Monday, January 13, 2020

Hexagram for Day 13

Filed under: General — m759 @ 11:00 AM

Wednesday, January 1, 2020

A Hexagram for Pauli*

Filed under: General — Tags: — m759 @ 9:21 AM

I Ching box symbol

Pictorial version
of Hexagram 20,
Contemplation  (View)

* See Pauli in the Dec. 30
post Number and Time.

Saturday, October 19, 2019

Hexagram 19

Filed under: General — m759 @ 6:55 AM

The image “http://www.log24.com/log/pix05B/051202-Hex19.jpg” cannot be displayed, because it contains errors.

Click the above image
for its source.

See also Hexagram 19
in this journal.

Thursday, March 16, 2017

Hexagram 64 in Context

Filed under: General,Geometry — Tags: — m759 @ 10:22 PM

"Always with a little humor." — Dr. Yen Lo

Wednesday, January 7, 2015

Midrash on Hexagram 22

Filed under: General — m759 @ 12:30 PM

See Instantia Crucis and Josefine Lyche's
One-Night-Only exhibition in Oslo Jan. 5.

Monday, July 14, 2014

Hexagram 14

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

in great measure

Related material:

Lead obituary in today’s online New York Times  and Los Angeles Times 

Maazel reportedly died on Sunday, July 13, 2014.

From a search in this journal for Iconic Notation,
a related image from August 14, 2010—


See also…


Geometry of the I Ching (Box Style)

Box-style I Ching , January 6, 1989

Saturday, October 20, 2012

Hexagram 20 Revisited

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

Meditations for October 20 :

Box Style,  Wand Work, and  Bowling in Diagon Alley.

Wednesday, August 29, 2012

Hexagram 18

Filed under: General — Tags: — m759 @ 9:14 AM

(Continued from June 14, 2007)

The late William P. Thurston on how mathematical knowledge may decay:

"There are several obvious mechanisms of decay. The experts in a subject retire and die, or simply move on to other subjects and forget. Mathematics is commonly explained and recorded in symbolic and concrete forms that are easy to communicate, rather than in conceptual forms that are easy to understand once communicated. Translation in the direction conceptual -> concrete and symbolic is much easier than translation in the reverse direction, and symbolic forms often replaces [sic ] the conceptual forms of understanding. And mathematical conventions and taken-for-granted knowledge change, so older texts may become hard to understand.

In short, mathematics only exists in a living community of mathematicians that spreads understanding and breaths [sic ] life into ideas both old and new. The real satisfaction from mathematics is in learning from others and sharing with others. All of us have clear understanding of a few things and murky concepts of many more. There is no way to run out of ideas in need of clarification. The question of who is the first person to ever set foot on some square meter of land is really secondary. Revolutionary change does matter, but revolutions are few, and they are not self-sustaining — they depend very heavily on the community of mathematicians."

At mathoverflow.net, October 30, 2010.
     The discussion has been "closed as no longer relevant."
     For another Thurston quote of interest, see a more recent
     mathoverflow discussion "closed as not a real question."

Monday, July 2, 2012

Hexagram 44 Revisited

Filed under: General — m759 @ 12:00 PM

Images from a Google search suggested by
last night's post Coming to Meet, by the recent
film "Archie's Final Project," and by a Thursday,
June 28, 2012, Times Higher Education  piece, 
"Raiders of the Lost Archives"—

IMAGE- Diana Rigg in 'The Hospital' and in 'The Avengers'

    Log24, December 8, 2008 —

David Carradine displays a yellow book-- the Princeton I Ching.

"Let the fingers do the research."
          — Archive Raiders

Friday, September 18, 2020

Holiday Horns

Filed under: General — m759 @ 6:00 PM

Readings for Rosh Hashanah from this journal on April 5, 2005

Compare the following two passages from Holy Scripture:

Genesis 22:13 —

“…behold behind him
a ram caught in a thicket by his horns”

I Ching Hexagram 34 —

“A goat butts against a hedge
And gets its horns entangled.”

Wednesday, April 22, 2020

Arts at Cambridge

Filed under: General — m759 @ 3:40 PM

Click here to enlarge. See also  Hexagram 59, Feng Shui.

Saturday, March 14, 2020


Filed under: General — Tags: — m759 @ 7:09 AM

Detail of Box Style I Ching: Hexagram 14.

Click the above image for details.

There was, however, a challenge by Cozzens himself:

Cozzens replies to Macdonald, March 1958

The apparent source:


Wednesday, March 4, 2020


Filed under: General — m759 @ 3:43 PM

The minute  in the previous post's timestamp
suggests a review

See also Post-It Aesthetics
and posts tagged Story of N.

Tuesday, February 25, 2020

Masks of the Illuminati:

Filed under: General — Tags: — m759 @ 9:46 AM

The Sternheim Portrait  (For Harlan Kane)

From last night's 1:01 AM post

Detail —

This portrait is of German playwright Carl Sternheim.

Steve Martin's version of Sternheim's 1910 play "The Underpants"
reportedly opened on November 3, 2006.

My own interests on that date lay elsewhere . . .

Related abstract art —

Thursday, November 7, 2019

Jagged Crest

Filed under: General — Tags: — m759 @ 9:01 AM

"The man touched the white bishop, queen and king,
and ran his finger over the jagged crest of the rook.
Then, sitting down before the chess set owner could nod
his head, he made his first move with the white pawn."

The late Stephen Dixon, "The Chess House," in
The Paris Review Winter-Spring 1963 (early in 1963).

I Ching chessboard (original 1989 arrangement)

Tuesday, October 29, 2019

Conceptual News

Filed under: General — m759 @ 8:37 PM

The New York Times  reports this evening the
death of a Conceptual artist on October 19

Conceptual art from October 19

Friday, August 16, 2019


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


IMAGE- 'Nocciolo': A 'kernel' for Pascal's Hexagrammum Mysticum: The 15 2-subsets of a 6-set as points in a Galois geometry.

A revision of the above diagram showing
the Galois-addition-table structure —

Related tables from August 10

See "Schoolgirl Space Revisited."

Saturday, July 27, 2019

Make America Strange Again

Filed under: General — Tags: — m759 @ 10:59 PM

Suggested by the previous post, "The Swarm" —

“‘Oracle, why did you write
The Grasshopper Lies Heavy?
What are we supposed to learn?'”

— Philip K. Dick

“She began throwing the coins.“

I Ching Hexagram 61: Inner Truth

Monday, July 22, 2019

Space, Time, Form

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

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

Click the image for some remarks on a related novel.

Thursday, May 23, 2019

Fire on the Water

Filed under: General — m759 @ 8:13 PM

Related literary remarks from The Crimson Abyss 
(a Log24 post of March 29, 2017) —

Prospero's Children  was first published by HarperCollins,
London, in 1999. A statement by the publisher provides
an instance of the famous "much-needed gap." —

"This is English fantasy at its finest. Prospero’s Children 
steps into the gap that exists between The Lion, the Witch
and the Wardrobe
  and Clive Barker’s Weaveworld , and
is destined to become a modern classic."

Related imagery from The Crimson Abyss —

See as well posts of June 6, 2004, and May 22, 2004.

Wednesday, December 5, 2018

The Hassenfeld Legacy

Filed under: General — m759 @ 11:30 AM

The Finkelstein Talisman —

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

"Before time began, there was the Cube."

— Optimus Prime in "Transformers" (Paramount, 2007)

Wikipedia on Hasbro

Three American Jewish brothers,[6] Herman, Hillel, and Henry Hassenfeld[7] 
founded Hassenfeld Brothers in Providence, Rhode Island in 1923 . . . .

The Hassenfeld Auction — 

Also on September 16, 2015 —


The Hindman Image —

The Hood Warenkorb —

Under the Hood —

Megan Fox in "Transformers" (2007) —


This Way to the Egress —

Thursday, November 1, 2018

Formation, Transformation . . . . Solution, Dissolution

Filed under: General,Geometry — m759 @ 8:40 PM

Monday, August 20, 2018

A Wheel for Ellmann

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

The title was suggested by Ellmann's roulette-wheel analogy
in the previous post, "The Perception of Coincidence."

I Ching hexagrams as a Singer 63-cycle, plus zero

Saturday, July 28, 2018


Filed under: General,Geometry — m759 @ 11:22 AM

Floyd:  "You're trying to figure out this length.
            That's the hypotenuse. So you have to
             know this angle."

