Wednesday, December 11, 2019

Miracle Octad Generator Structure

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

Miracle Octad Generator — Analysis of Structure

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

Sunday, December 25, 2016

Miracle Eight Generator

Filed under: General — m759 @ 9:00 AM

Merry Xmas to Katherine Neville.

Friday, March 21, 2014

Three Constructions of the Miracle Octad Generator

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

IMAGE- Two constructions, by Turyn/Curtis, and by Conway, of the Miracle Octad Generator

See also a Log24 post on this subject from Dec. 14, 2013,
especially (scroll down) the update of March 9, 2014.

Related material on the Turyn-Curtis construction
from the University of Cambridge —

— Slide by “Dr. Parker” — Apparently Richard A. Parker —
Lecture 4, “Discovering M24,” in slides for lectures 1-8 from lectures
at Cambridge in 2010-2011 on “Sporadic and Related Groups.”
See also the Parker lectures of 2012-2013 on the same topic.

A third construction of Curtis’s 35  4×6  1976 MOG arrays would use
Cullinane’s analysis of the 4×4 subarrays’ affine and projective structure,
and point out the fact that Conwell’s 1910 correspondence of the 35
4+4-partitions of an 8-set with the 35 lines of the projective 3-space
over the 2-element field, PG(3, 2), is essentially the same correspondence
as that constituting Curtis’s 1976 MOG.

See The Diamond Theorem,  Finite RelativityGalois Space,
Generating the Octad Generator, and The Klein Correspondence.

Update of March 22-March 23 —

Adding together as (0,1)-matrices over GF(2) the black parts (black
squares as 1’s, all other squares as 0’s) of the 35  4×6 arrays of the 1976
Curtis MOG would then reveal*  the symmetric role played in octads
by what Curtis called the heavy brick , and so reveal also the action of
S3 on the three Curtis bricks that leaves invariant the set of all 759
octads of the S(5, 8, 24) constructed from the 35  MOG arrays.  For more
details of this “by-hand” construction, see Geometry of the 4×4 Square.
For the mathematical properties of the S(5, 8, 24), it is convenient to
have a separate construction,  not  by hand (such as Turyn’s), of the
extended binary Golay code. See the Brouwer preprint quoted above.

* “Then a miracle occurs,” as in the classic 1977 Sidney Harris cartoon.

Illustration of array addition from March 23 —

IMAGE- Discovery of the S_3 action on bricks in the Conwell-Cullinane 'by-hand' approach to octad-building

Saturday, May 2, 2020

Turyn’s Octad Theorem: The Next Level*

Filed under: General — Tags: — m759 @ 11:24 AM

From  the obituary of a game inventor  who reportedly died
on Monday, February 25, 2013 —

” ‘He was hired because of the game,’ Richard Turyn,
a mathematician who worked at Sylvania, told
the Washington Post in a 2004 feature on Diplomacy.”

* For the theorem, see Wolfram Neutsch,  Coordinates .
(Published by de Gruyter, 1996. See pp. 761-766.)

Having defined (pp. 751-752) the Miracle Octad Generator (MOG)
as a 4×6 array to be used with Conway’s “hexacode,” Neutsch says . . .

“Apart from the three constructions of the Golay codes
discussed at length in this book (lexicographic and via
the MOG or the projective line), there are literally
dozens of alternatives. For lack of space, we have to
restrict our attention to a single example. It has been
discovered by Turyn and can be connected in a very
beautiful way with the Miracle Octad Generator….

To this end, we consider the natural splitting of the MOG into
three disjoint octads L, M, R (‘left’, ‘middle’, and ‘right’ octad)….”

— From page 761

“The theorem of Turyn”  is on page 764

Thursday, April 30, 2020

Walpurgisnacht Geometry

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

A version more explicitly connected to finite geometry —

For the six synthematic totals , see The Joy of Six.

Wednesday, April 29, 2020

Curtis at Pilsen, Thursday, July 5, 2018

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

For an account by R. T. Curtis of how he discovered the Miracle Octad Generator,
see slides by Curtis, “Graphs and Groups,” from his talk on July 5, 2018, at the
Pilsen conference on algebraic graph theory, “Symmetry vs. Regularity: The first
50 years since Weisfeiler-Leman stabilization” (WL2018).

See also “Notes to Robert Curtis’s presentation at WL2018,” by R. T. Curtis.

Meanwhile, here  on July 5, 2018

Simultaneous perspective does not look upon language as a path because it is not the search for meaning that orients it. Poetry does not attempt to discover what there is at the end of the road; it conceives of the text as a series of transparent strata within which the various parts—the different verbal and semantic currents—produce momentary configurations as they intertwine or break apart, as they reflect each other or efface each other. Poetry contemplates itself, fuses with itself, and obliterates itself in the crystallizations of language. Apparitions, metamorphoses, volatilizations, precipitations of presences. These configurations are crystallized time: although they are perpetually in motion, they always point to the same hour—the hour of change. Each one of them contains all the others, each one is inside the others: change is only the oft-repeated and ever-different metaphor of identity.

— Paz, Octavio. The Monkey Grammarian
(Kindle Locations 1185-1191).
Arcade Publishing. Kindle Edition.

The 2018 Log24 post containing the above Paz quote goes on to quote
remarks by Lévi-Strauss. Paz’s phrase “series of transparent strata”
suggests a review of other remarks by Lévi-Strauss in the 2016 post
Key to  All Mythologies.

Wednesday, February 19, 2020

Aitchison’s Octads

Filed under: General — Tags: , — m759 @ 11:36 AM

The 759 octads of the Steiner system S(5,8,24) are displayed
rather neatly in the Miracle Octad Generator of R. T. Curtis.

A March 9, 2018, construction by Iain Aitchison* pictures the
759 octads on the faces of a cube , with octad elements the
24 edges of a  cuboctahedron :

The Curtis octads are related to symmetries of the square.

See my webpage "Geometry of the 4×4 square" from March 2004.
Aitchison's p. 42 slide includes an illustration from that page —

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

Note that essentially the same model as Aitchison's can be pictured 
by using, instead of the 24 edges of a cuboctahedron, the 24 outer 
faces of subcubes in the eightfold cube .

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

   Image from Christmas Day 2005.

See also Aitchison in this journal.


Saturday, December 14, 2019

Colorful Tale

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


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

The above image is from 

"A Four-Color Theorem:
Function Decomposition Over a Finite Field,"

These partitions of an 8-set into four 2-sets
occur also in Wednesday night's post
Miracle Octad Generator Structure.

This  post was suggested by a Daily News
story from August 8, 2011, and by a Log24
post from that same date, "Organizing the
Mine Workers
" —


Thursday, October 31, 2019

56 Triangles

Filed under: General — Tags: — m759 @ 8:09 AM

The post "Triangles, Spreads, Mathieu" of October 29 has been
updated with an illustration from the Curtis Miracle Octad Generator.

Related material — A search in this journal for "56 Triangles."

Saturday, September 21, 2019

Annals of Random Fandom

Filed under: General — Tags: , — m759 @ 5:46 PM

For Dan Brown fans …

… and, for fans of The Matrix, another tale
from the above death date: May 16, 2019 —

An illustration from the above
Miracle Octad Generator post:

Related mathematics — Tetrahedron vs. Square.

Wednesday, July 24, 2019

The Batty Farewell

Filed under: General — Tags: — m759 @ 8:40 PM

Adam Rogers today on "Rutger Hauer in Blade Runner , playing
the artificial* person Roy Batty in his death scene." 

* See the word "Artifice" in this  journal,
   as well as Tears in Rain . . .

Game Over

The film "The Matrix," illustrated
Coordinates for generating the Miracle Octad Generator

and Adam Rogers in
    the previous post.


Saturday, May 4, 2019

The Chinese Jars of Shing-Tung Yau

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

The title refers to Calabi-Yau spaces.

T. S. Eliot —

Four Quartets

. . . Only by the form, the pattern,
Can words or music reach
The stillness, as a Chinese jar still
Moves perpetually in its stillness.

A less "cosmic" but still noteworthy code — The Golay code.

This resides in a 12-dimensional space over GF(2).

Related material from Plato and R. T. Curtis

Counting symmetries with the orbit-stabilizer theorem

A related Calabi-Yau "Chinese jar" first described in detail in 1905

Illustration of K3 surface related to Mathieu moonshine

A figure that may or may not be related to the 4x4x4 cube that
holds the classical  Chinese "cosmic code" — the I Ching


Wednesday, May 1, 2019

The Medium and the Message

Filed under: General — Tags: — m759 @ 6:45 PM

In memory of Quentin Fiore — from a Log24 search for McLuhan,
an item related to today's previous post . . .

Related material from Log24 on the above-reported date of death —

See also, from a search for Analogy in this journal . . .


Thursday, March 7, 2019

In Reality

Filed under: General — m759 @ 11:45 AM

The previous post, quoting a characterization of the R. T. Curtis
Miracle Octad Generator , describes it as a "hand calculator ."

Other views 

A "natural diagram " —


A geometric object

Counting symmetries with the orbit-stabilizer theorem.

Wednesday, March 6, 2019

A Hand Calculator

Filed under: General — m759 @ 10:28 PM

Tuesday, March 5, 2019

A Block Design 3-(16,4,1) as a Steiner Quadruple System:

Filed under: General — Tags: , — m759 @ 11:19 AM

A Midrash for Wikipedia 

Midrash —

Related material —


The Miracle Octad Generator (MOG), the affine 4-space over GF(2), and the Cullinane diamond theorem

Friday, March 1, 2019

Wikipedia Scholarship (Continued)

Filed under: General — Tags: , , — m759 @ 12:45 PM

This post continues a post from yesterday on the square model of
PG(3,2) that apparently first appeared (presented as such*) in . . .

Cullinane, "Symmetry invariance in a diamond ring,"
Notices of the AMS , pp. A193-194, Feb. 1979.

The Cullinane diamond theorem, AMS Notices, Feb. 1979, pp. A-193-194

Yesterday's Wikipedia presentation of the square model was today
revised by yet another anonymous author —

Revision history accounting for the above change from yesterday —

The jargon "rm OR" means "remove original research."

The added verbiage about block designs is a smokescreen having
nothing to do with the subject, which is square  representation
of the 35 points and lines.

* The 35 squares, each consisting of four 4-element subsets, appeared earlier
   in the Miracle Octad Generator (MOG) of R. T. Curtis (published in 1976).
  They were not at that time  presented as constituting a finite geometry, 
  either affine (AG(4,2)) or projective (PG(3,2)).

Friday, February 22, 2019

Back Issues of AMS Notices

Filed under: General — m759 @ 3:04 PM

From the online home page of the new March issue —

Feb. 22, 2019 — AMS Notices back issues now available.

For instance . . .

The Cullinane diamond theorem, AMS Notices, Feb. 1979, pp. A-193-194

Related material now at Wikipedia

The Miracle Octad Generator (MOG), the affine 4-space over GF(2), and the Cullinane diamond theorem

Thursday, February 7, 2019

Geometry of the 4×4 Square: The Kummer Configuration

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

From the series of posts tagged Kummerhenge

A Wikipedia article relating the above 4×4 square to the work of Kummer —

A somewhat more interesting aspect of the geometry of the 4×4 square
is its relationship to the 4×6 grid underlying the Miracle Octad Generator
(MOG) of R. T. Curtis.  Hudson's 1905 classic Kummer's Quartic Surface
deals with the Kummer properties above and also foreshadows, without
explicitly describing, the finite-geometry properties of the 4×4 square as
a finite affine 4-space — properties that are of use in studying the Mathieu
group M24  with the aid of the MOG.

Sunday, December 2, 2018

Symmetry at Hiroshima

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

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




Construction of highly symmetric Riemann surfaces, related manifolds, and some exceptional objects, I, II


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

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

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

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

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

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

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

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

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

Wednesday, October 3, 2018

Adamantine Meditation

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


A Catholic philosopher —

Related art —

Image result for mog miracle octad bricks

Sunday, September 23, 2018

Three Times Eight

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

The New York Times 's Sunday School today —

I prefer the three bricks of the Miracle Octad Generator —

Image result for mog miracle octad bricks

Wednesday, July 18, 2018


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

From "The Educated Imagination: A Website Dedicated
to Northrop Frye
" —

"In one of the notebooks for his first Bible book Frye writes,

'For at least 25 years I’ve been preoccupied by
the notion of a key to all mythologies.' . . . .

Frye made a valiant effort to provide a key to all mythology,
trying to fit everything into what he called the Great Doodle. . . ."

From a different page at the same website —

Here Frye provides a diagram of four sextets.

I prefer the Miracle Octad Generator of R. T. Curtis —

Counting symmetries with the orbit-stabilizer theorem.

Monday, June 25, 2018

The Trials of Device

Filed under: General — Tags: — m759 @ 9:34 AM

"A blank underlies the trials of device."

Wallace Stevens

"Designing with just a blank piece of paper is very quiet."

Kate Cullinane

Related material —

An image posted at 12 AM ET December 25, 2014:

The image stands for the
phrase "five by five,"
meaning "loud and clear."

