP News —

"Richard Wilbur, the Pulitzer Prize-winning poet and translator
who intrigued and delighted generations of readers and theatergoers
through his rhyming editions of Moliere and his own verse on memory,
writing and nature, died. He was 96.

Wilbur died Saturday night [Oct. 14, 2017] in Belmont, Massachusetts,
with his family by his side, according to friend and fellow poet, Dana Gioia."

Images from the post "Center" in this journal on Saturday afternoon —


"Things fall apart; the centre cannot hold"

William Butler Yeats (1865-1939)

Saturday, October 14, 2017


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

Rosalind Krauss in 1978

"To get inside the systems of this work,
whether LeWitt's or Judd's or Morris's,
is precisely to enter
a world without a center,
a world of substitutions and transpositions
nowhere legitimated by the revelations
of a transcendental subject. This is the strength
of this work, its seriousness, and its claim to modernity." 


"The center of
the quaternion group,
Q8 = {1, −1, i, −i, j, −j, k, −k} ,
is {1, −1}."

Illustration from a post of Feb. 3,  2011


Friday, October 13, 2017

Speak, Memra

Filed under: Geometry — Tags: — m759 @ 11:59 AM

The above was suggested by a Log24 review of October 13, 2002,
which in turn suggested a Log24 search for Carousel that yielded
(from Bloomsday Lottery) —

See as well Asimov's "prime radiant," and an illustration
of the number 13 as a radiant prime

"The Prime Radiant can be adjusted to your mind,
and all corrections and additions can be made
through mental rapport. There will be nothing to
indicate that the correction or addition is yours.
In all the history of the Plan there has been no
personalization. It is rather a creation of all of us 
together. Do you understand?"  

"Yes, Speaker!"

— Isaac Asimov, 
    Second Foundation , Ch. 8: Seldon's Plan

"Before time began, there was the Cube."
— Optimus Prime

See also Transformers in this journal.

Thursday, October 12, 2017

East Meets West

Filed under: Geometry — m759 @ 8:09 PM

Wednesday, October 11, 2017

IPFS Version

Filed under: Geometry — m759 @ 9:00 PM

Tuesday, October 10, 2017

Another 35-Year Wait

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

The title refers to today's earlier post "The 35-Year Wait."

A check of my activities 35 years ago this fall, in the autumn
of 1982, yields a formula I prefer to the nonsensical, but famous,
"canonical formula" of Claude Lévi-Strauss.

The Lévi-Strauss formula

My "inscape" formula, from a note of Sept. 22, 1982 —

S = f ( f ( X ) ) .

Some mathematics from last year related to the 1982 formula —

Koen Thas, 'Unextendible Mututally Unbiased Bases' (2016)

See also Inscape in this  journal and posts tagged Dirac and Geometry.

Dueling Formulas

Filed under: Geometry — Tags: , — m759 @ 12:35 PM

Continued from the previous post and from posts
now tagged Dueling Formulas

The four-diamond formula of Jung and
the four-dot "as" of Claude Lévi-Strauss:


Simplified versions of the diamonds and the dots

The Ring of the Diamond Theorem          ::

I prefer Jung. For those who prefer Lévi-Strauss —

     First edition, Cornell University Press, 1970.

A related tale — "A Meaning, Like."

The 35-Year Wait

Filed under: Geometry — m759 @ 11:17 AM

From the Web this morning —

A different 35-year wait:

A monograph of August 1976 —

Thirty-five years later, in a post of August 2011, "Coordinated Steps" —

'The Seven Dwarfs and their Diamond Mine

"SEE HEAR READ" — Walt Disney Productions

Some other diamond-mine productions —

 Image -- The cast of 1937's 'King Solomon's Mines' goes back to the future

Monday, October 9, 2017

Still Point for a Dance

Filed under: Geometry — m759 @ 12:00 PM

"At the still point of the turning world. Neither flesh nor fleshless;
Neither from nor towards; at the still point, there the dance is,
But neither arrest nor movement. And do not call it fixity,
Where past and future are gathered. Neither movement from nor towards,
Neither ascent nor decline. Except for the point, the still point,
There would be no dance, and there is only the dance."