Monday, June 25, 2018

The Gateway Device

Filed under: General,Geometry — m759 @ 6:24 PM

<title data-rh="true">Frank Heart, Who Linked Computers Before the Internet, Dies at 89 – The New York Times</title>
. . . .
<meta data-rh="true" name="description" itemprop="description" content="Mr. Heart’s team built the gateway device for the Arpanet, the precursor to the internet. Data networking was so new then, they made it up as they went."/>
. . . .
<meta data-rh="true" property="article:published" itemprop="datePublished" content="2018-06-25T19:16:17.000Z"/>

See also yesterday's "For 6/24" and 

IMAGE- 'Nocciolo': A 'kernel' for Pascal's Hexagrammum Mysticum: The 15 2-subsets of a 6-set as points in a Galois geometry.

Wednesday, January 17, 2018

At Heaven’s Gate

Filed under: General — Tags: — m759 @ 7:30 AM

(Continued from September 12, 2005)

The previous post contrasted the number-triple 11-7-8 below
with number triples 12-9-5 and 12-5-9.

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

A perhaps more logical counterpart of the triple 11-7-8, based
on opposite  locations of star-points or cube-edges, is
the triple 9-12-5. For a theological interpretation, see 9/12/05.

Tuesday, January 16, 2018

An Orison for Ha-Why

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

'Cloud Atlas' book cover illustrating the film

Lines from characters played in the film by Tom Hanks and Halle Berry —

— Cloud Atlas , by David Mitchell (2004).

An orison of sorts from a post on Martin Scorsese's
birthday, Sunday, Nov. 18, 2007 —

BlackBerry with pictures from Log24

Displayed on the BlackBerry are parts
of Log24 posts from October 25, 2007,
and October 24, 2007.

Related pattern geometry 

From a Log24 search for Angleton + Brotherhood:
A photo of Angleton in a post from 12/9/5

The image “http://www.log24.com/log/pix05B/051209-Angleton.jpg” cannot be displayed, because it contains errors.

From a post of 11/7/8


A cryptic note for Dan Brown:

The above dates 11/7/8 and 12/9/5 correspond to the corner-labels
(read clockwise and counter-clockwise) of the two large triangles
in the Finkelstein Talisman

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

Above: More symbology for Tom Hanks from
this morning's post The Pentagram Papers.

The above symbology is perhaps better suited to Hanks in his
role as Forrest Gump than in his current role as Ben Bradlee.

The image “http://www.log24.com/log/pix05B/051211-gump.jpg” cannot be displayed, because it contains errors.

For Hanks as Dan Brown's Harvard symbologist 
Robert Langdon, see the interpretation 12/5/9, rather
than 12/9/5, of the above triangle/cube-corner label.

The Pentagram Papers

Filed under: General,Geometry — Tags: — m759 @ 9:16 AM

Other intersection-points-counting material —

The Finkelstein Talisman:

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

See also Hanks + Cube in this journal —

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

Tuesday, August 29, 2017

The Finkelstein Talisman

Filed under: General — m759 @ 8:38 PM

An image in memory of a publisher* who reportedly died
on Saturday, August 26, 2017.  

He and his wife wrote a novel, The Twelve , that has been compared to
the classic film "Village of the Damned." (See a sequel in this journal.)

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

For more on the image, see posts now tagged The Finkelstein Talisman.


Sunday, August 27, 2017

Black Well

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

The “Black” of the title refers to the previous post.
For the “Well,” see Hexagram 48.

Related material —

The Galois Tesseract and, more generally, Binary Coordinate Systems.

Tuesday, August 15, 2017

Every Picture Tells a Story

Filed under: General,Geometry — m759 @ 11:48 AM

Hexagram 15:

See also remarks today by David Brooks at The New York Times .

Friday, July 14, 2017


Filed under: General,Geometry — m759 @ 8:42 AM

Box-style I Ching,  January 6, 1989 —

Geometry of the I Ching (Box Style)

(Click on images for background.)


Detail of Box Style I Ching: Hexagram 14.

See also yesterday's illustration of 
the 1965 paperback edition 
of Whittaker and Watson 



Thursday, July 13, 2017

Dead at 61

Filed under: General — m759 @ 11:22 AM

I Ching Hexagram 61: Inner Truth

See also Hexagram 61 in this journal.

Tuesday, April 11, 2017

Partitioning the Crimson Abyss

Filed under: General — Tags: , — m759 @ 6:00 PM

For the title, see Crimson + Abyss in this journal.

"Ready when you are, C. B."

Hexagram 63, "After Completion"

Sunday, April 9, 2017

23 for 23 for …

Filed under: General — m759 @ 11:01 PM

James McAvoy —

Saturday, April 1, 2017

Art Space

Filed under: General,Geometry — m759 @ 12:20 AM


Click image for some backstory.

“Whatever he drew was the platonic ideal
of what a cartoon should look like.”

— Bob Mankoff on Jack Ziegler, who reportedly
     died on Wednesday, March 29, 2017.

See also "Hexagram 64 in Context," March 16, 2017.

Wednesday, March 29, 2017

The Crimson Abyss

Filed under: General,Geometry — Tags: , , — m759 @ 3:19 PM

"And as the characters in the meme twitch into the abyss
that is the sky, this meme will disappear into whatever
internet abyss swallowed MySpace."

—Staff writer Kamila Czachorowski, Harvard Crimson , March 29


IMAGE- 'Affine Groups on Small Binary Spaces,' illustration


Logo design for Stack Exchange Math by Jin Yang

Recent posts now tagged Crimson Abyss suggest
the above logo be viewed in light of a certain page 29

"… as if into a crimson abyss …." —

Update of 9 PM ET March 29, 2017:

Prospero's Children  was first published by HarperCollins,
London, in 1999. A statement by the publisher provides
an instance of the famous "much-needed gap." —

"This is English fantasy at its finest. Prospero’s Children 
steps into the gap that exists between The Lion, the Witch
and the Wardrobe
  and Clive Barker’s Weaveworld , and
is destined to become a modern classic."

Related imagery —

See also "Hexagram 64 in Context" (Log24, March 16, 2017).

Design Abyss

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

Hexagram 29,
The Abyss (Water)

This post was suggested by an August 6, 2010, post by the designer
(in summer or fall, 2010) of the Stack Exchange math logo (see
the previous Log24 post, Art Space Illustrated) —

http://www.8164.org/☵☲/  .

In that post, the designer quotes the Wilhelm/Baynes I Ching  to explain
his choice of Hexagram 63, Water Over Fire, as a personal icon —

"When water in a kettle hangs over fire, the two elements
stand in relation and thus generate energy (cf. the
production of steam). But the resulting tension demands
caution. If the water boils over, the fire is extinguished
and its energy is lost. If the heat is too great, the water
evaporates into the air. These elements here brought in
to relation and thus generating energy are by nature
hostile to each other. Only the most extreme caution
can prevent damage."

See also this  journal on Walpurgisnacht (April 30), 2010 —


Hexagram 29:



Hexagram 30:

"Hates California,
it's cold and it's damp.

Image--'The Fire,' by Katherine Neville

A thought from another German-speaking philosopher

"Die Philosophie ist ein Kampf gegen die Verhexung
unsres Verstandes durch die Mittel unserer Sprache."

See also The Crimson 's abyss in today's 4:35 AM post Art Space, Continued.

Saturday, March 25, 2017

Like Decorations in a Cartoon Graveyard

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

Continued from April 11, 2016, and from

A tribute to Rothko suggested by the previous post

For the idea  of Rothko's obstacles, see Hexagram 39 in this journal.

Monday, March 6, 2017

Reading the Coins

Filed under: General — m759 @ 7:00 PM

From Didion’s Play It As It Lays :

Everything goes.  I am working very hard at
not thinking about how everything goes. 
I watch a hummingbird, throw the I Ching
but never read the coins, keep my mind in the now.
— Page 8

From Play It As It Lays :

I lie here in the sunlight, watch the hummingbird. 
This morning I threw the coins in the swimming pool,
and they gleamed and turned in the water in such a way
that I was almost moved to read them.  I refrained.
— Page 214

From a search in this journal for "The Southwest Furthers" —

Hexagram 39:

I Ching, Hexagram 39

The Judgment

Obstruction. The southwest furthers.

(See Zenna Henderson.) 

Saturday, February 18, 2017


Filed under: General,Geometry — m759 @ 3:00 AM

The Log24 version  (Nov. 9, 2005, and later posts) —



Escher's 'Verbum'

Escher's Verbum

Solomon's Cube

Solomon's Cube

I Ching hexagrams as parts of 4x4x4 cube

Geometry of the I Ching

The Warner Brothers version

The Paramount version

See also related material in the previous post, Transformers.