Other posts featuring the above 5×5 square with some added structure:

Saturday, June 16, 2018

For June 16

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

"But perhaps the desire for story
is what gets us into trouble to begin with."

Sarah Marshall on June 5, 2018

"Beckett wrote that Joyce believed fervently in
the significance of chance events and of
random connections. ‘To Joyce reality was a paradigm,
an illustration of a possibly unstateable rule
According to this rule, reality, no matter how much
we try to manipulate it, can only shift about
in continual movement, yet movement
limited in its possibilities’ giving rise to
‘the notion of the world where unexpected simultaneities
are the rule.’ In other words, a coincidence  is actually
just part of a continually moving pattern, like a kaleidoscope.
Or Joyce likes to put it, a ‘collideorscape’."

— Gabrielle Carey, "Breaking Up with James Joyce,"
Sydney Review of Books , 15 June 2018

Carey's carelessness with quotations suggests a look at another
author's quoting of Ellmann on Joyce

Friday, March 2, 2018

A Snowflake on the Iceberg

Filed under: General — m759 @ 11:00 AM

A scholium on the previous post, "Mother Ship Meets Mother Church" —

"Consider Saint Hedwig of Silesia (1174–1243), for example,
who is elegantly depicted in this catalogue, striking a pose
with her prayer book, rosary, and Virgin and child statue
(a reminder of the legitimacy of her sex), along with boots
slung over her elbow so she could walk barefoot like
the apostles. Between miracles, she was also known to 
supervise con­struction of new convents. Hedwig is but
a snowflake on the iceberg  of the extra­ordinary role of
actual women in the Middle Ages, to which more evidence
is added continually thanks the Feminae database."

— From "Wonder Women," by Matthew J. Milliner

Tuesday, May 2, 2017

Image Albums

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

Pinterest boards uploaded to the new m759.net/piwigo

Diamond Theorem 

Diamond Theorem Correlation

Miracle Octad Generator

The Eightfold Cube

Six-Set Geometry

Diamond Theory Cover

Update of May 2 —

Four-Color Decomposition

Binary Galois Spaces

The Galois Tesseract

Update of May 3 —

Desargues via Galois

The Tetrahedral Model

Solomon's Cube

Update of May 8 —

Art Space board created at Pinterest

Thursday, March 30, 2017

The Internet Abyss

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

Suggested by the previous post, The Crimson Abyss

“Wer mit Ungeheuern kämpft, mag zusehn, dass er nicht dabei
zum Ungeheuer wird. Und wenn du lange in einen Abgrund blickst,
blickt der Abgrund auch in dich hinein.”

“He who fights with monsters should look to it that he himself
does not become a monster. And if you gaze long into an abyss,
the abyss also gazes into you.”

—  Nietzsche, Beyond Good and Evil , Aphorism 146

From the Internet Abyss on Red October Day, October 25, 2010 —

An image reproduced in this  journal on that same day

Image-- 'Then a miracle occurs' cartoon

Cartoon by S.Harris

Wednesday, December 7, 2016

Spreads and Conwell’s Heptads

Filed under: General,Geometry — m759 @ 7:11 PM

For a concise historical summary of the interplay between
the geometry of an 8-set and that of a 16-set that is
involved in the the Miracle Octad Generator approach
to the large Mathieu group M24, see Section 2 of 

Alan R. Prince
A near projective plane of order 6 (pp. 97-105)
Innovations in Incidence Geometry
Volume 13 (Spring/Fall 2013).

This interplay, notably discussed by Conwell and
by Edge, involves spreads and Conwell's heptads .

Update, morning of the following day (7:07 ET) — related material:

See also "56 spreads" in this  journal.

Friday, December 2, 2016

A Small Witt Design*

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

The New York Times 's  online T Magazine  yesterday —

"A version of this article appears in print on December 4, 2016, on page
M263 of T Magazine with the headline: The Year of Magical Thinking."

* Thanks to Emily Witt for inadvertently publicizing the
   Miracle Octad Generator  of R. T. Curtis, which
   summarizes the 759 octads found in the large Witt design.

Tuesday, September 13, 2016

Parametrizing the 4×4 Array

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

The previous post discussed the parametrization of 
the 4×4 array as a vector 4-space over the 2-element 
Galois field GF(2).

The 4×4 array may also be parametrized by the symbol
0  along with the fifteen 2-subsets of a 6-set, as in Hudson's
1905 classic Kummer's Quartic Surface

Hudson in 1905:

These two ways of parametrizing the 4×4 array — as a finite space
and as an array of 2-element sets —  were related to one another
by Cullinane in 1986 in describing, in connection with the Curtis
"Miracle Octad Generator,"  what turned out to be 15 of Hudson's
1905 "Göpel tetrads":

A recap by Cullinane in 2013:

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

Click images for further details.

Friday, August 19, 2016

Ex Machina

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

For the Symmetry Dancers of CERN

1947 Opening

Filed under: General,Geometry — Tags: — m759 @ 1:06 PM
1. e4  See also Geometry of the I Ching
and "Miracle on 34th Street."

Tuesday, May 24, 2016

Rosenhain and Göpel Revisited

The authors Taormina and Wendland in the previous post
discussed some mathematics they apparently did not know was
related to a classic 1905 book by R. W. H. T. Hudson, Kummer's
Quartic Surface

"This famous book is a prototype for the possibility
of explaining and exploring a many-faceted topic of
research, without focussing on general definitions,
formal techniques, or even fancy machinery. In this
regard, the book still stands as a highly recommendable,
unparalleled introduction to Kummer surfaces, as a
permanent source of inspiration and, last but not least, 
as an everlasting symbol of mathematical culture."

— Werner Kleinert, Mathematical Reviews ,
     as quoted at Amazon.com

Some 4×4 diagrams from that book are highly relevant to the
discussion by Taormina and Wendland of the 4×4 squares within
the 1974 Miracle Octad Generator of R. T. Curtis that were later,
in 1987, described by Curtis as pictures of the vector 4-space over
the two-element Galois field GF(2).

Hudson did not think of his 4×4 diagrams as illustrating a vector space,
but he did use them to picture certain subsets of the 16 cells in each
diagram that he called Rosenhain and Göpel tetrads .

Some related work of my own (click images for related posts)—

Rosenhain tetrads as 20 of the 35 projective lines in PG(3,2)

IMAGE- Desargues's theorem in light of Galois geometry

Göpel tetrads as 15 of the 35 projective lines in PG(3,2)

Anticommuting Dirac matrices as spreads of projective lines

Related terminology describing the Göpel tetrads above

Ron Shaw on symplectic geometry and a linear complex in PG(3,2)

Tuesday, April 19, 2016

The Folding

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


A recent post about the eightfold cube  suggests a review of two
April 8, 2015, posts on what Northrop Frye called the ogdoad :

As noted on April 8, each 2×4 "brick" in the 1974 Miracle Octad Generator
of R. T. Curtis may be constructed by folding  a 1×8 array from Turyn's
1967 construction of the Golay code.

Folding a 2×4 Curtis array yet again  yields the 2x2x2 eightfold cube .

Those who prefer an entertainment  approach to concepts of space
may enjoy a video (embedded yesterday in a story on theverge.com) —
"Ghost in the Shell: Identity in Space." 

Monday, April 11, 2016

Combinatorial Spider

Filed under: General,Geometry — Tags: , — m759 @ 1:16 PM

“Chaos is order yet undeciphered.”

— The novel The Double , by José Saramago,
on which the film "Enemy" was based

Some background for the 2012 Douglas Glover
"Attack of the Copula Spiders" book
mentioned in Sunday's Synchronicity Check

  • "A vision of Toronto as Hell" — Douglas Glover in the
    March 25, 2011, post Combinatorial Delight
  • For Louise Bourgeois — a post from the date of Galois's death—


  • For Toronto — Scene from a film that premiered there
    on Sept. 8, 2013:

Sunday, April 10, 2016

In Memoriam

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

A great cartoonist died on Friday.

Related religious art — Ogdoads and Miracle Cartoon.

Friday, April 8, 2016

Ogdoads by Curtis

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

As was previously noted here, the construction of the Miracle Octad Generator
of R. T. Curtis in 1974 involved his "folding" the 1×8 octads constructed in 1967
by Turyn into 2×4 form.

This resulted in a way of picturing a well-known correspondence (Conwell, 1910)
between partitions of an 8-set and lines of the projective 3-space PG(3,2).

For some background related to the "ogdoads" of the previous post, see
A Seventh Seal (Sept. 15, 2014).

Monday, February 1, 2016

Religious Note

Filed under: General — m759 @ 1:00 PM

See also the previous post.

Historical Note

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

Possible title

A new graphic approach 
to an old geometric approach
to a new combinatorial approach
to an old algebraic approach
to M24

Friday, November 13, 2015

A Connection between the 16 Dirac Matrices and the Large Mathieu Group

Note that the six anticommuting sets of Dirac matrices listed by Arfken
correspond exactly to the six spreads in the above complex of 15 projective
lines of PG(3,2) fixed under a symplectic polarity (the diamond theorem
). As I noted in 1986, this correlation underlies the Miracle
Octad Generator of R. T. Curtis, hence also the large Mathieu group.


Arfken, George B., Mathematical Methods for Physicists , Third Edition,
Academic Press, 1985, pages 213-214

Cullinane, Steven H., Notes on Groups and Geometry, 1978-1986

Related material:

The 6-set in my 1986 note above also appears in a 1996 paper on
the sixteen Dirac matrices by David M. Goodmanson —

Background reading:

Ron Shaw on finite geometry, Clifford algebras, and Dirac groups 
(undated compilation of publications from roughly 1994-1995)—

Saturday, September 19, 2015

Geometry of the 24-Point Circle

Filed under: General,Geometry — Tags: — m759 @ 1:13 AM

The latest Visual Insight  post at the American Mathematical
Society website discusses group actions on the McGee graph,
pictured as 24 points arranged in a circle that are connected
by 36 symmetrically arranged edges.

Wikipedia remarks that

"The automorphism group of the McGee graph
is of order 32 and doesn't act transitively upon
its vertices: there are two vertex orbits of lengths
8 and 16."

The partition into 8 and 16 points suggests, for those familiar
with the Miracle Octad Generator and the Mathieu group M24,
the following exercise:

Arrange the 24 points of the projective line
over GF(23) in a circle in the natural cyclic order
, 1, 2, 3,  , 22, 0 ).  Can the McGee graph be
modeled by constructing edges in any natural way?

Image that may or may not be related to the extended binary Golay code and the large Witt design

In other words, if the above set of edges has no
"natural" connection with the 24 points of the
projective line over GF(23), does some other 
set of edges in an isomorphic McGee graph
have such a connection?

Update of 9:20 PM ET Sept. 20, 2015:

Backstory: A related question by John Baez
at Math Overflow on August 20.

Thursday, August 27, 2015

Tears in the Rain

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

For a Norwegian historian

Game Over

The film "The Matrix," illustrated
Coordinates for generating the Miracle Octad Generator

Monday, May 18, 2015

Ex Nihilo

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

Yesterday morning's quote from The Politics of Experience :

"Creation ex nihilo  has been pronounced impossible even for
God. But we are concerned with miracles."

Related material:

Curtis + Turyn in this journal and Sidney Harris on theology —

Sunday, May 17, 2015

Metaphysics for Sunday

Filed under: General — m759 @ 8:00 AM

From Chapter 1 of R. D. Laing's The Politics of Experience —

"An activity has to be understood in terms of the experience
from which it emerges. These arabesques that mysteriously
embody mathematical truths only glimpsed by a very few —
how beautiful, how exquisite — no matter that they were
the threshing and thrashing of a drowning man.

We are here beyond all questions except those of being
and nonbeing, incarnation, birth, life and death.

Creation ex nihilo  has been pronounced impossible even for
God. But we are concerned with miracles. We must hear the
music of those Braque guitars (Lorca*)."

See also Christmas Day, 2009.

*  Update of Sunday afternoon: A search for the Lorca quote yields
   no result, but Cocteau wrote that "Mon rêve, en musique, serait
   d'entendre la musique des guitares de Picasso. "
   (Oeuvres complètes , Vol. 10, p. 107)

   See also Stevens + "Blue Guitar" in this journal.

Monday, January 12, 2015

Points Omega*

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

The previous post displayed a set of
24 unit-square "points" within a rectangular array.
These are the points of the 
Miracle Octad Generator  of R. T. Curtis.

The array was labeled  Ω
because that is the usual designation for
a set acted upon by a group:

* The title is an allusion to Point Omega , a novel by
   Don DeLillo published on Groundhog Day 2010.
   See "Point Omega" in this journal.

Thursday, January 8, 2015


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


From the abstract of a talk, "Extremal Lattices," at TU Graz
on Friday, Jan. 11, 2013, by Prof. Dr. Gabriele Nebe
(RWTH Aachen) —

"I will give a construction of the extremal even
unimodular lattice Γ of dimension 72  I discovered
in summer 2010. The existence of such a lattice
was a longstanding open problem. The
construction that allows to obtain the
minimum by computer is similar to the one of the
Leech lattice from E8 and of the Golay code from
the Hamming code (Turyn 1967)."