— T. S. Eliot, Four Quartets

See also a recurrent image
from this journal —

IMAGE- The ninefold square .

Sunday, October 8, 2017

Patterns at Oxford

Filed under: Geometry — m759 @ 7:59 PM

See also W. Tecumseh Fitch in this journal.

From the publisher (click to enlarge) —

The above publication date, 01 September 2015, suggests a review
of posts now tagged A Mirror Darkly.


Filed under: Geometry — m759 @ 12:07 PM

'Origin' (NOT by Dan Brown)


"The Cardinal seemed a little preoccupied today."

Saturday, October 7, 2017

Byte Space

Filed under: Geometry — m759 @ 7:59 PM

The Eightfold Cube

"Before time began,
there was the Cube."

Optimus Prime

Four Walls

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

From a web page quoted here on the
Feast of St. Louis, 2003 —

Case 9 of  Hekiganroku: 
Joshu's Four Gates

A monk asked Joshu,
"What is Joshu?" (Chinese: Chao Chou)

Joshu said,
"East Gate, West Gate,
 North Gate, South Gate."

Setcho's Verse:

Its intention concealed,
    the question came;
The Diamond King's eye was
    as clear as a jewel.
There stood the gates,
    north, south, east, and west,
But the heaviest hammer blow
    could not open them.

Setcho (980-1052),
Hekiganroku, 9 (Blue Cliff Records)
(translated by Katsuki Sekida,

Two Zen Classics, 1977, p. 172)

The epigraph to Lefebvre's
The Production of Space   (1974, translated in 1991) —

Octavio Paz, 'Envoi'— 'Imprisoned by four walls....'

(Adapted from a prose poem, "La Higuera ,"
in ¿Águila o Sol?  (1951).)

Broken Symmetries

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

From posts tagged Design Deadline

A quotation from Lefebvre:

"… an epoch-making event so generally ignored
that we have to be reminded of it at every moment.
The fact is that around 1910 a certain space was shattered…
the space… of classical perspective and geometry…."

— Page 25 of The Production of Space 
    (Blackwell Publishing, 1991)

This suggests, for those who prefer Harvard's past glories
to its current state, a different Raum  from the Zeit  1910.

In January 1910 Annals of Mathematics , then edited at Harvard,
published George M. Conwell's "The 3-space PG (3, 2) and Its Group."
This paper, while perhaps neither epoch-making nor shattering, has
a certain beauty. For some background, see this journal on February 24, 2009.

Wednesday, October 4, 2017

Text and Context

Filed under: Geometry — m759 @ 11:00 AM

Text —

"A field is perhaps the simplest algebraic structure we can invent."

— Hermann Weyl, 1952

Context —

See also yesterday's Personalized Book Search.

Full text of Symmetry  – Internet Archive —


A field is perhaps the simplest algebraic 143 structure
we can invent. Its elements are numbers. Characteristic
for its structure are the operations of addition and 

From a Log24 search for Mathematics+Nutshell —

IMAGE- History of Mathematics in a Nutshell

Tuesday, October 3, 2017

Show Us Your Wall

Filed under: Geometry — m759 @ 12:10 PM

From Monday morning's post Advanced Study

"Mathematical research currently relies on
a complex system of mutual trust
based on reputations."

— The late Vladimir Voevodsky,
Institute for Advanced Study, Princeton,
The Institute Letter , Summer 2014, p. 8

Related news from today's online New York Times

A heading from the above screenshot: "SHOW US YOUR WALL."

This suggests a review of a concept from Galois geometry

On the wall— A Galois-geometry 'inscape'

(On the wall — a Galois-geometry inscape .)

Personalized Book Search

Filed under: Geometry — m759 @ 10:21 AM

Click to enlarge

The quote from Hermann Weyl on which the above search is based
is from a search within this journal for Springer + Knight.