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, October 15, 2016

Twelve and Twelve

Filed under: General — m759 @ 1:12 PM

See All Saints 2014 in this journal and listen to 
the new Stevie Nicks reissue of Bella Donna.

Related religious imagery —

Magic cube and corresponding hexagram, or Star of David, with faces mapped to lines and edges mapped to points (The 6 cube faces are mapped to the 6 hexagram lines.)

Friday, September 2, 2016

Raiders of the Lost Birthday

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

Some images from the posts of last July 13
(Harrison Ford's birthday) may serve as funeral
ornaments for the late Prof. David Lavery.

IMAGE- Massimo Vignelli, his wife Lella, and cube

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

See as well posts on "Silent Snow" and "Starlight Like Intuition."

Monday, August 29, 2016

Roll Credits

Filed under: General — m759 @ 9:29 AM

Click images for some backstories.



    Pink hexagram in cube

Related material: The Wet Hot Summa.

Wednesday, August 10, 2016

Logos and Logic

Filed under: General — m759 @ 8:11 PM

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

— Wallace Stevens,
    "Notes Toward a Supreme Fiction,"
     Section I, Canto VIII



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

The novel Blood on Snow , set in Oslo, was published
by Knopf on April 7, 2015.  This journal on that date —

Log24 on Tuesday, April 7, 2015


Filed under: Uncategorized — m759 @ 7:00 PM 

Seven years ago in this journal —

The above links:  the Stone,  the rules.

A related image —

Saturday, August 6, 2016

Mystic Correspondence:

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

The Cube and the Hexagram

The above illustration, by the late Harvey D. Heinz,
shows a magic cube* and a corresponding magic 
hexagram, or Star of David, with the six cube faces 
mapped to the six hexagram lines and the twelve  
cube edges mapped to the twelve hexagram points.
The eight cube vertices correspond to eight triangles
in the hexagram (six small and two large). 

Exercise:  Is this noteworthy mapping** of faces to lines, 
edges to points, and vertices to triangles an isolated 
phenomenon, or can it be viewed in a larger context?

* See the discussion at magic-squares.net of
   "perimeter-magic cubes"

** Apparently derived from the Cube + Hexagon figure
    discussed here in various earlier posts. See also
    "Diamonds and Whirls," a note from 1984.

Wednesday, July 13, 2016

Block That Metaphor

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

Magic cube and corresponding hexagram, or Star of David, with faces mapped to lines and edges mapped to points (The 6 cube faces are mapped to the 6 hexagram lines.)

Happy dies natalis  to the late Frida Kahlo.

Saturday, May 14, 2016

The Hourglass Code

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

version of the I Ching’s Hexagram 19:

I Ching Hexagram 19, 'Approach,' the box-style version

From Katherine Neville's The Eight , a book on the significance
of the date April 4 — the author's birthday —

Axe image from Katherine Neville's 'The Eight'

The Eight  by Katherine Neville —

    “What does this have to do with why we’re here?”
    “I saw it in a chess book Mordecai showed me.  The most ancient chess service ever discovered was found at the palace of King Minos on Crete– the place where the famous Labyrinth was built, named after this sacred axe.  The chess service dates to 2000 B.C.  It was made of gold and silver and jewels…. And in the center was carved a labrys.”
… “But I thought chess wasn’t even invented until six or seven hundred A.D.,” I added.  “They always say it came from Persia or India.  How could this Minoan chess service be so old?”
    “Mordecai’s written a lot himself on the history of chess,” said Lily…. “He thinks that chess set in Crete was designed by the same guy who built the Labyrinth– the sculptor Daedalus….”
    Now things were beginning to click into place….
    “Why was this axe carved on the chessboard?” I asked Lily, knowing the answer in my heart before she spoke.  “What did Mordecai say was the connection?”….
    “That’s what it’s all about,” she said quietly.  “To kill the King.”
     The sacred axe was used to kill the King.  The ritual had been the same since the beginning of time. The game of chess was merely a reenactment.  Why hadn’t I recognized it before?

Related material:  Posts now tagged Hourglass Code.

See also the hourglass in a search for Pilgrim's Progress Illustration.

Tuesday, April 5, 2016

Stevens Illustrated

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

From a Stevens poem, "The Public Square" —

"A slash of angular blacks."

I Ching hexagram 14, box style

See also "Hexagram 14."

Saturday, February 27, 2016

Back to the Blackboard

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

Related material —

Runes to Grave and

Thursday, January 28, 2016


Filed under: General,Geometry — m759 @ 9:57 AM

The title refers to a line by Louis Menand quoted
at the end of the previous post.

There "a6!" refers to the chessboard square in
column a, row 6.  In Geometry of the I Ching,
this square represents Hexagram 61, "Inner Truth."

See also "inner truth" in this journal.

Friday, January 22, 2016

Easter Footnote

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

A search from Easter 2013 for "Cremona synthemes" *

IMAGE- Google image search for 'cremona synthemes'

For some strictly mathematical background, see
Classical Geometry in Light of Galois Geometry.

* For more about Cremona and synthemes, 
   see a 1975 paper by W. L. Edge,
  "A Footnote on the Mystic Hexagram."

Thursday, January 21, 2016

Dividing the Indivisible

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

My statement yesterday morning that the 15 points
of the finite projective space PG(3,2) are indivisible 
was wrong.  I was misled by quoting the powerful
rhetoric of Lincoln Barnett (LIFE magazine, 1949).

Points of Euclidean  space are of course indivisible
"A point is that which has no parts" (in some translations).

And the 15 points of PG(3,2) may be pictured as 15
Euclidean  points in a square array (with one point removed)
or tetrahedral array (with 11 points added).

The geometry of  PG(3,2) becomes more interesting,
however, when the 15 points are each divided  into
several parts. For one approach to such a division,
see Mere Geometry. For another approach, click on the
image below.

IMAGE- 'Nocciolo': A 'kernel' for Pascal's Hexagrammum Mysticum: The 15 2-subsets of a 6-set as points in a Galois geometry.

Wednesday, December 23, 2015

Splitting Apart

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

Bleecker Street logo —

Click image for some background.

Related remarks on mathematics:

Boole vs. Galois

Tuesday, December 1, 2015

The Nutshell

Filed under: General,Geometry — m759 @ 1:13 PM

See a search for Nocciolo  in this journal.

An image from that search —

IMAGE- 'Nocciolo': A 'kernel' for Pascal's Hexagrammum Mysticum: The 15 2-subsets of a 6-set as points in a Galois geometry.

Recall also Hamlet's
"O God bad dreams."

Pascal’s Finite Geometry

Filed under: General,Geometry — Tags: — m759 @ 12:01 AM

See a search for "large Desargues configuration" in this journal.

The 6 Jan. 2015 preprint "Danzer's Configuration Revisited," 
by Boben, Gévay, and Pisanski, places this configuration,
which they call the Cayley-Salmon configuration , in the 
interesting context of Pascal's Hexagrammum Mysticum .

They show how the Cayley-Salmon configuration is, in a sense,
dual to something they call the Steiner-Plücker configuration .

This duality appears implicitly in my note of April 26, 1986,
"Picturing the smallest projective 3-space." The six-sets at
the bottom of that note, together with Figures 3 and 4
of Boben et. al. , indicate how this works.

The duality was, as they note, previously described in 1898.

Related material on six-set geometry from the classical literature—

Baker, H. F., "Note II: On the Hexagrammum Mysticum  of Pascal,"
in Principles of Geometry , Vol. II, Camb. U. Press, 1930, pp. 219-236  

Richmond, H. W., "The Figure Formed from Six Points in Space of Four Dimensions,"
Mathematische Annalen  (1900), Volume 53, Issue 1-2, pp 161-176

Richmond, H. W., "On the Figure of Six Points in Space of Four Dimensions," 
Quarterly Journal of Pure and Applied Mathematics , Vol. 31 (1900), pp. 125-160

Related material on six-set geometry from a more recent source —

Cullinane, Steven H., "Classical Geometry in Light of Galois Geometry," webpage

Thursday, November 12, 2015

For Bernstein and Horowitz

Filed under: General — m759 @ 12:17 AM

Detail of Box Style I Ching: Hexagram 14.

Click image for some backstory.

The Horowitz of the title reportedly died Nov. 2.