On an earlier talk by Nebe at Oberwolfach in 2011 —

"Exciting new developments were presented by
Gabriele Nebe (Extremal lattices and codes ) who
sketched the construction of her recently found
extremal lattice in 72 dimensions…."

Nebe's Oberwolfach slides include one on 
"The history of Turyn's construction" —

Nebe's list omits the year 1976. This was the year of
publication for "A New Combinatorial Approach to M24"
by R. T. Curtis, the paper that defined Curtis's 
"Miracle Octad Generator."

Turyn's 1967 construction, uncredited by Curtis,
was the basis for Curtis's octad-generator construction.

See Turyn in this journal.

Monday, November 3, 2014

Gods and Giants

Filed under: General — Tags: — m759 @ 2:01 PM

A weblog reports Chris Rock's remarks
on Saturday Night Live this past weekend:

"It’s America, we commercialize everything.
Look at what we did to Christmas.
Christmas.  Christmas is Jesus’ birthday.
It’s Jesus’ birthday.  Now, I don’t know Jesus
but from what I’ve read, Jesus is the least
materialistic person to ever roam the earth.
No bling on Jesus.
Jesus kept a low profile and we turned his
birthday into the most materialistic day of the
year.  Matter of fact, we have the Jesus birthday
season.  It’s a whole season of materialism.
Then, at the end of the Jesus birthday season
we have the nerve to have an economist come
on TV and tell you how horrible the Jesus birthday
season was this year.  Oh, we had a horrible Jesus’
birthday this year.  Hopefully, business will pick up
by his Crucifixion.”

Related music and image:

"Show us the way to the next little girl …"

Natalie Wood in "Miracle on 34th Street" (1947)

Related non-materialistic meditations:
The Rhetoric of Abstract Concepts and Gods and Giants.

Saturday, October 25, 2014

Foundation Square

Filed under: General,Geometry — Tags: — m759 @ 2:56 PM

In the above illustration of the 3-4-5 Pythagorean triangle,
the grids on each side may be regarded as figures of
Euclidean  geometry or of Galois  geometry.

In Euclidean geometry, these grids illustrate a property of
the inner triangle.

In elementary Galois geometry, ignoring the connection with
the inner triangle, the grids may be regarded instead as
illustrating vector spaces over finite (i.e., Galois) fields.
Previous posts in this journal have dealt with properties of
the 3×3 and 4×4 grids.  This suggests a look at properties of
the next larger grid, the 5×5 array, viewed as a picture of the
two-dimensional vector space (or affine plane) over the finite
Galois field GF(5) (also known as ℤ5).

The 5×5 array may be coordinatized in a natural way, as illustrated
in (for instance) Matters Mathematical , by I.N. Herstein and
Irving Kaplansky, 2nd ed., Chelsea Publishing, 1978, p. 171:

See Herstein and Kaplansky for the elementary Galois geometry of
the 5×5 array.

For 5×5 geometry that is not so elementary, see…

Hafner's abstract:

We describe the Hoffman-Singleton graph geometrically, showing that
it is closely related to the incidence graph of the affine plane over ℤ5.
This allows us to construct all automorphisms of the graph.

The remarks of Brouwer on graphs connect the 5×5-related geometry discussed
by Hafner with the 4×4 geometry related to the Steiner system S(5,8,24).
(See the Miracle Octad Generator of R. T. Curtis and the related coordinatization
by Cullinane of the 4×4 array as a four-dimensional vector space over GF(2).)

Monday, September 15, 2014

The Eight

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

The image at the end of today’s previous post A Seventh Seal
suggests a review of posts on Katherine Neville’s The Eight .

Update of 1:25 PM ET on Sept. 15, 2014:

Neville’s longtime partner is neurosurgeon and cognitive theorist
Karl H. Pribram. A quote from one of his books:

See also Sense and Sensibility.

Sunday, August 31, 2014

Sunday School

Filed under: General,Geometry — Tags: — m759 @ 9:00 AM

The Folding

Cynthia Zarin in The New Yorker , issue dated April 12, 2004—

“Time, for L’Engle, is accordion-pleated. She elaborated,
‘When you bring a sheet off the line, you can’t handle it
until it’s folded, and in a sense, I think, the universe can’t
exist until it’s folded — or it’s a story without a book.’”

The geometry of the 4×4 square array is that of the
3-dimensional projective Galois space PG(3,2).

This space occurs, notably, in the Miracle Octad Generator (MOG)
of R. T. Curtis (submitted to Math. Proc. Camb. Phil. Soc.  on
15 June 1974).  Curtis did not, however, describe its geometric
properties. For these, see the Cullinane diamond theorem.

Some history: 

Curtis seems to have obtained the 4×4 space by permuting,
then “folding” 1×8 binary sequences into 4×2 binary arrays.
The original 1×8 sequences came from the method of Turyn
(1967) described by van Lint in his book Coding Theory
(Springer Lecture Notes in Mathematics, No. 201 , first edition
published in 1971). Two 4×2 arrays form each 4×4 square array
within the MOG. This construction did not suggest any discussion
of the geometric properties of the square arrays.

[Rewritten for clarity on Sept. 3, 2014.]

Sunday, August 24, 2014

Symplectic Structure…

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

In the Miracle Octad Generator (MOG):

The above details from a one-page note of April 26, 1986, refer to the
Miracle Octad Generator of R. T. Curtis, as it was published in 1976:


From R. T. Curtis (1976). A new combinatorial approach to M24,
Mathematical Proceedings of the Cambridge Philosophical Society ,
79, pp 25-42. doi:10.1017/S0305004100052075.

The 1986 note assumed that the reader would be able to supply, from the
MOG itself, the missing top row of each heavy brick.

Note that the interchange of the two squares in the top row of each
heavy brick induces the diamond-theorem correlation.

Note also that the 20 pictured 3-subsets of a 6-set in the 1986 note
occur as paired complements  in two pictures, each showing 10 of the

This pair of pictures corresponds to the 20 Rosenhain tetrads  among
the 35 lines of PG(3,2), while the picture showing the 2-subsets
corresponds to the 15 Göpel tetrads  among the 35 lines.

See Rosenhain and Göpel tetrads in PG(3,2). Some further background:

Tuesday, June 17, 2014

Finite Relativity

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


Anyone tackling the Raumproblem  described here
on Feb. 21, 2014 should know the history of coordinatizations
of the 4×6 Miracle Octad Generator (MOG) array by R. T. Curtis
and J. H. Conway. Some documentation:

The above two images seem to contradict a statement by R. T. Curtis
in a 1989 paper.  Curtis seemed in that paper to be saying, falsely, that
his original 1973 and 1976 MOG coordinates were those in array M below—

This seemingly false statement involved John H. Conway's supposedly
definitive and natural canonical coordinatization of the 4×6 MOG
array by the symbols for the 24 points of the projective line over GF(23)—
{∞, 0, 1, 2, 3… , 21, 22}:

An explanation of the apparent falsity in Curtis's 1989 paper:

By "two versions of the MOG" Curtis seems to have meant merely that the
octads , and not the projective-line coordinates , in his earlier papers were
mirror images of the octads  that resulted later from the Conway coordinates,
as in the images below.

Thursday, May 15, 2014

Mathematics as Religion

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

On Edward Frenkel:

"Math is, for him, 'a narrative' of human endeavor
that shares much with art, music and religion.

For instance, he describes new mathematical insights
as 'revelations,' and the utterly unchanging truths of
mathematical ideas are 'nothing short of a miracle.'"


Thursday, April 24, 2014

The Inscape of 24

Filed under: General,Geometry — m759 @ 9:29 AM

“The more intellectual, less physical, the spell of contemplation
the more complex must be the object, the more close and elaborate
must be the comparison the mind has to keep making between
the whole and the parts, the parts and the whole.”

— The Journals and Papers of Gerard Manley Hopkins ,
edited by Humphry House, 2nd ed. (London: Oxford
University Press, 1959), p. 126, as quoted by Philip A.
Ballinger in The Poem as Sacrament 

Related material from All Saints’ Day in 2012:

Talk pointing out that R. T. Curtis's 1974 construction of the Steiner system S(5,8,24) is taken from Turyn.

Thursday, April 3, 2014

Better Late…

Filed under: General — m759 @ 4:00 PM

Last Sunday’s sermon from Princeton’s Nassau Presbyterian
Church is now online. It reveals the answer to the “One Thing”
riddle posted at the church site Sunday:

IMAGE- Sermon topic 'One Thing Do I Know'

The online sermon has been retitled “One Thing I Do Know.”
A related search yields a relevant example of the original
Yoda-like word order:

IMAGE- 'One thing do I know' in a religious book from 1843

From the online sermon —

“What comes into view is the bombarding cynicism,
the barrage of mistrust and questions, and the
flat out trial of the man born blind. The
interrogation coming not because of the miracle
that gave the man sight….”

Related material — “Then a miracle occurs.”

Friday, March 28, 2014

Blazing Thule

Filed under: General — Tags: — m759 @ 10:20 AM

The title is suggested by a new novel (see cover below),
and by an unwritten book by Nabokov —

Siri Hustvedt, 'The Blazing World'.

Related material:

Sunday, March 9, 2014

The Story Creeps Up

Filed under: General,Geometry — Tags: — m759 @ 11:01 PM

For Women’s History Month —

Conclusion of “The Storyteller,” a story
by Cynthia Zarin about author Madeleine L’Engle—

See also the exercise on the Miracle Octad Generator (MOG) at the end of
the previous post, and remarks on the MOG by Emily Jennings (non -fiction)
on All Saints’ Day, 2012 (the date the L’Engle quote was posted here).

Hesse’s Table

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

From “Quartic Curves and Their Bitangents,” by
Daniel Plaumann, Bernd Sturmfels, and Cynthia Vinzant,
arXiv:1008.4104v2  [math.AG] 10 Jan 2011 —

The table mentioned (from 1855) is…

Exercise: Discuss the relationship, if any, to
the Miracle Octad Generator of R. T. Curtis.

Friday, February 21, 2014


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

Despite the blocking of Doodles on my Google Search
screen, some messages get through.

Today, for instance —

"Your idea just might change the world.
Enter Google Science Fair 2014"

Clicking the link yields a page with the following image—

IMAGE- The 24-triangle hexagon

Clearly there is a problem here analogous to
the square-triangle coordinatization problem,
but with the 4×6 rectangle of the R. T. Curtis
Miracle Octad Generator playing the role of
the square.

I once studied this 24-triangle-hexagon
coordinatization problem, but was unable to
obtain any results of interest. Perhaps
someone else will have better luck.

* For a rather different use of this word,
see Hermann Weyl in the Stanford
Encyclopedia of Philosophy.

Friday, December 20, 2013

For Emil Artin

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

(On His Dies Natalis )

An Exceptional Isomorphism Between Geometric and
Combinatorial Steiner Triple Systems Underlies 
the Octads of the M24 Steiner System S(5, 8, 24).

This is asserted in an excerpt from… 

"The smallest non-rank 3 strongly regular graphs
​which satisfy the 4-vertex condition"
by Mikhail Klin, Mariusz Meszka, Sven Reichard, and Alex Rosa,

(Click for clearer image)

Note that Theorem 46 of Klin et al.  describes the role
of the Galois tesseract  in the Miracle Octad Generator
of R. T. Curtis (original 1976 version). The tesseract
(a 4×4 array) supplies the geometric  part of the above
exceptional geometric-combinatorial isomorphism.

Saturday, December 14, 2013

Beautiful Mathematics

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

The title, which I dislike, is taken from a 2011 publication
of the MAA, also sold by Cambridge University Press.

Some material relevant to the title adjective:

"For those who have learned something of higher mathematics, nothing could be more natural than to use the word 'beautiful' in connection with it. Mathematical beauty, like the beauty of, say, a late Beethoven quartet, arises from a combination of strangeness and inevitability. Simply defined abstractions disclose hidden quirks and complexities. Seemingly unrelated structures turn out to have mysterious correspondences. Uncanny patterns emerge, and they remain uncanny even after being underwritten by the rigor of logic."— Jim Holt, opening of a book review in the Dec. 5, 2013, issue of The New York Review of Books

Some relevant links—

The above list was updated on Jan. 31, 2014, to include the
"Strangeness" and "Hidden quirks" links.  See also a post of
​Jan. 31, 2014.

Update of March 9, 2014 —

The link "Simply defined abstractions" is to the construction of the Steiner
system S(5, 8, 24) described by R. T. Curtis in his 1976 paper defining the
Miracle Octad Generator. It should be noted that this construction is due
to Richard J. Turyn, in a 1967 Sylvania research report. (See Emily Jennings's
talk of 1 Nov. 2012.) Compare  the Curtis construction, written in 1974,
with the Turyn construction of 1967 as described in Sphere Packings, Lattices
and Groups , by J. H. Conway and N. J. A. Sloane (first published in 1988).

