**See also other posts now tagged
**

**Natural Diagram .**

**Related remarks by J. H. Conway —**

**See also other posts now tagged
**

**Related remarks by J. H. Conway —**

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From a Log24 search for Schwartz + “The Sun” —

**“Looking carefully at Golay’s code
is like staring into the sun.”**

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**Prosaic —**

**Poetic —**

**Prosaic —**

**“These devices may have some
theoretical as well as practical value.“**

**Poetic —**

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See also The Lexicographic Octad Generator (LOG) (July 13, 2020)

and Octads and Geometry (April 23, 2020).

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

**By Steven H. Cullinane, July 13, 2020**

Background —

The Miracle Octad Generator (MOG)

**of R. T. Curtis (Conway-Sloane version) —**

A basis for the Golay code, excerpted from a version of

the code generated in lexicographic order, in

**“Constructing the Extended Binary Golay Code“**

**Ben Adlam**

**Harvard University**

**August 9, 2011:**

000000000000000011111111

000000000000111100001111

000000000011001100110011

000000000101010101010101

000000001001011001101001

000000110000001101010110

000001010000010101100011

000010010000011000111010

000100010001000101111000

001000010001001000011101

010000010001010001001110

100000010001011100100100

Below, each vector above has been reordered within

a 4×6 array, by Steven H. Cullinane, to form twelve

independent Miracle Octad Generator vectors

(as in the Conway-Sloane SPLAG version above, in

which Curtis’s earlier heavy bricks are reflected in

their vertical axes) —

01 02 03 04 05 . . . 20 21 22 23 24 --> 01 05 09 13 17 21 02 06 10 14 18 22 03 07 11 15 19 23 04 08 12 16 20 24 0000 0000 0000 0000 1111 1111 --> 0000 11 0000 11 0000 11 0000 11 as in the MOG. 0000 0000 0000 1111 0000 1111 --> 0001 01 0001 01 0001 01 0001 01 as in the MOG. 0000 0000 0011 0011 0011 0011 --> 0000 00 0000 00 0011 11 0011 11 as in the MOG. 0000 0000 0101 0101 0101 0101 --> 0000 00 0011 11 0000 00 0011 11 as in the MOG. 0000 0000 1001 0110 0110 1001 --> 0010 01 0001 10 0001 10 0010 01 as in the MOG. 0000 0011 0000 0011 0101 0110 --> 0000 00 0000 11 0101 01 0101 10 as in the MOG. 0000 0101 0000 0101 0110 0011 --> 0000 00 0101 10 0000 11 0101 01 as in the MOG. 0000 1001 0000 0110 0011 1010 --> 0100 01 0001 00 0001 11 0100 10 as in the MOG. 0001 0001 0001 0001 0111 1000 --> 0000 01 0000 10 0000 10 1111 10 as in the MOG. 0010 0001 0001 0010 0001 1101 --> 0000 01 0000 01 1001 00 0110 11 as in the MOG. 0100 0001 0001 0100 0100 1110 --> 0000 01 1001 11 0000 01 0110 00 as in the MOG. 1000 0001 0001 0111 0010 0100 --> 10 00 00 00 01 01 00 01 10 01 11 00 as in the MOG (heavy brick at center).

**Update at 7:41 PM ET the same day —**

A check of SPLAG shows that the above result is not new:

**And at 7:59 PM ET the same day —**

Conway seems to be saying that at some unspecified point in the past,

M.J.T. Guy, examining the lexicographic Golay code, found (as I just did)

that weight-8 lexicographic Golay codewords, when arranged naturally

in 4×6 arrays, yield certain intriguing visual patterns. If the MOG existed

at the time of his discovery, he would have identified these patterns as

those of the MOG. (Lexicographic codes have apparently been

known since 1960, the MOG since the mid-1970s.)

* Addendum at 4 AM ET the next day —

See also **Logline** (Walpurgisnacht 2013).

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(Adapted from Eightfold Geometry, a note of April 28, 2010.

See also the recent post Geometry of 6 and 8.)

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Possible title:

**A new graphic approach
to an old geometric approach
to a new combinatorial approach
to an old algebraic approach
to M**

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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.

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A professor at Harvard has written about

“the urge to seize and display something

real beyond artifice.”

He reportedly died on January 3, 2015.

An image from this journal on that date:

Another *Gitterkrieg* image:

*The 24-set * Ω *of R. T. Curtis*

Click on the images for related material.