Friday, September 29, 2017

Principles Before Personalities*

Filed under: Geometry — m759 @ 12:00 PM

(Some Remarks for Science Addicts)

Principles —

IMAGE- The large Desargues configuration in light of Galois geometry

Personalities —

* See "Tradition Twelve."

Tuesday, January 1, 2013

The Simplest Situation

Filed under: Geometry — m759 @ 6:00 PM

Thanks to a Harvard math major for the following V. I. Arnold quote 
in a weblog post yesterday titled "Abstraction and Generality"—

"… the author has attempted to adhere to the principle of
minimal generality, according to which every idea should first
be clearly understood in the simplest situation;*
only then can the method developed be extended to
more complicated cases.

— Vladimir I. Arnold, Lectures on Partial Differential Equations
(Russian edition 1997; English translation 2004),
Preface to the second Russian edition

Thanks also to the math major for his closing post today.

* For instance… Natalie Angier's New Year's meditation
    on a Buddha Field

"… the multiverse as envisioned in Tibetan Buddhism,
'a vast system of 1059 [sic ; corrected to 10^59 on Jan. 3]
universes, that together are called a Buddha Field,' said
Jonathan C. Gold, who studies Buddhist philosophy at

— versus a search in this journal for "Japanese character" that yields

 Japanese character
          for "field"

Thursday, December 5, 2002

Thursday December 5, 2002

Filed under: Geometry — Tags: , — m759 @ 3:17 AM

Sacerdotal Jargon

From the website

Abstracts and Preprints in Clifford Algebra [1996, Oct 8]:

Paper:  clf-alg/good9601
From:  David M. Goodmanson
Address:  2725 68th Avenue S.E., Mercer Island, Washington 98040

Title:  A graphical representation of the Dirac Algebra

Abstract:  The elements of the Dirac algebra are represented by sixteen 4×4 gamma matrices, each pair of which either commute or anticommute. This paper demonstrates a correspondence between the gamma matrices and the complete graph on six points, a correspondence that provides a visual picture of the structure of the Dirac algebra.  The graph shows all commutation and anticommutation relations, and can be used to illustrate the structure of subalgebras and equivalence classes and the effect of similarity transformations….

Published:  Am. J. Phys. 64, 870-880 (1996)

The following is a picture of K6, the complete graph on six points.  It may be used to illustrate various concepts in finite geometry as well as the properties of Dirac matrices described above.

"The Relations between Poetry and Painting,"
by Wallace Stevens:

"The theory of poetry, that is to say, the total of the theories of poetry, often seems to become in time a mystical theology or, more simply, a mystique. The reason for this must by now be clear. The reason is the same reason why the pictures in a museum of modern art often seem to become in time a mystical aesthetic, a prodigious search of appearance, as if to find a way of saying and of establishing that all things, whether below or above appearance, are one and that it is only through reality, in which they are reflected or, it may be, joined together, that we can reach them. Under such stress, reality changes from substance to subtlety, a subtlety in which it was natural for Cézanne to say: 'I see planes bestriding each other and sometimes straight lines seem to me to fall' or 'Planes in color. . . . The colored area where shimmer the souls of the planes, in the blaze of the kindled prism, the meeting of planes in the sunlight.' The conversion of our Lumpenwelt went far beyond this. It was from the point of view of another subtlety that Klee could write: 'But he is one chosen that today comes near to the secret places where original law fosters all evolution. And what artist would not establish himself there where the organic center of all movement in time and space—which he calls the mind or heart of creation— determines every function.' Conceding that this sounds a bit like sacerdotal jargon, that is not too much to allow to those that have helped to create a new reality, a modern reality, since what has been created is nothing less."

Tuesday, November 26, 2002

Tuesday November 26, 2002

Filed under: Geometry — m759 @ 10:00 PM

Notes toward a Supreme Fact

In "Notes toward a Supreme Fiction," Wallace Stevens lists criteria for a work of the imagination:

  • It Must Be Abstract
  • It Must Change
  • It Must Give Pleasure.