Wednesday, September 4, 2013


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

Unexpected connections between areas of mathematics
previously thought to be unrelated are sometimes referred
to as "moonshine."  An example—  the apparent connections
between parts of complex analysis and groups related to the 
large Mathieu group M24. Some recent work on such apparent
connections, by Anne Taormina and Katrin Wendland, among
others (for instance, Miranda C.N. Cheng and John F.R. Duncan),
involves structures related to Kummer surfaces .
In a classic book, Kummer's Quartic Surface  (1905),
R.W.H.T. Hudson pictured a set of 140 structures, the 80
Rosenhain tetrads and the 60 Göpel tetrads, as 4-element
subsets of a 16-element 4×4 array.  It turns out that these
140 structures are the planes of the finite affine geometry
AG(4,2) of four dimensions over the two-element Galois field.
(See Diamond Theory in 1937.) 

A Google search documents the moonshine
relating Rosenhain's and Göpel's 19th-century work
in complex analysis to M24  via the book of Hudson and
the geometry of the 4×4 square.

Monday, August 12, 2013


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

The Galois tesseract  appeared in an early form in the journal
Computer Graphics and Art , Vol. 2, No. 1, February 1977—

IMAGE- Hypercube and 4x4 matrix from the 1976 'Diamond Theory' preprint, as excerpted in 'Computer Graphics and Art'

The Galois tesseract is the basis for a representation of the smallest 
projective 3-space, PG(3,2), that differs from the representation at
Wolfram Demonstrations Project. For the latter, see yesterday's post.

The tesseract representation underlies the diamond theorem, illustrated
below in its earliest form, also from the above February 1977 article—

IMAGE- Steven H. Cullinane, diamond theorem, from 'Diamond Theory,' Computer Graphics and Art, Vol. 2 No. 1, Feb. 1977, pp. 5-7

As noted in a more recent version, the group described by
the diamond theorem is also the group of the 35 square
patterns within the 1976 Miracle Octad Generator  (MOG) of
R. T. Curtis.

Monday, August 5, 2013

Wikipedia Updates

Filed under: General,Geometry — m759 @ 12:30 PM

I added links today in the following Wikipedia articles:

The links will probably soon be deleted,
but it seemed worth a try.

Saturday, July 13, 2013


Filed under: General,Geometry — Tags: , — m759 @ 2:22 AM

A sort of poem
by Gauss and Weyl —

Click the circle for the context in Weyl's Symmetry .

For related remarks, see the previous post.

A literary excursus—

Brad Leithauser in a New Yorker  post of July 11, 2013:

Reading Poems Backward

If a poet determines that a poem should begin at point A and conclude at point D, say, the mystery of how to get there—how to pass felicitously through points B and C—strikes me as an artistic task both genuine and enlivening. There are fertile mysteries of transition, no less than of termination.

And I’d like to suppose that Frost himself would recognize that any ingress into a poem is better than being locked out entirely. His little two-liner, “The Secret,” suggests as much: “We dance round in a ring and suppose / But the Secret sits in the middle and knows.” Most truly good poems might be said to contain a secret: the little sacramental miracle by which you connect, intimately, with the words of a total stranger. And whether you come at the poem frontward, or backward, or inside out—whether you approach it deliberately, word by word and line by line, or you parachute into it borne on a sudden breeze from the island of Serendip—surely isn’t the important thing. What matters is whether you achieve entrance into its inner ring, and there repose companionably beside the Secret.

One should try, of course, to avoid repose in an inner circle of Hell .

Tuesday, July 9, 2013

Vril Chick

Filed under: General,Geometry — m759 @ 4:30 AM

Profile picture of "Jo Lyxe" (Josefine Lyche) at Vimeo

Profile picture for "Jo Lyxe" (Josefine Lyche) at Vimeo

Compare to an image of Vril muse Maria Orsitsch.

From the catalog of a current art exhibition
(25 May – 31 August, 2013) in Norway,

Josefine Lyche
Born in 1973 in Bergen, Norway.
Lives and works in Oslo and Berlin.

Keywords (to help place my artwork in the
proper context): Aliens, affine geometry, affine
planes, affine spaces, automorphisms, binary
codes, block designs, classical groups, codes,
coding theory, collineations, combinatorial,
combinatorics, conjugacy classes, the Conwell
correspondence, correlations, Cullinane,
R. T. Curtis, design theory, the diamond theorem,
diamond theory, duads, duality, error correcting
codes, esoteric, exceptional groups,
extraterrestrials, finite fields, finite geometry, finite
groups, finite rings, Galois fields, generalized
quadrangles, generators, geometry, GF(2),
GF(4), the (24,12) Golay code, group actions,
group theory, Hadamard matrices, hypercube,
hyperplanes, hyperspace, incidence structures,
invariance, Karnaugh maps, Kirkman’s schoolgirls
problem, Latin squares, Leech lattice, linear
groups, linear spaces, linear transformations,
Magick, Mathieu groups, matrix theory, Meno,
Miracle Octad Generator, MOG, multiply transitive
groups, occultism, octahedron, the octahedral
group, Orsic, orthogonal arrays, outer automorphisms,
parallelisms, partial geometries,
permutation groups, PG(3,2), Plato, Platonic
solids, polarities, Polya-Burnside theorem, projective
geometry, projective planes, projective
spaces, projectivities, Pythagoras, reincarnation,
Reed-Muller codes, the relativity problem,
reverse engineering, sacred geometry, Singer
cycle, skew lines, Socrates, sporadic simple
groups, Steiner systems, Sylvester, symmetric,
symmetry, symplectic, synthemes, synthematic,
Theosophical Society tesseract, Tessla, transvections,
Venn diagrams, Vril society, Walsh
functions, Witt designs.

(See also the original catalog page.)

Clearly most of this (the non-highlighted parts) was taken
from my webpage Diamond Theory. I suppose I should be
flattered, but I am not thrilled to be associated with the
(apparently fictional) Vril Society.

For some background, see (for instance) 
Conspiracy Theories and Secret Societies for Dummies .

Tuesday, July 2, 2013

Diamond Theorem Updates

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

My diamond theorem articles at PlanetMath and at 
Encyclopedia of Mathematics have been updated
to clarify the relationship between the graphic square
patterns of the diamond theorem and the schematic
square patterns of the Curtis Miracle Octad Generator.

Tuesday, May 28, 2013


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

The hypercube  model of the 4-space over the 2-element Galois field GF(2):

IMAGE- A hyperspace model of the 4D vector space over GF(2)

The phrase Galois tesseract  may be used to denote a different model
of the above 4-space: the 4×4 square.

MacWilliams and Sloane discussed the Miracle Octad Generator
(MOG) of R. T. Curtis further on in their book (see below), but did not
seem to realize in 1977 that the 4×4 structures within the MOG are
based on the Galois-tesseract model of the 4-space over GF(2).

IMAGE- Octads within the Curtis MOG, which uses a 4x4-array model of the 4D vector space over GF(2)

The thirty-five 4×4 structures within the MOG:

IMAGE- The 35 square patterns within the Curtis MOG

Curtis himself first described these 35 square MOG patterns
combinatorially, (as his title indicated) rather than
algebraically or geometrically:

IMAGE- R. T. Curtis's combinatorial construction of 4x4 patterns within the Miracle Octad Generator

A later book co-authored by Sloane, first published in 1988,
did  recognize the 4×4 MOG patterns as based on the 4×4
Galois-tesseract model.

Between the 1977 and 1988 Sloane books came the diamond theorem.

Update of May 29, 2013:

The Galois tesseract appeared in an early form in the journal
Computer Graphics and Art , Vol. 2, No. 1, February 1977
(the year the above MacWilliams-Sloane book was first published):

IMAGE- Hypercube and 4x4 matrix from the 1976 'Diamond Theory' preprint, as excerpted in 'Computer Graphics and Art'

Sunday, May 19, 2013

Priority Claim

Filed under: General,Geometry — Tags: , , , — m759 @ 9:00 AM

From an arXiv preprint submitted July 18, 2011,
and last revised on March 11, 2013 (version 4):

"By our construction, this vector space is the dual
of our hypercube F24 built on I \ O9. The vector space
structure of the latter, to our knowledge, is first
mentioned by Curtis
in [Cur89]. Hence altogether
our proposition 2.3.4 gives a novel geometric
meaning in terms of Kummer geometry to the known
vector space structure on I \ O9."

[Cur89] reference:
 R. T. Curtis, "Further elementary techniques using
the miracle octad generator," Proc. Edinburgh
Math. Soc. 
32 (1989), 345-353 (received on
July 20, 1987).

— Anne Taormina and Katrin Wendland,
    "The overarching finite symmetry group of Kummer
      surfaces in the Mathieu group 24 ,"
     arXiv.org > hep-th > arXiv:1107.3834

"First mentioned by Curtis…."

No. I claim that to the best of my knowledge, the 
vector space structure was first mentioned by me,
Steven H. Cullinane, in an AMS abstract submitted
in October 1978, some nine years before the
Curtis article.

Update of the above paragraph on July 6, 2013—

No. The vector space structure was described by
(for instance) Peter J. Cameron in a 1976
Cambridge University Press book —
Parallelisms of Complete Designs .
See the proof of Theorem 3A.13 on pages 59 and 60.

The vector space structure as it occurs in a 4×4 array
of the sort that appears in the Curtis Miracle Octad
Generator may first have been pointed out by me,
Steven H. Cullinane,
 in an AMS abstract submitted in
October 1978, some nine years before the Curtis article.

See Notes on Finite Geometry for some background.

See in particular The Galois Tesseract.

For the relationship of the 1978 abstract to Kummer
geometry, see Rosenhain and Göpel Tetrads in PG(3,2).

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.

Thursday, April 25, 2013

Rosenhain and Göpel Revisited

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

Some historical background for today's note on the geometry
underlying the Curtis Miracle Octad Generator (MOG):

IMAGE- Bateman in 1906 on Rosenhain and Göpel tetrads

The above incidence diagram recalls those in today's previous post
on the MOG, which is used to construct the large Mathieu group M24.

For some related material that is more up-to-date, search the Web
for Mathieu + Kummer .

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).

Wednesday, February 13, 2013


Filed under: General,Geometry — Tags: , — m759 @ 9:29 PM

Story, Structure, and the Galois Tesseract

Recent Log24 posts have referred to the 
"Penrose diamond" and Minkowski space.

The Penrose diamond has nothing whatever
to do with my 1976 monograph "Diamond Theory,"
except for the diamond shape and the connection
of the Penrose diamond to the Klein quadric—

IMAGE- The Penrose diamond and the Klein quadric

The Klein quadric occurs in the five-dimensional projective space
over a field. If the field is the two-element Galois field GF(2), the
quadric helps explain certain remarkable symmetry properties 
of the R. T. Curtis Miracle Octad Generator  (MOG), hence of
the large Mathieu group M24. These properties are also 
relevant to the 1976 "Diamond Theory" monograph.

For some background on the quadric, see (for instance)

IMAGE- Stroppel on the Klein quadric, 2008

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

Related material:

"… one might crudely distinguish between philosophical
and mathematical motivation. In the first case one tries
to convince with a telling conceptual story; in the second
one relies more on the elegance of some emergent
mathematical structure. If there is a tradition in logic
it favours the former, but I have a sneaking affection for
the latter. Of course the distinction is not so clear cut.
Elegant mathematics will of itself tell a tale, and one with
the merit of simplicity. This may carry philosophical
weight. But that cannot be guaranteed: in the end one
cannot escape the need to form a judgement of significance."

– J. M. E. Hyland. "Proof Theory in the Abstract." (pdf)
Annals of Pure and Applied Logic 114, 2002, 43-78.

Those who prefer story to structure may consult 

  1. today's previous post on the Penrose diamond
  2. the remarks of Scott Aaronson on August 17, 2012
  3. the remarks in this journal on that same date
  4. the geometry of the 4×4 array in the context of M24.

Monday, December 24, 2012

All Over Again

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

Octavio Paz —

"… the movement of analogy
begins all over once again."

See A Reappearing Number in this journal.


Figure 1 —

Background: MOG in this journal.

Figure 2 —

Image-- 'Then a miracle occurs' cartoon
Cartoon by S.Harris

Background —

Image-- Google search on 'miracle octad'-- top 3 results

Monday, November 19, 2012

Poetry and Truth

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

From today's noon post

"In all his poems with all their enchantments
for the poet himself, there is the final enchantment
that they are true. The significance of the poetic act
then is that it is evidence. It is instance and illustration.
It is an illumination of a surface,
the movement of a self in the rock.
Above all it is a new engagement with life.
It is that miracle to which the true faith of the poet
attaches itself."

— Wallace Stevens at Bard College, March 30, 1951

Stevens also said at Bard that

"When Joan of Arc said: 

Have no fear: what I do, I do by command.
My brothers of Paradise tell me what I have to do.

these words were the words of an hallucination.
No matter what her brothers of Paradise drove her to do,
what she did was never a poetic act of faith in reality
because it could not be."

There are those who would dispute this.

