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.