Log24

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.

Singer 7-Cycles

Click on images for details.

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

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.)

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.