Some related material:

"Ageometretos me eisito."—
"Let no one ignorant of geometry enter."—
Said to be a saying of Plato, part of the
seal of the American Mathematical Society—

A poetic approach to geometry—

"A surface" and "the rock," from All Saints' Day, 2012

Spaces as Hypercubes

— and from 1981—


Some mathematical background for poets in Purgatory—

"… the Klein correspondence underlies Conwell's discussion 
of eight heptads. These play an important role in another
correspondence, illustrated in the Miracle Octad Generator
of R. T. Curtis, that may be used to picture actions
of the large Mathieu group M24."


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

Speech by Wallace Stevens upon accepting
an honorary degree from Bard College in 1951

(Click to enlarge.)

Transcription of conclusion:

"In all his poems with all their enchantments
for the poet himself, there is the final enchantment
that they are true. The significance of the poetic act
then is that it is evidence. It is instance and illustration.
It is an illumination of a surface,
the movement of a self in the rock.
Above all it is a new engagement with life.
It is that miracle to which the true faith of the poet
attaches itself."

— Wallace Stevens at Bard College, March 30, 1951

Friday, August 17, 2012


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

Detail from last night's 1.3 MB image
"Search for the Lost Tesseract"—

The lost tesseract appears here on the cover of Wittgenstein's
Zettel  and, hidden in the form of a 4×4 array, as a subarray 
of the Miracle Octad Generator on the cover of Griess's
Twelve Sporadic Groups  and in a figure illustrating
the geometry of logic.

Another figure—

IMAGE- Serbian chess innovator Svetozar Gligoric dies at 89

Gligoric died in Belgrade, Serbia, on Tuesday, August 14.

From this journal on that date

"Visual forms, he thought, were solutions to 
specific problems that come from specific needs."

— Michael Kimmelman in The New York Times
    obituary of E. H. Gombrich (November 7th, 2001)

Monday, July 9, 2012

Taps for Fatso

Filed under: General — m759 @ 7:00 AM
IMAGE- Donna Reed and Montgomery Clift in 'From Here to Eternity'

A Story That Works

“There is the dark, eternally silent, unknown universe;
there are the friend-enemy minds shouting and whispering
their tales and always seeking the three miracles

  • that minds should really touch, or
  • that the silent universe should speak, tell minds a story,
  • or (perhaps the same thing) that there should be a story
    that works, that is all hard facts, all reality, with
    no illusions and no fantasy;

and lastly, there is lonely, story-telling, wonder-questing,
mortal me.”

Dichtung ist Stiftung.

Sunday, September 18, 2011

Midnight in Dostoevsky

Filed under: General — m759 @ 12:00 AM

"But Thou knewest not, it seems, that no sooner would man reject
miracle than he would reject God likewise, for he seeketh less
God than 'a sign' from Him." —The Grand Inquisitor

Update of 1:44 AM—

Seek and Ye Shall Find …

(Click images for larger context)


  The Holy Office —


Saturday, September 3, 2011

The Galois Tesseract (continued)

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

A post of September 1, The Galois Tesseract, noted that the interplay
of algebraic and geometric properties within the 4×4 array that forms
two-thirds of the Curtis Miracle Octad Generator (MOG) may first have
been described by Cullinane (AMS abstract 79T-A37, Notices , Feb. 1979).

Here is some supporting material—


The passage from Carmichael above emphasizes the importance of
the 4×4 square within the MOG.

The passage from Conway and Sloane, in a book whose first edition
was published in 1988, makes explicit the structure of the MOG's
4×4 square as the affine 4-space over the 2-element Galois field.

The passage from Curtis (1974, published in 1976) describes 35 sets
of four "special tetrads" within the 4×4 square of the MOG. These
correspond to the 35 sets of four parallel 4-point affine planes within
the square. Curtis, however, in 1976 makes no mention of the affine
structure, characterizing his 140 "special tetrads" rather by the parity
of their intersections with the square's rows and columns.

The affine structure appears in the 1979 abstract mentioned above—

IMAGE- An AMS abstract from 1979 showing how the affine group AGL(4,2) of 322,560 transformations acts on a 4x4 square

The "35 structures" of the abstract were listed, with an application to
Latin-square orthogonality, in a note from December 1978

IMAGE- Projective-space structure and Latin-square orthogonality in a set of 35 square arrays

See also a 1987 article by R. T. Curtis—

Further elementary techniques using the miracle octad generator
, by R. T. Curtis. Abstract:

“In this paper we describe various techniques, some of which are already used by devotees of the art, which relate certain maximal subgroups of the Mathieu group M24, as seen in the MOG, to matrix groups over finite fields. We hope to bring out the wealth of algebraic structure* underlying the device and to enable the reader to move freely between these matrices and permutations. Perhaps the MOG was mis-named as simply an ‘octad generator’; in this paper we intend to show that it is in reality a natural diagram of the binary Golay code.”

(Received July 20 1987)

Proceedings of the Edinburgh Mathematical Society (Series 2) (1989), 32: 345-353

* For instance:

Algebraic structure in the 4x4 square, by Cullinane (1985) and Curtis (1987)

Update of Sept. 4— This post is now a page at finitegeometry.org.

Saturday, August 6, 2011


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

Comme de longs échos qui de loin se confondent
Dans une ténébreuse et profonde unité….

— Baudelaire, "Correspondances "

From "A Four-Color Theorem"


Figure 1

Note that this illustrates a natural correspondence

(A) the seven highly symmetrical four-colorings
      of the 4×2 array at the left of Fig. 1, and

(B) the seven points of the smallest
      projective plane at the right of Fig. 1.

To see the correspondence, add, in binary
fashion, the pairs of projective points from the
"points" section that correspond to like-colored
squares in a four-coloring from the left of Fig. 1.
(The correspondence can, of course, be described
in terms of cosets rather than of colorings.)

A different correspondence between these 7 four-coloring
structures and these 7 projective-line structures appears in
a structural analysis of the Miracle Octad Generator
(MOG) of R.T. Curtis—


Figure 2

Here the correspondence between the 7 four-coloring structures (left section) and the 7 projective-line structures (center section) is less obvious, but more fruitful.  It yields, as shown, all of the 35 partitions of an 8-element set  (an 8-set ) into two 4-sets. The 7 four-colorings in Fig. 2 also appear in the 35 4×4 parts of the MOG that correspond, in a way indicated by Fig. 2, to the 35 8-set paritions. This larger correspondence— of 35 4×2 arrays with 35 4×4 arrays— is  the MOG, at least as it was originally defined. See The MOG, Generating the Octad Generator, and Eightfold Geometry.


For some applications of the Curtis MOG, see
(for instance) Griess's Twelve Sporadic Groups .

Wednesday, July 6, 2011

Nordstrom-Robinson Automorphisms

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

A 2008 statement on the order of the automorphism group of the Nordstrom-Robinson code—

"The Nordstrom-Robinson code has an unusually large group of automorphisms (of order 8! = 40,320) and is optimal in many respects. It can be found inside the binary Golay code."

— Jürgen Bierbrauer and Jessica Fridrich, preprint of "Constructing Good Covering Codes for Applications in Steganography," Transactions on Data Hiding and Multimedia Security III, Springer Lecture Notes in Computer Science, 2008, Volume 4920/2008, 1-22

A statement by Bierbrauer from 2004 has an error that doubles the above figure—

The automorphism group of the binary Golay code G is the simple Mathieu group M24 of order |M24| = 24 × 23 × 22 × 21 × 20 × 48 in its 5-transitive action on the 24 coordinates. As M24 is transitive on octads, the stabilizer of an octad has order |M24|/759 [=322,560]. The stabilizer of NR has index 8 in this group. It follows that NR admits an automorphism group of order |M24| / (759 × 8 ) = [?] 16 × 7! [=80,640]. This is a huge symmetry group. Its structure can be inferred from the embedding in G as well. The automorphism group of NR is a semidirect product of an elementary abelian group of order 16 and the alternating group A7.

— Jürgen Bierbrauer, "Nordstrom-Robinson Code and A7-Geometry," preprint dated April 14, 2004, published in Finite Fields and Their Applications , Volume 13, Issue 1, January 2007, Pages 158-170

The error is corrected (though not detected) later in the same 2004 paper—

In fact the symmetry group of the octacode is a semidirect product of an elementary abelian group of order 16 and the simple group GL(3, 2) of order 168. This constitutes a large automorphism group (of order 2688), but the automorphism group of NR is larger yet as we saw earlier (order 40,320).

For some background, see a well-known construction of the code from the Miracle Octad Generator of R.T. Curtis—

Click to enlarge:

IMAGE - The 112 hexads of the Nordstrom-Robinson code

For some context, see the group of order 322,560 in Geometry of the 4×4 Square.

Sunday, June 19, 2011

The London Piracy Project

Filed under: General,Geometry — m759 @ 6:25 PM

My work has been pirated by an artist in London.

An organization there, AND Publishing, sponsors what it calls
"The Piracy Project." The artist's piracy was a contribution
to the project.

The above material now reflects the following update:

UPDATE of June 21, 2011, 10:00 PM ET:

The organization's weblog (a post for 19th June)
has now been updated, and this  post, which originally
discussed that weblog, has been altered to reflect the
changes that were made at AND Publishing's weblog.

In this  weblog, changes have been made to correct my
earlier incorrect statements that the Piracy Project was
sponsored by the art school where it takes place.
It was not. The organization has informed me that

"AND Publishing is not sponsored by the art school.
We are an independent artist's publishing house,
kindly hosted by the art school. While we are offered
office space on campus, our program and website
are funded, directed and managed by ourselves –
we are an independent entity running an
autonomous program."

As this post originally stated…

The web pages from the site finitegeometry.org/sc that
the artist, Steve Richards, copied as part of his contribution to
the AND Publishing Piracy Project have had the author's name,
Steven H. Cullinane, and the date of composition systematically removed.

See a sample (jpg, 2.1 MB).

Here is some background on Richards.

Abracadabra (continued)

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

Yesterday's post Ad Meld featured Harry Potter (succeeding in business),
a 4×6 array from a video of the song "Abracadabra," and a link to a post
with some background on the 4×6 Miracle Octad Generator  of R.T. Curtis.

A search tonight for related material on the Web yielded…

(Click to enlarge.)

IMAGE- Art by Steven H. Cullinane displayed as his own in Steve Richards's Piracy Project contribution

   Weblog post by Steve Richards titled "The Search for Invariants:
   The Diamond Theory of Truth, the Miracle Octad Generator
   and Metalibrarianship." The artwork is by Steven H. Cullinane.
   Richards has omitted Cullinane's name and retitled the artwork.

The author of the post is an artist who seems to be interested in the occult.

His post continues with photos of pages, some from my own work (as above), some not.

My own work does not  deal with the occult, but some enthusiasts of "sacred geometry" may imagine otherwise.

The artist's post concludes with the following (note also the beginning of the preceding  post)—


"The Struggle of the Magicians" is a 1914 ballet by Gurdjieff. Perhaps it would interest Harry.

Saturday, June 18, 2011

Ad Meld

Filed under: General — m759 @ 2:22 PM

Step One


Step Two

IMAGE- Abracadabra 4x6

Step Three

Image-- 'Then a miracle occurs' cartoon

For further details, click on step two.

Sunday, June 5, 2011

Edifice Complex

Filed under: General,Geometry — Tags: , — m759 @ 7:00 PM

"Total grandeur of a total edifice,
Chosen by an inquisitor of structures
For himself. He stops upon this threshold,
As if the design of all his words takes form
And frame from thinking and is realized."

— Wallace Stevens, "To an Old Philosopher in Rome"

The following edifice may be lacking in grandeur,
and its properties as a configuration  were known long
before I stumbled across a description of it… still…

"What we do may be small, but it has
 a certain character of permanence…."
 — G.H. Hardy, A Mathematician's Apology

The Kummer 166 Configuration
as seen by Kantor in 1969— (pdf, 2.5 MB)

IMAGE-- 16_6 configuration from '2-Transitive Symmetric Designs,' by William M. Kantor (AMS Transactions, 1969)

For some background, see Configurations and Squares.

For some quite different geometry of the 4×4 square that  is
original with me, see a page with that title. (The geometry's
importance depends in part on its connection with the
Miracle Octad Generator (MOG) of R.T. Curtis. I of course
had nothing to do with the MOG's discovery, but I do  claim credit
for discovering some geometric properties of the 4×4 square
that constitutes two-thirds of the MOG as originally defined .)

Related material— The Schwartz Notes of June 1.

Wednesday, June 1, 2011

The Schwartz Notes

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

A Google search today for material on the Web that puts the diamond theorem
in context yielded a satisfyingly complete list. (See the first 21 results.)
(Customization based on signed-out search activity was disabled.)

The same search limited to results from only the past month yielded,
in addition, the following—


This turns out to be a document by one Richard Evan Schwartz,
Chancellor’s Professor of Mathematics at Brown University.

Pages 12-14 of the document, which is untitled, undated, and
unsigned, discuss the finite-geometry background of the R.T.
Curtis Miracle Octad Generator (MOG) . As today’s earlier search indicates,
this is closely related to the diamond theorem. The section relating
the geometry to the MOG is titled “The MOG and Projective Space.”
It does not mention my own work.

See Schwartz’s page 12, page 13, and page 14.