Monday, September 21, 2015

Happy Birthday, Stephen King

Filed under: General — Tags: — m759 @ 2:02 PM


Sunday, August 30, 2015


Filed under: General — m759 @ 12:42 PM

(A companion-piece to the previous post, Bolt)

" In one of her more memorable roles, Ms. Craig
played Marta, a green-skinned slave girl, in the
'Star Trek' episode 'Whom Gods Destroy.' She
performed a seductive, loose-limbed dance that
seemed to nearly overwhelm William Shatner’s
red-blooded Captain Kirk, while Leonard Nimoy’s
Mr. Spock pronounced it 'mildly interesting.' "

Obituary by Katie Rogers in the online
New York Times  of Aug. 19, 2015.  Rogers was
describing actress Yvonne Craig, who reportedly
died on Monday, August 17, 2015.

Related material from this morning's online Times


Filed under: General — m759 @ 11:47 AM

(Quoted here in Annals of Consciousness, June 20, 2014.)

Sunday, August 9, 2015

The Cauldron

Filed under: General — Tags: — m759 @ 7:19 AM

From a review of the 2013 film "The Wolverine" —

"The rituals, culture and hierarchies of Japan
have intrigued and baffled the typical Westerner
for centuries …."

Not to mention those of China 

 Hexagram 50:
The Cauldron

Sunday, July 26, 2015

Sunday Review

Filed under: General — m759 @ 10:00 AM

"Little emblems of eternity 

— Oliver Sacks, contemplating his own impending death,
in The New York Times  Sunday Review section today.  

Sacks's phrase refers to elements of the periodic table —

Another approach to "emblems of eternity" — The I Ching .

Sunday School

Filed under: General — m759 @ 9:00 AM

Hexagram 51:

"I woke last night to the sound of thunder,
How far off, I sat and wondered.
Started humming a song from 1962.
Ain't it funny how the night moves?"

Saturday, July 25, 2015

Strange Loop

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

From an explanation of the Web app IFTTT —
"IF This Then That" —

"If you are a programmer you can think of it as a loop*
that checks for a certain condition… to run one or
multiple actions if the condition is met."

After Completion  (from Friday night, and 1989) —

Advertisement —

Wikipedia —

"On February 19, 2015, IFTTT renamed
their original application to IF…."

This journal —

From Tuesday's post on the death of E. L. Doctorow —

“…right through hell
     there is a path…”
  — Malcolm Lowry

* More precisely, a conditional  or conditional loop 

Friday, July 24, 2015

After Completion

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

Obituaries, including one for George Coe of 'The Dove' (1968)


Tuesday, May 26, 2015

Colorful Tales

Filed under: General — Tags: , — m759 @ 9:50 PM

See also the Log24 post from May 18,
the date of Eric Caidin's reported death,
as well as Hexagram 50 and May 14, 2014—
Death in Mathmagic Land.

Thursday, March 19, 2015

“Divisive Rhetoric”

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

I Ching hexagram 14, box style

An example for 'Jews on Style'- Kipnis on the separatrix

     — Jeffrey Kipnis, "Twisting the Separatrix"
     Assemblage  No. 14 (Apr., 1991), pp. 30-61
     Published by: The MIT Press
     Article Stable URL: http://www.jstor.org/stable/3171098

Monday, March 2, 2015

For Turing’s Cathedral

Filed under: General — m759 @ 10:30 PM

Item from the British press on Oct. 17, 2014:

Item from this journal on that same date:

Raiders of the Inarticulate.

A related "slash" —

Elementary, my dear Watson.

Tuesday, November 18, 2014

Our Most Important Product

Filed under: General — Tags: — m759 @ 12:45 AM

Hexagram 35:

"Then came a 'Robot Psychologist,' known as a Psychological Matrix Rotator,
developed for the Department of Defense. It is still used to literally 'see' that
the right man gets the right Army job."

Ronald Reagan, 1961 GE Sales Meeting

"Always with a little humor." — Yen Lo

In memory of Dr. Irving Peress,
who reportedly died on Thursday,
November 13, 2014.

Thursday, November 13, 2014

Progressive Matrix

Filed under: General,Geometry — Tags: , — m759 @ 6:00 AM

Yesterday's post and recent Hollywood news suggest
a meditation on a Progressive Matrix —

Oct. 12-14, 2005:

'A Poem for Pinter,' conclusion: 'Tick Tick Hash.'

'The Interpreter'-- Sean Penn to Nicole Kidman-- 'My Card.'

Click to enlarge.

"My card."

Structurally related images —

A sample Raven's Progressive Matrices  test item
(such items share the 3×3 structure of the hash symbol above):

IMAGE- Raven's Progressive Matrices item with symbols from Cullinane's box-style I Ching

Structural background —

Sunday, November 9, 2014


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

The Ideas

“We tell ourselves stories in order to live….
We interpret what we see, select the most workable
of multiple choices. We live entirely, especially if we
are writers, by the imposition of a narrative line upon
disparate images, by the ‘ideas’  with which we have
learned to freeze the shifting phantasmagoria
which is our actual experience.”
— Joan Didion

See Didion and the I Ching  and posts tagged Plato in China .

Tuesday, August 5, 2014


Filed under: General — Tags: — m759 @ 8:15 PM

"Philosophy is a battle against the bewitchment [Verhexung ]
of our intelligence by means of our language."

— Wittgenstein, Philosophical Investigations , Section 109

"The philosophy of logic speaks of sentences and words
in exactly the sense in which we speak of them in ordinary life
when we say e.g. 'Here is a Chinese sentence,' or 'No, that only
looks like writing; it is actually just an ornament' and so on."

— Wittgenstein, Philosophical Investigations , Section 108

Monday, June 30, 2014

High Concept

Tags:  — m759 @ 5:24 PM

For the title, see a post of Nov. 4, 2007.

Related material:

Hexagram 29, Water, and a pattern resembling
the symbol for Aquarius:

http://www.log24.com/images/IChing/hexagram29.gif          .

For some backstory about the former,
see the June 21 post Hallmark.

For some backstory about the latter,
see today’s post Toward Evening.

Tom Wolfe has supplied some scaffolding*
to support the concept.

* A reference to Grossman and Descartes.

Monday, June 30, 2014

High Concept

Filed under: General — Tags: , — m759 @ 5:24 PM

For the title, see a post of Nov. 4, 2007.

Related material:

Hexagram 29, Water, and a pattern resembling
the symbol for Aquarius:

http://www.log24.com/images/IChing/hexagram29.gif          .

For some backstory about the former,
see the June 21 post Hallmark.

For some backstory about the latter,
see today’s post Toward Evening.

Tom Wolfe has supplied some scaffolding*
to support the concept.

* A reference to Grossman and Descartes.

Saturday, June 21, 2014


Filed under: General — Tags: , — m759 @ 5:01 PM

A suitable hallmark for
the previous post, Logical Death:



Hexagram 29: “K’an represents…
the principle of light inclosed in the dark.”
— The Richard Wilhelm I Ching

A related page from Stanford:

IMAGE- Page 29 of 'Radical Atheism,' by Martin Hägglund, Stanford U. Press, 2008

Friday, March 28, 2014

Chinese Rune

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

"The Geometry of the I Ching introduces something called the Cullinane sequence
for the hexagrams, and uses a notation based on the four sides and two diagonals
in a square to indicate the yin and yang lines. The resulting rune-like symbols
are intriguing…."

— Andreas Schöter's  I Ching  home page

Actually, the geometry is a bit deeper than the rune-like symbols.

" 'Harriet Burden has been really great to me,'
Rune says in an interview, 'not only as a collector
of my work but as a true supporter. And I think of her
as a muse for the project … ' "

— In The Blazing World , the artist known as Rune

Tuesday, February 11, 2014

Everybody Comes to Rick’s

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

Continued from a post of January 28 —
In Memory of Pete Seeger.

The spiritual tribute link in that post suggests a review
of the following page from a pop-philosophy novel —

"Turn, turn, turn." — Pete Seeger.  See also this morning's news.