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The *Galois tesseract* appeared in an early form in the journal

*Computer Graphics and Art* , Vol. 2, No. 1, February 1977—

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—

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.

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The *hypercube * model of the 4-space over the 2-element Galois field 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).

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

Curtis himself first described these 35 square MOG patterns

combinatorially, (as his title indicated) rather than

algebraically or geometrically:

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):

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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—

The “35 structures” of the abstract were listed, with an application to

Latin-square orthogonality, in a note from December 1978—

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 *M*_{24}, 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:

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

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*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

between

(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 |

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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 M_{24},” *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.

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**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.

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**Singer 7-Cycles**

Click on images for details.

The 1985 Cullinane version gives some algebraic background for the 1987 Curtis version.

The Singer referred to above is *James* Singer. See his “A Theorem in Finite Projective Geometry and Some Applications to Number Theory,” *Transactions of the American Mathematical Society* **43** (1938), 377-385.For other singers, see Art Wars and today’s obituaries.

Some background: the Log24 entry of this date seven years ago, and the entries preceding it on Las Vegas and painted ponies.

Comments Off on Wednesday October 14, 2009

**A Wealth of**

**Algebraic Structure**

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 *M*_{24}, 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 M_{24}**, by R. T. Curtis. Abstract:

“In this paper, we define *M*_{24} from scratch as the subgroup of *S*_{24} 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.)

Comments Off on Monday January 5, 2009

*Elements*

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 –

*Stoicheia*

“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:–

*stix*, gen.*stichos*, a row, a line of soldiers*stichos*, a row, a line;*distich*, a couplet*steicho*,*estichon*, to march in order, step by step; to mount*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.”

**Example:**

**The Miracle Octad Generator of R. T. Curtis**

For the geometry of these *stoicheia*, see

The Smallest Perfect Universe and

Finite Geometry of the Square and Cube.

Comments Off on Wednesday February 28, 2007

**Big Rock**

Thanks to Ars Mathematica, a link to everything2.com:

“In mathematics, a big rock is a result which is vastly more powerful than is needed to solve the problem being considered. Often it has a difficult, technical proof whose methods are not related to those of the field in which it is applied. You say ‘I’m going to hit this problem with a big rock.’ Sard’s theorem is a good example of a big rock.”

** Another example:**

Properties of the Monster Group of R. L. Griess, Jr., may be investigated with the aid of the Miracle Octad Generator, or MOG, of R. T. Curtis. See the MOG on the cover of a book by Griess about some of the 20 sporadic groups involved in the Monster:

The MOG, in turn, illustrates (via Abstract 79T-A37, *Notices of the American Mathematical Society*, February 1979) the fact that the group of automorphisms of the affine space of four dimensions over the two-element field is also the natural group of automorphisms of an arbitrary 4×4 array.

This affine group, of order 322,560, is also the natural group of automorphisms of a family of graphic designs similar to those on traditional American quilts. (See the diamond theorem.)

This top-down approach to the diamond theorem may serve as an illustration of the “big rock” in mathematics.

For a somewhat simpler, bottom-up, approach to the theorem, see Theme and Variations.

For related literary material, see Mathematics and Narrative and The Diamond as Big as the Monster.

Comments Off on Saturday July 29, 2006

**Bagombo Snuff Box
(in memory of
Burt Kerr Todd)
**

“Well, it may be the devil

or it may be the Lord

But you’re gonna have to

serve somebody.”

— “Bob Dylan”

(pseudonym of Robert Zimmerman),

quoted by “Bob Stewart”

on July 18, 2005

“Bob Stewart” may or may not be the same person as “crankbuster,” author of the “Rectangular Array Theorem” or “RAT.” This “theorem” is intended as a parody of the “Miracle Octad Generator,” or “MOG,” of R. T. Curtis. (See the Usenet group sci.math, “Steven Cullinane is a Crank,” July 2005, messages 51-60.)

“Crankbuster” has registered at Math Forum as a teacher in Sri Lanka (formerly Ceylon). For a tall tale involving Ceylon, see the short story “Bagombo Snuff Box” in the book of the same title by Kurt Vonnegut, who has at times embodied– like Martin Gardner and “crankbuster“– “*der Geist, der stets verneint*.”

Here is my own version (given the alleged Ceylon background of “crankbuster”) of a Bagombo snuff box:

Related material:

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