Log24

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.)