Tuesday, January 7, 2014

See More Glass

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

The first story of "The Snow Queen, in Seven Stories"
by Hans Christian Andersen (1845) (see yesterday morning)—

Story the First,
Which Describes a Looking-Glass
and the Broken Fragments

You must attend to the commencement of this story, for when we get to the end we shall know more than we do now about a very wicked hobgoblin; he was one of the very worst, for he was a real demon.  One day, when he was in a merry mood, he made a looking-glass which had the power of making everything good or beautiful that was reflected in it almost shrink to nothing, while everything that was worthless and bad looked increased in size and worse than ever. The most lovely landscapes appeared like boiled spinach, and the people became hideous, and looked as if they stood on their heads and had no bodies. Their countenances were so distorted that no one could recognize them, and even one freckle on the face appeared to spread over the whole of the nose and mouth. The demon said this was very amusing. When a good or pious thought passed through the mind of any one it was misrepresented in the glass; and then how the demon laughed at his cunning invention. All who went to the demon’s school—for he kept a school—talked everywhere of the wonders they had seen, and declared that people could now, for the first time, see what the world and mankind were really like. They carried the glass about everywhere, till at last there was not a land nor a people who had not been looked at through this distorted mirror. They wanted even to fly with it up to heaven to see the angels, but the higher they flew the more slippery the glass became, and they could scarcely hold it, till at last it slipped from their hands, fell to the earth, and was broken into millions of pieces. But now the looking-glass caused more unhappiness than ever, for some of the fragments were not so large as a grain of sand, and they flew about the world into every country. When one of these tiny atoms flew into a person’s eye, it stuck there unknown to him, and from that moment he saw everything through a distorted medium, or could see only the worst side of what he looked at, for even the smallest fragment retained the same power which had belonged to the whole mirror. Some few persons even got a fragment of the looking-glass in their hearts, and this was very terrible, for their hearts became cold like a lump of ice. A few of the pieces were so large that they could be used as window-panes; it would have been a sad thing to look at our friends through them. Other pieces were made into spectacles; this was dreadful for those who wore them, for they could see nothing either rightly or justly. At all this the wicked demon laughed till his sides shook—it tickled him so to see the mischief he had done. There were still a number of these little fragments of glass floating about in the air, and now you shall hear what happened with one of them.

 "Was there more to come? Was I done?
I wondered if I had dreamed
the connectedness of Being
the night before, or if now, awake,
I dreamed distinctions.
I didn’t know where I was for an instant."

"Alethia," by Charles Johnson, as
     quoted by Eve Tushnet on Aug. 22, 2013

Tushnet on Johnson —

"Somebody–I hope a commenter will remind me who it was–
has suggested that the Left typically thinks in terms of an
opposition between oppression and liberation, whereas
the right typically thinks in terms of an opposition between
civilization and barbarism. I would reframe the latter opposition
as order vs. chaos; if we do that, it’s obvious that both
oppositions are unrelentingly relevant, yet few thinkers or artists
are able to hold both conflicts before our eyes at once.

I just finished Charles Johnson’s 1986 short-story collection 
The Sorcerer’s Apprentice: Tales and Conjurations ,
a bag of broken glass which is equal parts liberationist and
reactionary, yearning for freedom and knuckling under to fatalism." 

Related material —

Saturday Night Live, Dec. 11, 1976

Consumer Reporter: Alright. Fine. Fine. Well, we'd like to show you another  one of Mr. Mainway's products. It retails for $1.98, and it's called Bag O' Glass. [ holds up bag of glass ] Mr. Mainway, this is simply a bag of jagged, dangerous, glass bits.

Irwin Mainway: Yeah, right, it's you know, it's glass, it's broken glass, you know? It sells very well, as a matter of fact, you know? It's just broken glass, you know?

Consumer Reporter: [ laughs ] I don't understand. I mean, children could seriously cut themselves on any one of these pieces!

Irwin Mainway: Yeah, well, look – you know, the average kid, he picks up, you know, broken glass anywhere, you know? The beach, the street, garbage cans, parking lots, all over the place in any big city. We're just packaging what the kids want! I mean, it's a creative toy, you know? If you hold this up, you know, you see colors, every color of the rainbow! I mean, it teaches him about light refraction, you know? Prisms, and that stuff! You know what I mean?

Tommy Lee Jones perhaps knows what Mainway means.
Kristen Wiig (see Aug. 22, 2013, in this  journal) perhaps does not.

See also Tushnet on The Man in the High Castle  as well as
Tommy Lee Jones and Hexagram 61.

Saturday, December 14, 2013

Bend Sinister

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

I Ching hexagram 14, box style

Click image for background.
See also related posts.

Monday, November 11, 2013

The Mystic Hexastigm…

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

Or: The Nutshell

What about Pascal?

For some background on Pascal's mathematics,
not his wager, see

Richmond, H. W., 
"On the Figure of Six Points in Space of Four Dimensions," 
Quarterly Journal of Pure and Applied Mathematics , 
Volume 31 (1900), pp. 125-160,
dated by Richmond March 30,1899

Richmond, H. W.,
"The Figure Formed from Six Points in Space of Four Dimensions,"
Mathematische Annalen , 
Volume 53 (1900), Issue 1-2, pp 161-176,
dated by Richmond February 1, 1899

See also Nocciolo  in this journal.

Recall as well that six points in space may,
if constrained to lie on a circle, be given
a religious interpretation.  Richmond's
six points are secular and more general.

Tuesday, August 20, 2013

The 20

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

In memory of author Elmore Leonard

A graphic symbol and a search for "Nowhere"*
in this journal yield

Box symbol

Pictorial version
of Hexagram 20,
Contemplation (View)

"Cotton Mather died
when I was a boy.
The books/ He read,
all day, all night
and all the nights,/
Had got him nowhere."

— Wallace Stevens,
"The Blue Buildings
in the Summer Air"

* See previous post.

Monday, August 19, 2013


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

Last midnight's post quoted poet John Hollander
on Cervantes—

"… the Don’s view of the world is correct at midnight,
and Sancho’s at noon."

The post concluded with a figure that might, if
rotated slightly, be regarded as a sort of Star of
David or Solomon's Seal. The figure's six vertices
may be viewed as an illustration of Pascal's
"mystic hexagram."

Pacal's hexagram is usually described
as a hexagon inscribed in a conic
(such as a circle). Clearly the hexagon
above may be so inscribed.

The figure suggests that last midnight's Don be
played by the nineteenth-century mathematician
James Joseph Sylvester. His 1854 remarks on
the nature of geometry describe a different approach
to the Pascal hexagram

"… the celebrated theorem of Pascal known under the name of the Mystic Hexagram, which is, that if you take two straight lines in a plane, and draw at random other straight lines traversing in a zigzag fashion between them, from A in the first to B in the second, from B in the second to C in the first, from C in the first to D in the second, from D in the second to E in the first, from E in the first to F in the second and finally from F in the second back again to A the starting point in the first, so as to obtain ABCDEF a twisted hexagon, or sort of cat's-cradle figure and if you arrange the six lines so drawn symmetrically in three couples: viz. the 1st and 4th in one couple, the 2nd and 5th in a second couple, the 3rd and 6th in a third couple; then (no matter how the points ACE have been selected upon one of the given lines, and BDF upon the other) the three points through which these three couples of lines respectively pass, or to which they converge (as the case may be) will lie all in one and the same straight line."

For a Sancho view of Sylvester's "cat's cradle," see some twentieth-century
remarks on "the most important configuration of all geometry"—

"Now look, your grace," said Sancho,
"what you see over there aren't giants,
but windmills, and what seems to be arms
are just their sails, that go around in the wind
and turn the millstone."
"Obviously," replied Don Quijote,
"you don't know much about adventures.”

― Miguel de Cervantes Saavedra

Friday, August 16, 2013

Six-Set Geometry

Filed under: General,Geometry — Tags: — m759 @ 5:24 AM

From April 23, 2013, in
​"Classical Geometry in Light of Galois Geometry"—

Click above image for some background from 1986.

Related material on six-set geometry from the classical literature—

Baker, H. F., "Note II: On the Hexagrammum Mysticum  of Pascal,"
in Principles of Geometry , Vol. II, Camb. U. Press, 1930, pp. 219-236  

Richmond, H. W., "The Figure Formed from Six Points in Space of Four Dimensions,"
Mathematische Annalen  (1900), Volume 53, Issue 1-2, pp 161-176

Richmond, H. W., "On the Figure of Six Points in Space of Four Dimensions," 
Quarterly Journal of Pure and Applied Mathematics , Vol. 31 (1900), pp. 125-160

Sunday, July 7, 2013

Mere 61

Filed under: General — m759 @ 9:00 AM

Today is the 61st anniversary of the publication
of the book Mere Christianity , by C. S. Lewis.

In its honor, here is a link to "Hexagram 61
in this journal.

See also "Moonshine and Lion."

Sunday, June 16, 2013

Mathematical Review

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

From a weblog post on June 11, 2013, by one Pete Trbovich:

Diamond Theory

Here again, I don't think Steven Cullinane is really unhinged per se. At the very least, his geometric study is fun to play with, particularly when you find this toy. And I'm not really sure that anything he says is wrong per se. But you might find yourself asking "So what?" or more to the point, "Why is this supposed to be the central theory to explaining life, the universe, and everything?"