Compare to the web pages from today’s earlier search.

There are no references at the end of the Schwartz document,
but there is this at the beginning—

These are some notes on error correcting codes. Two good sources for
this material are
From Error Correcting Codes through Sphere Packings to Simple Groups ,
by Thomas Thompson.
Sphere Packings, Lattices, and Simple Groups  by J. H. Conway and N.
Planet Math (on the internet) also some information.

It seems clear that these inadequate remarks by Schwartz on his sources
can and should be expanded.

Thursday, May 26, 2011

Life’s Persistent Questions

Filed under: General — m759 @ 4:01 PM

This afternoon's online New York Times  reviews "The Tree of Life," a film that opens tomorrow.

With disarming sincerity and daunting formal sophistication “The Tree of Life” ponders some of the hardest and most persistent questions, the kind that leave adults speechless when children ask them. In this case a boy, in whispered voice-over, speaks directly to God, whose responses are characteristically oblique, conveyed by the rustling of wind in trees or the play of shadows on a bedroom wall. Where are you? the boy wants to know, and lurking within this question is another: What am I doing here?

Persistent answers… Perhaps conveyed by wind, perhaps by shadows, perhaps by the New York Lottery.

For the nihilist alternative— the universe arose by chance out of nothing and all is meaningless— see Stephen Hawking and Jennifer Ouellette.

Update of 10:30 PM EDT May 26—

Today's NY Lottery results: Midday 407, Evening 756. The first is perhaps about the date April 7, the second about the phrase "three bricks shy"— in the context of the number 759 and the Miracle Octad Generator. (See also Robert Langdon and The Poetics of Space.)

Tuesday, May 24, 2011

Noncontinuous (or Non-Continuous) Groups

Filed under: General,Geometry — Tags: — m759 @ 2:56 PM

The web page has been updated.

An example, the action of the Mathieu group M24
on the Miracle Octad Generator of R.T. Curtis,
was added, with an illustration from a book cover—


Wednesday, March 2, 2011

Labyrinth of the Line

Filed under: General,Geometry — Tags: — m759 @ 11:24 AM

“Yo sé de un laberinto griego que es una línea única, recta.”
—Borges, “La Muerte y la Brújula”

“I know of one Greek labyrinth which is a single straight line.”
—Borges, “Death and the Compass”

Another single-line labyrinth—

Robert A. Wilson on the projective line with 24 points
and its image in the Miracle Octad Generator (MOG)—

IMAGE- Robert Wilson on the projective line with 24 points and its image in the MOG

Related material —

The remarks of Scott Carnahan at Math Overflow on October 25th, 2010
and the remarks at Log24 on that same date.

A search in the latter for miracle octad is updated below.


This search (here in a customized version) provides some context for the
Benedictine University discussion described here on February 25th and for
the number 759 mentioned rather cryptically in last night’s “Ariadne’s Clue.”

Update of March 3— For some historical background from 1931, see The Mathieu Relativity Problem.

Tuesday, January 25, 2011

For Your Consideration…

Filed under: General — m759 @ 1:20 PM

Early Nothing

Manohla Dargis on film director Fritz Lang in The New York Times  (online Jan. 21, 2011, printed Jan. 23)—

"Hollywood endings can be beautiful fibs, but in Lang’s movies the glossy smiles and fade-outs feel forced. You can almost feel him pulling at them, trying to bring them back into the dark where they belong. The miracle of his Hollywood era is that, even when the screenplays tried to force his work in one direction, he managed to take them into richer, more complex realms with a style that was alternately baroque and stripped down and peopled with characters whose cynicism was earned. Every so often, though, he did strike screenwriting gold, notably in 'The Big Heat,' his 1953 crime masterwork. 'Say, I like this, early nothing,' a mink-swaddled Gloria Grahame says of a hotel room. Everyone really is a critic."

Here comes everyone .

Monday, October 25, 2010

The Embedding*

Filed under: General,Geometry — m759 @ 4:04 PM

A New York Times  "The Stone" post from yesterday (5:15 PM, by John Allen Paulos) was titled—

Stories vs. Statistics

Related Google searches—

"How to lie with statistics"— about 148,000 results

"How to lie with stories"— 2 results

What does this tell us?

Consider also Paulos's phrase "imbedding the God character."  A less controversial topic might be (with the spelling I prefer) "embedding the miraculous." For an example, see this journal's "Mathematics and Narrative" entry on 5/15 (a date suggested, coincidentally, by the time of Paulos's post)—

Image-- 'Then a miracle occurs' cartoon
Cartoon by S.Harris

Image-- Google search on 'miracle octad'-- top 3 results


* Not directly  related to the novel The Embedding  discussed at Tenser, said the Tensor  on April 23, 2006 ("Quasimodo Sunday"). An academic discussion of that novel furnishes an example of narrative as more than mere entertainment. See Timothy J. Reiss, "How can 'New' Meaning Be Thought? Fictions of Science, Science Fictions," Canadian Review of Comparative Literature , Vol. 12, No. 1, March 1985, pp. 88-126. Consider also on this, Picasso's birthday, his saying that "Art is a lie that makes us realize truth…."

Saturday, July 24, 2010

Playing with Blocks

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

"Many of the finite simple groups can be described as symmetries of finite geometries, and it remains a hot topic in group theory to expand our knowledge of the Classification of Finite Simple Groups using finite geometry."

Finite geometry page at the Centre for the Mathematics of
   Symmetry and Computation at the University of Western Australia
   (Alice Devillers, John Bamberg, Gordon Royle)

For such symmetries, see Robert A. WIlson's recent book The Finite Simple Groups.

The finite simple groups are often described as the "building blocks" of finite group theory.

At least some of these building blocks have their own building blocks. See Non-Euclidean Blocks.

For instance, a set of 24 such blocks (or, more simply, 24 unit squares) appears in the Miracle Octad Generator (MOG) of R.T. Curtis, used in the study of the finite simple group M24.

(The octads  of the MOG illustrate yet another sort of mathematical blocks— those of a block design.)

Sunday, July 4, 2010

Brightness at Noon (continued)

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

Today's sermon mentioned the phrase "Omega number."

Other sorts of Omega numbers— 24 and 759— occur
in connection with the set named Ω by R. T. Curtis in 1976—

Image-- In a 1976 paper, R.T. Curtis names the 24-set of his Miracle Octad Generator 'Omega.'

— R. T. Curtis, "A New Combinatorial Approach to M24,"
Math. Proc. Camb. Phil. Soc. (1976), 79, 25-42

Friday, June 11, 2010

Toward the Light

Filed under: General — m759 @ 11:01 AM

The title is a reference to yesterday's noon post.

For the late Vladimir Igorevich Arnold

"All things began in order, so shall they end, and so shall they begin again; according to the ordainer of order and mystical Mathematicks of the City of Heaven."

— Sir Thomas Browne, The Garden of Cyrus, Chapter V

Arnold's own mystical mathematics may be found in his paper

"Polymathematics: Is Mathematics a Single Science or a Set of Arts?"

Page 13–
"In mathematics we always encounter mysterious analogies, and our trinities [page 8] represent only a small part of these miracles."

Also from that paper—

Page 5, footnote 2–
"The Russian way to formulate problems is to mention the first nontrivial case (in a way that no one would be able to simplify it). The French way is to formulate it in the most general form making impossible any further generalization."

Arnold died in Paris on June 3. A farewell gathering was held there on June 8—

"Celles et ceux qui le souhaitent pourront donner un dernier adieu à Vladimir Igorevitch
mardi 8 juin, de 14h a 16h, chambre mortuaire de l'hopital Saint Antoine…."

An International Blue Diamond

In Arnold's memory—  Here, in the Russian style, is a link to a "first nontrivial case" of a blue diamond— from this journal on June 8 (feast of St. Gerard Manley Hopkins). For those who prefer French style, here is a link to a blue diamond from May 18

From French cinema—


"a 'non-existent myth' of a battle between
goddesses of the sun and the moon
for a mysterious blue diamond
that has the power to make
mortals immortal and vice versa"

Saturday, May 15, 2010

Mathematics and Narrative continued…

Filed under: General,Geometry — m759 @ 4:16 PM

Step Two

Image-- 'Then a miracle occurs' cartoon
Cartoon by S.Harris

Image-- Google search on 'miracle octad'-- top 3 results

Friday, May 14, 2010

Competing MOG Definitions

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

A recently created Wikipedia article says that  "The Miracle Octad Generator [MOG] is an array of coordinates, arranged in four rows and six columns, capable of describing any point in 24-dimensional space…." (Clearly any  array with 24 parts is so capable.) The article ignores the fact that the MOG, as defined by R.T. Curtis in 1976, is not  an array of coordinates, but rather a picture of a correspondence between two sets, each containing 35 structures. (As a later commentator has remarked, this correspondence is a well-known one that preserves a certain incidence property. See Eightfold Geometry.)

From the 1976 paper defining the MOG—

"There is a correspondence between the two systems of 35 groups, which is illustrated in Fig. 4 (the MOG or Miracle Octad Generator)." —R.T. Curtis, "A New Combinatorial Approach to M24," Mathematical Proceedings of the Cambridge Philosophical Society  (1976), 79: 25-42


Curtis's 1976 Fig. 4. (The MOG.)

The Wikipedia article, like a similar article at PlanetMath, is based on a different definition, from a book first published in 1988—


I have not seen the 1973 Curtis paper, so I do not know whether it uses the 35-sets correspondence definition or the 6×4 array definition. The remarks of Conway and Sloane on page 312 of the 1998 edition of their book about "Curtis's original way of finding octads in the MOG [Cur2]" indicate that the correspondence definition was the one Curtis used in 1973—


Here the picture of  "the 35 standard sextets of the MOG"
is very like (modulo a reflection) Curtis's 1976 picture
of the MOG as a correspondence between two 35-sets.

A later paper by Curtis does  use the array definition. See "Further Elementary Techniques Using the Miracle Octad Generator," Proceedings of the Edinburgh Mathematical Society  (1989) 32, 345-353.

The array definition is better suited to Conway's use of his hexacode  to describe octads, but it obscures the close connection of the MOG with finite geometry. That connection, apparent in the phrases "vector space structure in the standard square" and "parallel 2-spaces" (Conway and Sloane, third ed., p. 312, illustrated above), was not discussed in the 1976 Curtis paper.  See my own page on the MOG at finitegeometry.org.

Wednesday, April 28, 2010

Eightfold Geometry

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

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

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

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

Related web pages:

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

Related folklore:

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

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

Update of August 20, 2010–

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

Thursday, August 6, 2009

Thursday August 6, 2009

Filed under: General,Geometry — Tags: — m759 @ 1:44 PM
A Fisher of Men
Cover, Schulberg's novelization of 'Waterfront,' Bantam paperback
Update: The above image was added
at about 11 AM ET Aug. 8, 2009.

Dove logo, First United Methodist Church of Bloomington, Indiana

From a webpage of the First United Methodist Church of Bloomington, Indiana–


Dr. Joe Emerson, April 24, 2005–

"The Ultimate Test"

— Text: I Peter 2:1-9

Dr. Emerson falsely claims that the film "On the Waterfront" was based on a book by the late Budd Schulberg (who died yesterday). (Instead, the film's screenplay, written by Schulberg– similar to an earlier screenplay by Arthur Miller, "The Hook"–  was based on a series of newspaper articles by Malcolm Johnson.)

"The movie 'On the Waterfront' is once more in rerun. (That’s when Marlon Brando looked like Marlon Brando.  That’s the scary part of growing old when you see what he looked like then and when he grew old.)  It is based on a book by Budd Schulberg."


Emerson goes on to discuss the book, Waterfront, that Schulberg wrote based on his screenplay–

"In it, you may remember a scene where Runty Nolan, a little guy, runs afoul of the mob and is brutally killed and tossed into the North River.  A priest is called to give last rites after they drag him out."


Hook on cover of Budd Schulberg's novel 'Waterfront' (NY Times obituary, detail)

New York Times today

Dr. Emerson flunks the test.


Dr. Emerson's sermon is, as noted above (Text: I Peter 2:1-9), not mainly about waterfronts, but rather about the "living stones" metaphor of the Big Fisherman.

My own remarks on the date of Dr. Emerson's sermon

The 4x6 array used in the Miracle Octad Generator of R. T. Curtis

Those who like to mix mathematics with religion may regard the above 4×6 array as a context for the "living stones" metaphor. See, too, the five entries in this journal ending at 12:25 AM ET on November 12 (Grace Kelly's birthday), 2006, and today's previous entry.

Wednesday, May 20, 2009

Wednesday May 20, 2009

Filed under: General,Geometry — Tags: — m759 @ 4:00 PM
From Quilt Blocks to the
Mathieu Group


(a traditional
quilt block):

Illustration of a diamond-theorem pattern


Octads formed by a 23-cycle in the MOG of R.T. Curtis


Click on illustrations for details.

The connection:

The four-diamond figure is related to the finite geometry PG(3,2). (See "Symmetry Invariance in a Diamond Ring," AMS Notices, February 1979, A193-194.) PG(3,2) is in turn related to the 759 octads of the Steiner system S(5,8,24). (See "Generating the Octad Generator," expository note, 1985.)