For a work that seems to satisfy these criteria, see the movable images at my diamond theory website. Central to these images is the interplay of rational sides and irrational diagonals in square subimages.

"Logos and logic, crystal hypothesis,
 Incipit and a form to speak the word
 And every latent double in the word…."

— "Notes toward a Supreme Fiction," Section 1, Canto VIII

Recall that "logos" in Greek means "ratio," as well as (human or divine) "word." Thus when I read the following words of Simone Weil today, I thought of Stevens.

"The beautiful in mathematics resides in contradiction.   Incommensurability, logoi alogoi , was the first splendor in mathematics."

— Simone Weil, Oeuvres Choisies , éd. Quarto, Gallimard, 1999, p. 100



In the conclusion of Section 3, Canto X, of "Notes," Stevens says

"They will get it straight one day at the Sorbonne.
 We shall return at twilight from the lecture
 Pleased that the irrational is rational…."

This is the logoi alogoi  of Simone Weil.

Thursday, October 31, 2002

Thursday October 31, 2002

Filed under: Geometry — m759 @ 11:07 PM


From The Unknowable (1999), by Gregory J. Chaitin, who has written extensively about his constant, which he calls Omega:

"What is Omega? It's just the diamond-hard distilled and crystallized essence of mathematical truth! It's what you get when you compress tremendously the coal of redundant mathematical truth…" 

Charles H. Bennett has written about Omega as a cabalistic number.

Here is another result with religious associations which, historically, has perhaps more claim to be called the "diamond-hard essence" of mathematical truth: The demonstration in Plato's Meno that a diamond inscribed in a square has half the area of the square (or that, vice-versa, the square has twice the area of the diamond).

From Ivars Peterson's discussion of Plato's diamond and the Pythagorean theorem:

"In his textbook The History of Mathematics, Roger Cooke of the University of Vermont describes how the Babylonians might have discovered the Pythagorean theorem more than 1,000 years before Pythagoras.

Basing his account on a passage in Plato's dialogue Meno, Cooke suggests that the discovery arose when someone, either for a practical purpose or perhaps just for fun, found it necessary to construct a square twice as large as a given square…."

From "Halving a Square," a presentation of Plato's diamond by Alexander Bogomolny, the moral of the story:

SOCRATES: And if the truth about reality is always in our soul, the soul must be immortal….

From "Renaissance Metaphysics and the History of Science," at The John Dee Society website:

Galileo on Plato's diamond:

"Cassirer, drawing attention to Galileo's frequent use of the Meno, particularly the incident of the slave's solving without instruction a problem in geometry by 'natural' reason stimulated by questioning, remarks, 'Galileo seems to accept all the consequences drawn by Plato from this fact…..'"

Roger Bacon on Plato's diamond:

"Fastening on the incident of the slave in the Meno, which he had found reproduced in Cicero, Bacon argued from it 'wherefore since this knowledge (of mathematics) is almost innate and as it were precedes discovery and learning or at least is less in need of them than other sciences, it will be first among sciences and will precede others disposing us towards them.'"

It is perhaps appropriate to close this entry, made on All Hallows' Eve, with a link to a page on Dr. John Dee himself.

Tuesday, September 3, 2002

Tuesday September 3, 2002

Filed under: Geometry — Tags: — m759 @ 6:00 PM

Today's birthday: James Joseph Sylvester

"Mathematics is the music of reason." — J. J. Sylvester

Sylvester, a nineteenth-century mathematician, coined the phrase "synthematic totals" to describe some structures based on 6-element sets that R. T. Curtis has called "rather unwieldy objects." See Curtis's abstract, Symmetric Generation of Finite Groups, John Baez's essay, Some Thoughts on the Number 6, and my website, Diamond Theory. See also the abstract of a December 7, 2000, talk, Mathematics and the Art of M. C. Escher, in which Curtis notes that graphic designs can "often convey a mathematical idea more eloquently than pages of symbolism."

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