It isn't  supposed to be such a theory.
I do not know why Trbovich thinks it is 

— Steven H. Cullinane

Update of 11 PM June 16:

For one such central theory of everything, see
the I Ching .  Diamond theory is, unlike that
Chinese classic, pure mathematics, but the larger
of the binary-coordinate structures  it is based on
are clearly isomorphic, simply as structures , to
the I Ching 's 
64 hexagrams.

Make of this what you will.

Sunday, May 5, 2013

Night of Lunacy*

Filed under: General — m759 @ 9:00 PM

Structure vs. Character continued

   IMAGE- The 3x3 square   


IMAGE- Chinese character for 'well' and I Ching Hexagram 48, 'The Well'


Related vocabulary:

Nick Tosches on the German word "Quell "

and Heidegger on Hölderlin.

* The title is from Heidegger.

Sunday, April 28, 2013

The Octad Generator

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

… And the history of geometry  
Desargues, Pascal, Brianchon and Galois
in the light of complete n-points in space.

(Rewritten for clarity at about 10 AM ET April 29, with quote from Dowling added.
Updated with a reference to a Veblen and Young exercise (on p. 53) on April 30.)

Veblen and Young, Projective Geometry, Vol. I ,
Ginn and Company, 1910, page 39:

"The Desargues configuration. A very important configuration
is obtained by taking the plane section of a complete space five-point."

Each of figures 14 and 15 above has 15 points and 20 lines.
The Desargues configuration within each figure is denoted by
10 white points and 10 solid lines, with 3 points on each line and
3 lines on each point. Black  points and dashed  lines indicate the
complete space five-point and lines connecting it to the plane section
containing the Desargues configuration.

In a 1915 University of Chicago doctoral thesis, Archibald Henderson
used a complete space six -point to construct a configuration of
15 points and 20 lines in the context not of Desargues '  theorem, but
rather of Brianchon 's theorem and of the Pascal  hexagram.
Henderson's 1915 configuration is, it turns out, isomorphic to that of
the 15 points and 20 lines in the configuration constructed via a
complete space five -point five years earlier by Veblen and Young.
(See, in Veblen and Young's 1910 Vol. I, exercise 11, page 53:
"A plane section of a 6-point in space can  be considered as
3 triangles perspective in pairs from 3 collinear points with
corresponding sides meeting in 3 collinear points." This is the
large  Desargues configuration. See Classical Geometry in Light of 
Galois Geometry

For this large  Desargues configuration see April 19.
For Henderson's complete six –point, see The Six-Set (April 23).
That post ends with figures relating the large  Desargues configuration
to the Galois  geometry PG(3,2) that underlies the Curtis
Miracle Octad Generator  and the large Mathieu group M24 —

IMAGE- Geometry of the Six-Set, Steven H. Cullinane, April 23, 2013

See also Note on the MOG Correspondence from April 25, 2013.

That correspondence was also discussed in a note 28 years ago, on this date in 1985.

Saturday, April 27, 2013

Mark and Remark

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

“Fact and fiction weave in and out of novels like a shell game.” —R.B. Kitaj

Not just novels.


IMAGE- Anne Taormina on 'Mathieu Moonshine' and the 'super overarching symmetry group'

The mark preceding A in the above denotes the semidirect product.

Symbol from the box-style
I Ching  (Cullinane, 1/6/89).
This is Hexagram 55,
“Abundance [Fullness].”

The mathematical quote, from last evening’s Symmetry, is from Anne Taormina.

The I Ching  remark is not.

Another version of Abbondanza 

IMAGE- Taormina sunset from inabbondanza.com on June 22, 2009


Found in Translation and the giorno  June 22, 2009here.

Monday, April 8, 2013

Magic for Jews

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

A commenter on Saturday's "Seize the Dia" has
suggested a look at the work of one Mark Collins.

Here is such a look (click to enlarge):

I find attempts to associate pure mathematics with the words
"magic" or "mystic" rather nauseating. (H. F. Baker's work
on Pascal's mystic hexagram  is no exception; Baker was
stuck with Pascal's obnoxious adjective, but had no truck
with any mystic aspects of the hexagram.)

The remarks above by Clifford Pickover on Collins, Dürer, and
binary representations may interest some non-mathematicians,
who should not  be encouraged to waste their time on this topic.

For the mathematics underlying the binary representation of
Dürer's square, see, for instance, my 1984 article "Binary
Coordinate Systems

Those without the background to understand that article
may enjoy, instead of Pickover's abortive attempts above at
mathematical vulgarization, his impressively awful 2009 novel
Jews in Hyperspace .

Pickover's 2002 book on magic squares was, unfortunately,
published by the formerly reputable Princeton University Press.

Related material from today's Daily Princetonian :

See also Nash + Princeton in this journal.

Sunday, April 7, 2013

Pascal Inscape

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

Click to enlarge.

IMAGE- A Galois-geometry key to the mystic hexagram of Pascal

Background: Inscapes and The 2-subsets of a 6-set are the points of a PG(3,2).

Related remarks: Classical Geometry in Light of Galois Geometry.

Saturday, April 6, 2013

Pascal via Curtis

Filed under: General,Geometry — Tags: , — m759 @ 9:17 AM

Click image for some background.

IMAGE- The Miracle Octad Generator (MOG) of R.T. Curtis

Shown above is a rearranged version of the
Miracle Octad Generator (MOG) of R. T. Curtis
("A new combinatorial approach to M24,"
Math. Proc. Camb. Phil. Soc., 79 (1976), 25-42.)

The 8-subcell rectangles in the left part of the figure may be
viewed as illustrating (if the top left subcell is disregarded)
the thirty-five 3-subsets of a 7-set.

Such a view relates, as the remarks below show, the
MOG's underlying Galois geometry, that of PG(3,2), to
the hexagrammum mysticum  of Pascal.

On Danzer's 354 Configuration:

IMAGE- Branko Grünbaum on Danzer's configuration

"Combinatorially, Danzer’s configuration can be interpreted
as defined by all 3-sets and all 4-sets that can be formed
by the elements of a 7-element set; each 'point' is represented
by one of the 3-sets, and it is incident with those lines
(represented by 4-sets) that contain the 3-set."

— Branko Grünbaum, "Musings on an Example of Danzer's,"
European Journal of Combinatorics , 29 (2008),
pp. 1910–1918 (online March 11, 2008)

"Danzer's configuration is deeply rooted in
Pascal's Hexagrammum Mysticum ."

— Marko Boben, Gábor Gévay, and Tomaž Pisanski,
"Danzer's Configuration Revisited," arXiv.org, Jan. 6, 2013

For an approach to such configurations that differs from
those of Grünbaum, Boben, Gévay, and Pisanski, see

Classical Geometry in Light of Galois Geometry.

Grünbaum has written little about Galois geometry.
Pisanski has recently touched on the subject;
see Configurations in this journal (Feb. 19, 2013).

Tuesday, April 2, 2013

Baker on Configurations

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

The geometry posts of Sunday and Monday have been
placed in finitegeometry.org as

Classical Geometry in Light of Galois Geometry.

Some background:

See Baker, Principles of Geometry , Vol. II, Note I
(pp. 212-218)—

On Certain Elementary Configurations, and
on the Complete Figure for Pappus's Theorem

and Vol. II, Note II (pp. 219-236)—

On the Hexagrammum Mysticum  of Pascal.

Monday's elucidation of Baker's Desargues-theorem figure
treats the figure as a 15420configuration (15 points, 
4 lines on each, and 20 lines, 3 points on each).

Such a treatment is by no means new. See Baker's notes
referred to above, and 

"The Complete Pascal Figure Graphically Presented,"
a webpage by J. Chris Fisher and Norma Fuller.

What is new in the Monday Desargues post is the graphic
presentation of Baker's frontispiece figure using Galois geometry :
specifically, the diamond theorem square model of PG(3,2).

See also Cremona's kernel, or nocciolo :

Baker on Cremona's approach to Pascal—

"forming, in Cremona's phrase, the nocciolo  of the whole."

IMAGE- Definition of 'nocciolo' as 'kernel'

A related nocciolo :

IMAGE- 'Nocciolo': A 'kernel' for Pascal's Hexagrammum Mysticum: The 15 2-subsets of a 6-set as points in a Galois geometry.

Click on the nocciolo  for some
geometric background.

Monday, December 24, 2012

Eternal Recreation

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

Memories, Dreams, Reflections
by C. G. Jung

Recorded and edited By Aniela Jaffé, translated from the German
by Richard and Clara Winston, Vintage Books edition of April 1989

From pages 195-196:

“Only gradually did I discover what the mandala really is:
‘Formation, Transformation, Eternal Mind’s eternal recreation.’*
And that is the self, the wholeness of the personality, which if all
goes well is harmonious, but which cannot tolerate self-deceptions.”