The relationship of S(5,8,24) to the finite geometry PG(3,2) has also been discussed in–
  • "A Geometric Construction of the Steiner System S(4,7,23)," by Alphonse Baartmans, Walter Wallis, and Joseph Yucas, Discrete Mathematics 102 (1992) 177-186.

Abstract: "The Steiner system S(4,7,23) is constructed from the geometry of PG(3,2)."

  • "A Geometric Construction of the Steiner System S(5,8,24)," by R. Mandrell and J. Yucas, Journal of Statistical Planning and Inference 56 (1996), 223-228.

Abstract: "The Steiner system S(5,8,24) is constructed from the geometry of PG(3,2)."

For the connection of S(5,8,24) with the Mathieu group M24, see the references in The Miracle Octad Generator.

Tuesday, May 19, 2009

Tuesday May 19, 2009

Filed under: General,Geometry — Tags: , — m759 @ 7:20 PM
Exquisite Geometries

"By far the most important structure in design theory is the Steiner system S(5, 8, 24)."

"Block Designs," 1995, by Andries E. Brouwer

"The Steiner system S(5, 8, 24) is a set S of 759 eight-element subsets ('octads') of a twenty-four-element set T such that any five-element subset of T is contained in exactly one of the 759 octads. Its automorphism group is the large Mathieu group M24."

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

"… in 1861 Mathieu… discovered five multiply transitive permutation groups…. In a little-known 1931 paper of Carmichael… they were first observed to be automorphism groups of exquisite finite geometries."

William M. Kantor, 1981

The 1931 paper of Carmichael is now available online from the publisher for $10.

Tuesday, January 6, 2009

Tuesday January 6, 2009

Filed under: General,Geometry — m759 @ 12:00 AM
Archetypes, Synchronicity,
and Dyson on Jung

The current (Feb. 2009) Notices of the American Mathematical Society has a written version of Freeman Dyson’s 2008 Einstein Lecture, which was to have been given in October but had to be canceled. Dyson paraphrases a mathematician on Carl Jung’s theory of archetypes:

“… we do not need to accept Jung’s theory as true in order to find it illuminating.”

The same is true of Jung’s remarks on synchronicity.

For example —

Yesterday’s entry, “A Wealth of Algebraic Structure,” lists two articles– each, as it happens, related to Jung’s four-diamond figure from Aion as well as to my own Notes on Finite Geometry. The articles were placed online recently by Cambridge University Press on the following dates:

R. T. Curtis’s 1974 article defining his Miracle Octad Generator (MOG) was published online on Oct. 24, 2008.

Curtis’s 1987 article on geometry and algebraic structure in the MOG was published online on Dec. 19, 2008.

On these dates, the entries in this journal discussed…

Oct. 24:
Cube Space, 1984-2003

Material related to that entry:

Dec. 19:
Art and Religion: Inside the White Cube

That entry discusses a book by Mark C. Taylor:

The Picture in Question: Mark Tansey and the Ends of Representation (U. of Chicago Press, 1999).

In Chapter 3, “Sutures of Structures,” Taylor asks —

“What, then, is a frame, and what is frame work?”

One possible answer —

Hermann Weyl on the relativity problem in the context of the 4×4 “frame of reference” found in the above Cambridge University Press articles.

“Examples are the stained-glass
windows of knowledge.”
— Vladimir Nabokov 

Monday, January 5, 2009

Monday January 5, 2009

Filed under: General,Geometry — Tags: — m759 @ 9:00 PM
A Wealth of
Algebraic Structure

A 4x4 array (part of chessboard)

A 1987 article by R. T. Curtis on the geometry of his Miracle Octad Generator (MOG) as it relates to the geometry of the 4×4 square is now available online ($20):

Further elementary techniques using the miracle octad generator
, by R. T. Curtis. Abstract:

"In this paper we describe various techniques, some of which are already used by devotees of the art, which relate certain maximal subgroups of the Mathieu group M24, as seen in the MOG, to matrix groups over finite fields. We hope to bring out the wealth of algebraic structure* underlying the device and to enable the reader to move freely between these matrices and permutations. Perhaps the MOG was mis-named as simply an 'octad generator'; in this paper we intend to show that it is in reality a natural diagram of the binary Golay code."


(Received July 20 1987)

Proceedings of the Edinburgh Mathematical Society (Series 2) (1989), 32: 345-353, doi:10.1017/S0013091500004600.

(Published online by Cambridge University Press 19 Dec 2008.)

In the above article, Curtis explains how two-thirds of his 4×6 MOG array may be viewed as the 4×4 model of the four-dimensional affine space over GF(2).  (His earlier 1974 paper (below) defining the MOG discussed the 4×4 structure in a purely combinatorial, not geometric, way.)

For further details, see The Miracle Octad Generator as well as Geometry of the 4×4 Square and Curtis's original 1974 article, which is now also available online ($20):

A new combinatorial approach to M24, by R. T. Curtis. Abstract:

"In this paper, we define M24 from scratch as the subgroup of S24 preserving a Steiner system S(5, 8, 24). The Steiner system is produced and proved to be unique and the group emerges naturally with many of its properties apparent."


(Received June 15 1974)

Mathematical Proceedings of the Cambridge Philosophical Society (1976), 79: 25-42, doi:10.1017/S0305004100052075.

(Published online by Cambridge University Press 24 Oct 2008.)

* For instance:

Algebraic structure in the 4x4 square, by Cullinane (1985) and Curtis (1987)

Click for details.

Saturday, December 27, 2008

Saturday December 27, 2008

Filed under: General — m759 @ 9:00 PM


Monday, November 24, 2008

Monday November 24, 2008

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

Frame Tale

'Brick' octads in the Miracle Octad Generator (MOG) of R. T. Curtis

Click on image for details.

Saturday, October 25, 2008

Saturday October 25, 2008

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

The New York Times Book Review online today has a review by Sam Tanenhaus of a new John Updike book.

The title of the review (not the book) is "Mr. Wizard."

"John Updike is the great genial sorcerer of American letters. His output alone (60 books, almost 40 of them novels or story collections) has been supernatural. More wizardly still is the ingenuity of his prose. He has now written tens of thousands of sentences, many of them tiny miracles of transubstantiation whereby some hitherto overlooked datum of the human or natural world– from the anatomical to the zoological, the socio-economic to the spiritual– emerges, as if for the first time, in the complete­ness of its actual being."

Rolling Stone interview with Sting, February 7, 1991:

"'I was brought up in a very strong Catholic community,' Sting says. 'My parents were Catholic, and in the Fifties and Sixties, Catholicism was very strong. You know, they say, "Once a Catholic, always a Catholic." In a way I'm grateful for that background. There's a very rich imagery in Catholicism: blood, guilt, death, all that stuff.' He laughs."


RS 597, Feb. 7, 1991

Last night's 12:00 AM
Log24 entry:

Midnight Bingo

From this date six years ago:

It All Adds Up.

From this morning's newspaper,
a religious meditation I had not
seen last night:

The image “http://www.log24.com/log/pix08A/081025-WizardOfIdSm.jpg” cannot be displayed, because it contains errors.

Related material:

Juneteenth through
Midsummer Night, 2007


Church of the Forbidden Planet

Thursday, July 31, 2008

Thursday July 31, 2008

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

“Put bluntly, who is kidding whom?”

Anthony Judge, draft of
“Potential Psychosocial Significance
of Monstrous Moonshine:
An Exceptional Form of Symmetry
as a Rosetta Stone for
Cognitive Frameworks,”
dated September 6, 2007.

Good question.

Also from
September 6, 2007 —
the date of
Madeleine L’Engle‘s death —

Pavarotti takes a bow
Related material:

1. The performance of a work by
Richard Strauss,
Death and Transfiguration,”
(Tod und Verklärung, Opus 24)
by the Chautauqua Symphony
at Chautauqua Institution on
July 24, 2008

2. Headline of a music review
in today’s New York Times:

Welcoming a Fresh Season of
Transformation and Death

3. The picture of the R. T. Curtis
Miracle Octad Generator
on the cover of the book
Twelve Sporadic Groups:

Cover of 'Twelve Sporadic Groups'

4. Freeman Dyson’s hope, quoted by
Gorenstein in 1986, Ronan in 2006,
and Judge in 2007, that the Monster
group is “built in some way into
the structure of the universe.”

5. Symmetry from Plato to
the Four-Color Conjecture

6. Geometry of the 4×4 Square

7. Yesterday’s entry,
Theories of Everything


There is such a thing

     as a tesseract.

— Madeleine L’Engle

Cover of The New Yorker, April 12, 2004-- Roz Chast, Easter Eggs

For a profile of
L’Engle, click on
the Easter eggs.

Wednesday, July 2, 2008

Wednesday July 2, 2008

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

Sacerdotal Jargon

Wallace Stevens, from
"Credences of Summer" in
Transport to Summer (1947):

"Three times the concentred
     self takes hold, three times
The thrice concentred self,
     having possessed
The object, grips it
     in savage scrutiny…."

In memory of the former
first lady of Brazil,
who died on June 24 —

Emily Dickinson:

Till Summer folds her miracle
As Women — do — their Gown —
Or Priests — adjust the Symbols —
When Sacrament — is done —

Symbols of the
thrice concentred self:

Symbols of the Thrice Concentred Self

The circular symbol is from July 1.
The square symbol is from June 24,
the date of death for the former
first lady of Brazil.

Wallace Stevens quotes Paul Klee:

"'… 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…."

— "The Relations between Poetry and Painting" in The Necessary Angel (Knopf, 1951)

Monday, October 1, 2007

Monday October 1, 2007

Filed under: General,Geometry — Tags: — m759 @ 7:20 AM
Bright as Magnesium


— The New York Times,  
Sept. 30, 2007, on
Blade Runner:
The Final Cut

Institute for Advanced Study, Princeton, N.J.

"The art historian Kirk Varnedoe died on August 14, 2003, after a long and valiant battle with cancer. He was 57. He was a faculty member in the Institute for Advanced Study’s School of Historical Studies, where he was the fourth art historian to hold this prestigious position, first held by the German Renaissance scholar Erwin Panofsky in the 1930s."

Hal Crowther

"His final lecture was an eloquent, prophetic flight of free association….

Varnedoe chose to introduce his final lecture with the less-quoted last words of the android Roy Batty (Rutger Hauer) in Ridley Scott's film Blade Runner: 'I've seen things you people wouldn't believe– attack ships on fire off the shoulder of Orion, bright as magnesium; I rode on the back decks of a blinker and watched C-beams glitter in the dark near the Tannhauser Gate. All those moments will be lost in time, like tears in the rain. Time to die.'"

Related material: 
tears in the rain–

Game Over
(Nov. 5, 2003):

The film "The Matrix," illustrated

Coordinates for generating the Miracle Octad Generator

Thursday, June 28, 2007

Thursday June 28, 2007

Filed under: General — m759 @ 12:06 PM
Cornelius Uhlenbeck

Oct. 18, 1866 – Aug. 12, 1951

“… born at Voorburg near The Hague in Holland, and studied philology at the University of Leiden…. Though he would actually have preferred to graduate in Basque, Uhlenbeck in 1888, when only 22 years old, took his doctor’s degree in Dutch.  It must be here noted that for this degree the requirements in comparative philology were very considerable….” —International Journal of American Linguistics, Jan. 1953

From Uhlenbeck’s A Manual of Sanskrit Phonetics (1898):

The Indogermanic family of languages. The great family of languages, in which Sanskrit belongs, is called the Indogermanic, Indoceltic or Aryan…. The word Indogermanic dates from a time, when it was not yet proved, that the Celtic dialects also make part of our family of languages, and indicates by the combined name of the utmost branches, Indian and Germanic, the whole territory of speech, to which they belong. Now that it is certain, that Celtic also is a member of our family, it would be accurate to replace the word Indogermanic by Indoceltic, because not Germanic, but Celtic is the utmost branch in the Occident. The name Indogermanic however is generally adopted and it would be impossible to supplant it by another. By the word Aryan is generally understood a certain subdivision of the Indogermanic family, viz. the Indo-Iranian, and therefore it would seem unsuitable to use this name also for the whole Indogermanic family.”

An unsuitable Santa:

The image “http://www.log24.com/log/pix07/070628-Santa.jpg” cannot be displayed, because it contains errors.

A Santa understudy:

The image “http://www.log24.com/log/pix07/070628-christianus_cornelius_uhlenbeck.jpg” cannot be displayed, because it contains errors.

Transcript of
Miracle on 34th Street”

KRIS: Bye. Merry Christmas!   

Well, young lady,
what's your name?

MOTHER: I'm sorry.
She doesn't speak English.

She's Dutch. She just came over.

She's been living
in an orphans home...

in Rotterdam ever since...

We've adopted her.

I told her you wouldn't
be able to speak to her...

but when she saw you
in the parade yesterday...

she said you were
"Sinter Claes"...

and you could talk to her.

I didn't know what to do.