* Faust , Part Two, trans. by Philip Wayne (Harmondsworth,
England, Penguin Books Ltd., 1959), p. 79. The original:

                   … Gestaltung, Umgestaltung, 
  Des ewigen Sinnes ewige Unterhaltung….

Jung’s “Formation, Transformation” quote is from the realm of
the Mothers (Faust Part Two, Act 1, Scene 5: A Dark Gallery).
The speaker is Mephistopheles.

See also Prof. Bruce J. MacLennan on this realm
in a Web page from his Spring 2005 seminar on Faust:

“In alchemical terms, F is descending into the dark, formless
primary matter from which all things are born. Psychologically
he is descending into the deepest regions of the
collective unconscious, to the source of life and all creation.
Mater (mother), matrix (womb, generative substance), and matter
all come from the same root. This is Faust’s next encounter with
the feminine, but it’s obviously of a very different kind than his
relationship with Gretchen.”

The phrase “Gestaltung, Umgestaltung ” suggests a more mathematical
approach to the Unterhaltung . Hence

Part I: Mothers

“The ultimate, deep symbol of motherhood raised to
the universal and the cosmic, of the birth, sending forth,
death, and return of all things in an eternal cycle,
is expressed in the Mothers, the matrices of all forms,
at the timeless, placeless originating womb or hearth
where chaos is transmuted into cosmos and whence
the forms of creation issue forth into the world of
place and time.”

— Harold Stein Jantz, The Mothers in Faust:
The Myth of Time and Creativity 
Johns Hopkins Press, 1969, page 37

Part II: Matrices


Part III: Spaces and Hypercubes

Click image for some background.

Part IV: Forms

Forms from the I Ching :

Click image for some background.

Forms from Diamond Theory :

Click image for some background.

Wednesday, November 7, 2012


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

In several posts now tagged Chessboard
an I Ching chessboard
image in which adjacent squares 
have the Karnaugh property

— has been replaced by a picture of
the original 1989 version in which
the Karnaugh property applies only to cells
that are adjacent in a cubic, not square,

I Ching chessboard (original 1989 arrangement)

Tuesday, November 6, 2012

Bend Sinister

Filed under: General — m759 @ 6:30 AM

This morning's New York Times  obituaries—

These suggest a look at Solving Nabokov's Lolita Riddle ,
by Joanne Morgan (Sydney: Cosynch Press, 2005).

That book discusses Lolita as a character like Lewis Carroll's Alice.

(The Red Queen and Alice of course correspond to figures in
the first two thumbnails above.)

From the obituary associated with the third thumbnail above:

"Front-page headlines combined concision and dark humor." 

The title of this post, Bend Sinister , is not unlike such a headline.
It is the title of a novel by Nabokov (often compared with Orwell's 1984 )
that is discussed in the Lolita Riddle  book.

Related material— The bend sinister found in Log24 searches
for Hexagram 14 and for the phrase Hands-On

IMAGE- Magician's hands on his wand, viewed as a diagonal of a square

Monday, October 29, 2012

Opening Bell

Filed under: General — m759 @ 9:00 AM

Related material:

Hexagram 29: Water

Hexagram 29 and the life of
Robert Palmer Dilworth, who
died on this date in 1993.

Sunday, September 16, 2012


Filed under: General — m759 @ 11:00 AM

"I'm 18."

— Spoken by a very beautiful girl
      in the summer of 1991

"Work on what has been spoiled [ Decay ]."

Hexagram 18

IMAGE- Matchbook with monogram ROT from 'North by Northwest'

Click image for some background.

Monday, July 2, 2012


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

Heidegger, "Hölderlin and the Essence of Poetry,"
translated by Douglas Scott, in Existence and Being  ,
Regnery, 1949, pp. 291-316—

IMAGE- Page 304 of Heidegger's 'Existence and Being' - Heidegger's essay 'Hölderlin and the Essence of Poetry,' tr. by Douglas Scott, publ. by Henry Regnery Company, Chicago, in 1949

See also Hexagram 36.

Sunday, July 1, 2012

Coming to Meet

Filed under: General — m759 @ 10:30 PM

Hexagram 44:
Coming to Meet

"This hexagram indicates a situation in which
the principle of darkness, after having been eliminated,
furtively and unexpectedly obtrudes again from within
and below. Of its own accord the female principle
comes to meet the male. It is an unfavourable and
dangerous situation, and we must understand and
promptly prevent the possible consequences.

The hexagram is linked with the fifth month
[June-July], because at the summer solstice
the principle of darkness gradually becomes
ascendant again."

— Richard Wilhelm 

To counteract the principle of darkness—

The Uploading (Friday— St. Peter's Day, 2012),

Thor's Light Bulb Joke, and …


Friday, June 22, 2012

Wand Work

Filed under: General — m759 @ 7:59 PM

The New York Times  today—
 "Reality and our perception of it are incommensurate…."

IMAGE- NY Times Wire item- 'Your Mind on Magic,' by Alex Stone

The above New York Times Wire  item from 3:35 PM ET today
mentions two topics touched on in today's earlier Log24 post
Bowling in Diagon Alley— magic (implied by the title) and
incommensurability. The connection in that post
between the two topics is the diagonal  of a square.

The  wire item shows one detail from a Times  illustration
of the linked article— a blindfolded woman.

Another detail from the same illustration—

IMAGE- Magician's hands on his wand, viewed as a diagonal of a square (or as Hexagram 14 in the box-style I Ching

Hands-on Wand Work

See also remarks on Magic in this journal and on Harry Potter.

I dislike both topics.

Tuesday, June 12, 2012

Meet Max Black (continued)

Filed under: General,Geometry — Tags: — m759 @ 11:59 PM

Background— August 30, 2006—

The Seventh Symbol:

The image “http://www.log24.com/log/pix06A/060830-Algebra.jpg” cannot be displayed, because it contains errors.

In the 2006 post, the above seventh symbol  110000 was
interpreted as the I Ching hexagram with topmost and
next-to-top lines solid, not broken— Hexagram 20, View .

In a different interpretation, 110000 is the binary for the decimal
number 48— representing the I Ching's Hexagram 48, The Well .

“… Max Black, the Cornell philosopher, and 
others have pointed out how ‘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 U. Press, 1962, page 242, as quoted
in Dramas, Fields, and Metaphors,
by Victor Witter Turner, Cornell U. Press,
paperback, 1975, page 25

The algebra is certainly clearer than either I Ching
metaphor, but is in some respects less interesting.

For a post that combines both the above I Ching
metaphors, View  and Well  , see Dec. 14, 2007.

In memory of scholar Elinor Ostrom,
who died today—

"Time for you to see the field."
Bagger Vance

Tuesday, June 5, 2012

Chinese Epiphany

Filed under: General,Geometry — m759 @ 4:38 AM

References here yesterday to Epiphany 
and to Chinese logic suggest two observations—

First, a political interpretation of the number "64.89" 
from Monday's Shanghai stock market index
yielded the date  6/4 in 1989—

IMAGE- Google search on '64.89 Shanghai'

Second, an interpretation of 64.89 as the number  64 in 1989
(on the Feast of the Epiphany, January 6)—

Note of January 6, 1989 showing 
the 64 hexagrams in the von Franz style

No connection of the number  64 with the date  6/4 is implied.

Thursday, April 5, 2012

In the Details

Filed under: General — m759 @ 6:00 PM

An image from 62 years ago:

Click image for details.

For the religious significance of "62," see Strong Emergence Illustrated.

See also a related search.

Sunday, February 5, 2012

Sunday Morning

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

CBS Sunday Morning 's piece on
the number pi  today suggested…

Hexagram 20, Contemplation/View,
from the website Rightreading.com

Saturday, February 4, 2012

Ein Satz

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


4 × 5 =  ̲2̲0̲  .


Related material—

The link "Ten is a Hen"
in today's 10 AM post, and
Carl Schoettler on Barbara Wilson's
novel Gaudi Afternoon

"She's told a detective story without violence,
murder, mayhem, massacre, or even explicit sex
although there are a couple of sly double entendres.
She pays an homage to Dashiell Hammett
when Cassandra tells the gay saxophonist
 she's Brigid O'Shaughnessy.
Brigid O'Shaughnessy, of course, is the woman
Sam Spade won't play the sap for
in 'The Maltese Falcon.'"

Judy Davis, shown above in
"One Against the Wind," was the star
of the film version of Wilson's novel.

Sunday, October 23, 2011

Accentuate the Positive–

Filed under: General,Geometry — m759 @ 3:59 PM

A Date with Erin


Related material suggested by today's
midday New York Lottery— 032 and 7537—

Richard Wilhelm
on I Ching  Hexagram 32:

Hexagram 32, Duration, of the I Ching


“Duration is… not a state of rest, for mere standstill
is regression. Duration is rather the self-contained
and therefore self-renewing movement of
an organized, firmly integrated whole
[click on link for an example], taking place
in accordance with immutable laws
and beginning anew at every ending.”

and the date 7/5/37—


Wednesday, October 19, 2011


Filed under: General,Geometry — m759 @ 4:10 PM

Midday NY lottery on Oct. 19, 2011— 043 and 7531.

The latter is the birth date (7/5/31) of Jerry Slocum,
Hughes Aircraft designer and puzzle enthusiast.

For the former, see Hexagram 43 in Geometry of the I Ching.

Thursday, October 6, 2011


Filed under: General — m759 @ 9:29 PM

From post 4017 in this journal (do not click links)—

"Thanks to University Diaries  for an entry on Clancy Martin,
a philosophy professor in the 'show me' state, and his experiences with AA."

Neither link in this quote works anymore.
See instead Martin in the London Review of Books .

Lottery hermeneutics, however, still seems usable.


Today's midday NY lottery "163" may be taken as a sequel
to both the page number "162" in today's noon post


Humboldt's Gift , page 163 (Penguin Classics, 1996)

— and a sequel to University Diaries ' meditation today on the Nobel literature prize,
which includes a quote from the winner:

"At last my life returns. My name appears like an angel.
Outside the walls a trumpet signal blows…. It is I! It is I!"

Tomas Tranströmer, "The Name"

As for the evening NY numbers 014 and 5785, see Hexagram 14,
Not Even Wrong , and 5/7/85.

Friday, August 19, 2011


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

"Francis Bacon used the phrase instantia crucis,  'crucial instance,' to refer to something in an experiment that proves one of two hypotheses and disproves the other. Bacon's phrase was based on a sense of the Latin word crux,  'cross,' which had come to mean 'a guidepost that gives directions at a place where one road becomes two,' and hence was suitable for Bacon's metaphor."

The American Heritage® Dictionary of the English Language, Fourth Edition

Such a cross: St. Andrew's.  Some context—


X Marks the Spot  scene, "The Last Crusade"

Related symbology for Dan Brown—

Neville's Labrys and Notes on Mathematics and Narrative.

Saturday, July 2, 2011

Mass Appeal

Filed under: General — m759 @ 3:57 AM

A portrait reproduced here on October 11, 2007

The image “http://www.log24.com/log/pix07A/071011-vonNeumann.jpg” cannot be displayed, because it contains errors.



The above was suggested by yesterday's New York Lottery and by
items dated February  8, 1958 (date of receipt of an AMS Bulletin  article)
and February 1, 1953 (newspaper article on applied game theory).

(This covers three of the four relevant lottery numbers.
For the fourth, see 9/19 and recent discussions of
The End of Mr. Y  (Sunday Dinner and A Link for Sunrise)).

See also Saint  John Neumann.

Friday, June 3, 2011

Brightness at Noon (continued)

Filed under: General — m759 @ 12:00 PM



Hexagram 55
Abundance (Fullness)  

"Be not sad.
 Be like the sun at midday."

Sunday, May 22, 2011


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

A year ago today—

2:02 AM EDT

   Art Space

Box symbol

Pictorial version
of Hexagram 20,
Contemplation (View)


Space: what you damn well have to see.
– James Joyce, Ulysses

10:31 AM EDT

Image-- The Case of the Lyche Gate Asterisk

* See Vonnegut.

Friday, May 20, 2011

Lottery Hermeneutics (continued)

Filed under: General — Tags: — m759 @ 3:09 PM

Recent New York Lottery numbers—


The interpretation of "056" in yesterday's
The Aleph, the Lottery, and the Eightfold Way
was not without interest, but the interpretation there
of "236" was somewhat lacking in poetic resonance.

For aspiring students of lottery hermeneutics,
here are some notes that may help. The "236" may
be reinterpreted as a page number in Stevens's
Collected Poems . It then resonates rather nicely
("answers when I ask," "visible and responsive")
with yesterday evening's "434"—


For today's midday "022," see Hexagram 22: Grace in the context of the following—


As for yesterday afternoon's 609, see a particular Stevens-related page with that number…

IMAGE- Review of 'The Dome and the Rock'

For "a body of thought or poetry larger than the subject's," see The Dome of  the Rock.

Saturday, April 30, 2011

Sabato Tombstone

Filed under: General — m759 @ 8:28 PM

IMAGE- Sabato on his own tombstone in 'Angel of Darkness'

Related material:
  (Click images for details) —

IMAGE- The number 11 formed by twin switchblades, '12 Angry Men'

Exhibit B

IMAGE- Julie Taymor

Julie Taymor

  IMAGE- Hexagram 11: PEACE
Plato, Pegasus, and
the Evening Star

Monday, April 25, 2011

The Kristen Effect

Filed under: General — m759 @ 10:31 PM

From the author of The Abacus Conundrum


Harlan Kane's sequel to The Apollo Meme


IMAGE- Kristen Wiig, 'Cock and Bull Story'

"Thus the universal mutual attraction between the sexes is represented."
Hexagram 31

Friday, April 8, 2011

Concepts of Space

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

Part I — Roberta Smith in today's New York Times

"… the argument that painting may ultimately be about
little more than the communication of some quality of
light and space, however abstract or indirect."

– Review of "Rooms With a View" at the Met

Box symbol

Pictorial version
of Hexagram 20,
Contemplation (View)


Space: what you damn well have to see.
– James Joyce, Ulysses

Part II — Window from A Crooked House

"Teal lifted the blind a few inches. He saw nothing, and raised it a little more—still nothing. Slowly he raised it until the window was fully exposed. They gazed out at—nothing.

Nothing, nothing at all. What color is nothing? Don't be silly! What shape is it? Shape is an attribute of something . It had neither depth nor form. It had not even blackness. It was nothing ."

Part III — Not So Crooked: The Cabinet of Dr. Montessori

An April 5 Wall Street Journal  article on Montessori schools, and…


A cabinet from Dr. Montessori's own
explanation of her method

Part IV — Pilate Goes to Kindergarten and The Seven

Wednesday, March 16, 2011

Time and Chance (continued)

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

Accidental Time and Space

New York Lottery today— midday 987, evening 522.


The midday 987 may be interpreted as "…nine, eight, seven, …."—

"The countdown as we know it, 10-9-8-u.s.w.,
was invented by Fritz Lang in 1929 for
the Ufa film Die Frau im Mond . He put it into
the launch scene to heighten the suspense.
'It is another of my damned "touches,"' Fritz Lang said."

Gravity's Rainbow


The evening 522 suggests the date 5/22. From that date last year

Art Space (2:02 AM EDT)

Box symbol

Pictorial version
of Hexagram 20,
Contemplation (View)


Space: what you damn well have to see.
– James Joyce, Ulysses

Wednesday, February 9, 2011

An Abstract Window

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

The sliding window in blue below


Click for the web page shown.

is an example of a more general concept.

Such a sliding window,* if one-dimensional of length n , can be applied to a sequence of 0's and 1's to yield a sequence of n-dimensional vectors. For example— an "m-sequence" (where the "m" stands for "maximum length") of length 63 can be scanned by a length-6 sliding window to yield all possible 6-dimensional binary vectors except (0,0,0,0,0,0).

For details, see A Galois Field


The image is from Bert Jagers at his page on the Galois field GF(64) that he links to as "A Field of Honor."

For a discussion of the m-sequence shown in circular form above, see Jagers's  "Pseudo-Random Sequences from GF(64)." Here is a noncircular version of the length-63 m-sequence described by Jagers (with length scale below)—


This m-sequence may be viewed as a condensed version of 63 of the 64 I Ching  hexagrams. (See related material in this journal.)

For a more literary approach to the window concept, see The Seventh Symbol (scroll down after clicking).

* Moving windows also appear (in a different way) In image processing, as convolution kernels .

Wednesday, January 26, 2011

Requiem for a Screenwriter*

Filed under: General — m759 @ 1:00 PM

Click for obit.


Hexagram 61

Still Point.

* Notably, of the film "Downfall" (Der Untergang ).
   "Has time rewritten every line?" —Streisand

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