KRIS: Hello. [Speaking Dutch]

[Speaking Dutch]

[Singing in Dutch]

DORIS: Now do you understand?

Related material:

Pope Approves Wider
Use of Latin Mass

The image “http://www.log24.com/log/pix07/070628-Sanctus.jpg” cannot be displayed, because it contains errors.

(Click on image for details),


Seminar für Klassische Philologie
 der Universität Basel

The image “http://www.log24.com/log/pix07/070628-UnivBasel.gif” cannot be displayed, because it contains errors.

Indogermanistische Bibliothek

Sunday, June 10, 2007

Sunday June 10, 2007

Filed under: General — m759 @ 2:00 AM
Like a Melody

An excerpt from
The Miracle of the Bells
quoted in
A Mass for Lucero

“‘A pretty girl–          
 is like a melody—- !’
But that was always
Bill Dunnigan’s      
Song of Victory….  
Thus thought the…
press agent for     
       ‘The Garden of the Soul.'”

Ay que bonito es volar  
    A las dos de la mañana
— “La Bruja

For a rendition by
Salma Hayek, click
on the picture below.

Related material:
Log24 entries for
May 18, 2007.

Wednesday, May 30, 2007

Wednesday May 30, 2007

Filed under: General — m759 @ 10:00 PM

Al Gore and the
Absence of Truth

"Evil is a negation, because
it is the absence of truth."

— Mary Baker Eddy,
founder of Christian Science,
in Science and Health
with Key to the Scriptures,
(Boston, 1906,
page 186, line 11)

 M. Scott Peck on evil:

"There are quite popular
systems of thought these days,
such as Christian Science
or the Course in Miracles,
which define evil as unreality.
It is a half-truth. The spirit of evil
is one of unreality, but it itself
is real. It really exists."

"We must not fall back into Saint
Augustine's now discarded doctrine
of the 'privatio boni,' whereby evil
was defined as the absence of good.
Satan's personality cannot be
characterized simply by
an absence, a nothingness."

People of the Lie:
The Hope for Healing Human Evil,
by Morgan Scott Peck, 1986.
(Touchstone paperback,
2nd ed., 1998, page 208)

Al Gore on M. Scott Peck:

Al Gore trains a global army – USATODAY.com

"Peck wrote that 'Evil is the absence
of truth,' " Gore says, his fingers laced
together at the waist, eyes scanning
eager faces as he wraps up his remarks …

— www.usatoday.com/news/nation/
2007-04-24-gore-trainees_N.htm – 55k
— Google search 5/30/07

He did?

The image “http://www.log24.com/log/pix07/070531-Gore.jpg” cannot be displayed, because it contains errors.

"The greatest trick the Devil ever pulled
was convincing the world he didn't exist."
Verbal Kint in "The Usual Suspects"

Monday, May 28, 2007

Monday May 28, 2007

Filed under: General,Geometry — Tags: , — m759 @ 5:00 PM

and a Finite Model

Notes by Steven H. Cullinane
May 28, 2007

Part I: A Model of Space-Time

The following paper includes a figure illustrating Penrose’s model of  “complexified, compactified Minkowski space-time as the Klein quadric in complex projective 5-space.”
The image “http://www.log24.com/log/pix07/070528-Twistor.jpg” cannot be displayed, because it contains errors.

Click on picture to enlarge.

For some background on the Klein quadric and space-time, see Roger Penrose, “On the Origins of Twistor Theory,” from Gravitation and Geometry: A Volume in Honor of Ivor Robinson, Bibliopolis, 1987.

Part II: A Corresponding Finite Model


The Klein quadric also occurs in a finite model of projective 5-space.  See a 1910 paper:

G. M. Conwell, The 3-space PG(3,2) and its group, Ann. of Math. 11, 60-76.

Conwell discusses the quadric, and the related Klein correspondence, in detail.  This is noted in a more recent paper by Philippe Cara:

The image “http://www.log24.com/log/pix07/070528-Quadric.jpg” cannot be displayed, because it contains errors.


As Cara goes on to explain, the Klein correspondence underlies Conwell’s discussion of eight heptads.  These play an important role in another correspondence, illustrated in the Miracle Octad Generator of R. T. Curtis, that may be used to picture actions of the large Mathieu group M24.

Related material:

The projective space PG(5,2), home of the Klein quadric in the finite model, may be viewed as the set of 64 points of the affine space AG(6,2), minus the origin.

The 64 points of this affine space may in turn be viewed as the 64 hexagrams of the Classic of Transformation, China’s I Ching.

There is a natural correspondence between the 64 hexagrams and the 64 subcubes of a 4x4x4 cube.  This correspondence leads to a natural way to generate the affine group AGL(6,2).  This may in turn be viewed as a group of over a trillion natural transformations of the 64 hexagrams.

Geometry of the I Ching.
“Once Knecht confessed to his teacher that he wished to learn enough to be able to incorporate the system of the I Ching into the Glass Bead Game.  Elder Brother laughed.  ‘Go ahead and try,’ he exclaimed.  ‘You’ll see how it turns out.  Anyone can create a pretty little bamboo garden in the world.  But I doubt that the gardener would succeed in incorporating the world in his bamboo grove.'”
— Hermann Hesse, The Glass Bead Game,
  translated by Richard and Clara Winston

Saturday, May 26, 2007

Saturday May 26, 2007

Filed under: General — m759 @ 1:09 PM

A Baffled Reader

A reader this morning commented on my first Xanga entry (July 20, 2002):

“To set one up (which I have not done because I don’t want anyone to know what I think),” … William Safire regarding “blogs”.

I still don’t know what you think.  Yet … I try, try, try.

Here’s one thing that I think– today– based on my “Hate Speech for Harvard,” on “Devil in the Details” (Log24, May 18 and 23), and, more recently, on

  1. last evening’s PA lottery number 005,
  2. the I Ching hexagram of the same number, and
  3. the New Yorker issue linked to at the end of the previous entry:
Revised New Yorker cover from 5/21/07
Revised version of the
New Yorker cover of 5/21/07
Commentary on the cover
by the PA lottery
on 5/25/07
in the form of the
evening number, 005.
In the I Ching, this
is the number of


See also the previous entry
and Natalie Angier’s sneer
at a politician’s call for
prayer, which, she
said, involved the
“assumption that prayer is
some sort of miracle
Vicks VapoRub.”

Detail from the
5/21/07 New Yorker:

Detail, New Yorker cover, 5/21/07


Hexagram 5: Waiting (Nourishment)

Clouds rise up to heaven:
The image of WAITING.
Thus the superior man
eats and drinks,
Is joyous and
of good cheer.


Friday, May 25, 2007

Friday May 25, 2007

Filed under: General,Geometry — Tags: , , , — m759 @ 7:11 AM
Dance and the Soul

From Log24 on
this date last year:

"May there be an ennui
of the first idea?
What else,
prodigious scholar,
should there be?"

— Wallace Stevens,
"Notes Toward a
Supreme Fiction"

The Associated Press,
May 25, 2007–

Thought for Today:
"I hate quotations.
 Tell me what you know."
— Ralph Waldo Emerson

[Journals, on May 3, 1849]

The First Idea:

The Line, by S. H. Cullinane

Four Elements:

Four Elements (Diamond)

Square Dance:

Square Dance (Diamond Theorem)

This "telling of what
I know" will of course
mean little to those
who, like Emerson,
have refused to learn
through quotations.

For those less obdurate
than Emerson —Harold Bloom
on Wallace Stevens

and Paul Valery's
   "Dance and the Soul"–

"Stevens may be playful, yet seriously so, in describing desire, at winter's end, observing not only the emergence of the blue woman of early spring, but seeing also the myosotis, whose other name is 'forget-me-not.' Desire, hearing the calendar hymn, repudiates the negativity of the mind of winter, unable to bear what Valery's Eryximachus had called 'this cold, exact, reasonable, and moderate consideration of human life as it is.' The final form of this realization in Stevens comes in 1950, in The Course of a Particular, in the great monosyllabic line 'One feels the life of that which gives life as it is.' But even Stevens cannot bear that feeling for long. As Eryximachus goes on to say in Dance and the Soul:

A cold and perfect clarity is a poison impossible to combat. The real, in its pure state, stops the heart instantaneously….[…] To a handful of ashes is the past reduced, and the future to a tiny icicle. The soul appears to itself as an empty and measurable form. –Here, then, things as they are come together, limit one another, and are thus chained together in the most rigorous and mortal* fashion…. O Socrates, the universe cannot for one instant endure to be only what it is.

Valery's formula for reimagining the First Idea is, 'The idea introduces into what is, the leaven of what is not.' This 'murderous lucidity' can be cured only by what Valery's Socrates calls 'the intoxication due to act,' particularly Nietzschean or Dionysiac dance, for this will rescue us from the state of the Snow Man, 'the motionless and lucid observer.'" —Wallace Stevens: The Poems of Our Climate

* "la sorte… la plus mortelle":
    mortal in the sense
   "deadly, lethal"

Other quotations

(from March 28,
the birthday of
Reba McEntire):

Logical Songs

Reba McEntire, Saturday Evening Post, Mar/Apr 1995

Logical Song I

"When I was young, it seemed that
Life was so wonderful, a miracle,
Oh it was beautiful, magical
And all the birds in the trees,
Well they'd be singing so happily,
Joyfully, playfully watching me"

Logical Song II

"You make me feel so young,
You make me feel like
Spring has sprung
And every time I see you grin
I'm such a happy in-

You and I are
Just like a couple of tots
Running across the meadow
Picking up lots
Of forget-me-nots"

Wednesday, March 28, 2007

Wednesday March 28, 2007

Filed under: General — m759 @ 1:00 PM
Logical Songs

Reba McEntire, Saturday Evening Post, Mar/Apr 1995

Logical Song I

“When I was young, it seemed that
Life was so wonderful, a miracle,
Oh it was beautiful, magical
And all the birds in the trees,
Well they’d be singing so happily,
Joyfully, playfully watching me”

Logical Song II

“You make me feel so young,
You make me feel like
Spring has sprung
And every time I see you grin
I’m such a happy in-

You and I are
Just like a couple of tots
Running across the meadow
Picking up lots
Of forget-me-nots

Wednesday, February 28, 2007

Wednesday February 28, 2007

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

The title of Euclid’s Elements is, in Greek, Stoicheia.

From Lectures on the Science of Language,
by Max Muller, fellow of All Souls College, Oxford.
New York: Charles Scribner’s Sons, 1890, pp. 88-90 –


“The question is, why were the elements, or the component primary parts of things, called stoicheia by the Greeks? It is a word which has had a long history, and has passed from Greece to almost every part of the civilized world, and deserves, therefore, some attention at the hand of the etymological genealogist.

Stoichos, from which stoicheion, means a row or file, like stix and stiches in Homer. The suffix eios is the same as the Latin eius, and expresses what belongs to or has the quality of something. Therefore, as stoichos means a row, stoicheion would be what belongs to or constitutes a row….

Hence stoichos presupposes a root stich, and this root would account in Greek for the following derivations:–

  1. stix, gen. stichos, a row, a line of soldiers
  2. stichos, a row, a line; distich, a couplet
  3. steichoestichon, to march in order, step by step; to mount
  4. stoichos, a row, a file; stoichein, to march in a line

In German, the same root yields steigen, to step, to mount, and in Sanskrit we find stigh, to mount….

Stoicheia are the degrees or steps from one end to the other, the constituent parts of a whole, forming a complete series, whether as hours, or letters, or numbers, or parts of speech, or physical elements, provided always that such elements are held together by a systematic order.”

Saturday, February 3, 2007

Saturday February 3, 2007

Filed under: General — m759 @ 11:07 PM
Catholic Schools Week,

Jan. 28 – Feb. 3, 2007,

 PA Lottery 2/3/07: Day 373, Evening 401

For 373, see

For 401, see 4/01:
April 1 at Noon.

“Feel lucky? 
Well, do you?”

Tuesday, December 5, 2006

Tuesday December 5, 2006

Filed under: General — m759 @ 5:01 AM

Today in History
(via The Associated Press)

On this date (Dec. 5):

In 1776, the first scholastic fraternity in America, Phi Beta Kappa, was organized at the College of William and Mary in Williamsburg, Va.

In 1791, composer Wolfgang Amadeus Mozart died in Vienna, Austria, at age 35.

In 2006, author Joan Didion is 72.

Joan Didion, The White Album:

“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.

Or at least we do for a while. I am talking here about a time when I began to doubt the premises of all the stories I had ever told myself, a common condition but one I found troubling.”

An Alternate History

(based on entries of
the past three days):


England, 932 A.D. —
A kingdom divided….”

Introduction to “Spamalot”

A Story That Works

  • “There is the dark, eternally silent, unknown universe;
  • there are the friend-enemy minds shouting and whispering their tales and always seeking the three miracles
    • that minds should really touch, or
    • that the silent universe should speak, tell minds a story, or (perhaps the same thing)
    • that there should be a story that works, that is all hard facts, all reality, with no illusions and no fantasy;
  • and lastly, there is lonely, story-telling, wonder-questing, mortal me.”

    Fritz Leiber in “The Button Molder

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