In mathematics, the Miracle Octad Generator, or MOG, is a mathematical tool introduced by Rob T. Curtis for studying the

Mathieu group In group theory, a topic in abstract algebra, the Mathieu groups are the five sporadic simple groups ''M''11, ''M''12, ''M''22, ''M''23 and ''M''24 introduced by . They are multiply transitive permutation groups on 11, 12, 22, 23 or 24 objec ...

binary Golay code In mathematics and electronics engineering, a binary Golay code is a type of linear error-correcting code used in digital communications. The binary Golay code, along with the ternary Golay code, has a particularly deep and interesting connection ...

and

Leech lattice In mathematics, the Leech lattice is an even unimodular lattice Λ24 in 24-dimensional Euclidean space which is one of the best models for the kissing number problem. It was discovered by . It may also have been discovered (but not published) by Er ...

Description

The Miracle Octad Generator is a 4x6 array of

combination In mathematics, a combination is a selection of items from a set that has distinct members, such that the order of selection does not matter (unlike permutations). For example, given three fruits, say an apple, an orange and a pear, there are ...

s describing any point in 24-dimensional space. It preserves all of the symmetries and

maximal subgroup In mathematics, the term maximal subgroup is used to mean slightly different things in different areas of algebra. In group theory, a maximal subgroup ''H'' of a group ''G'' is a proper subgroup, such that no proper subgroup ''K'' contains ''H'' ...

s of the

M₂₄, namely the monad group, duad group, triad group, octad group, octern group, sextet group, trio group and duum group. It can therefore be used to study all of these symmetries.

Golay code

Another use for the Miracle Octad Generator is to quickly verify codewords of the

. Each element of the Miracle Octad Generator can store either a '1' or a '0', usually displayed as an

asterisk The asterisk ( ), from Late Latin , from Ancient Greek , , "little star", is a Typography, typographical symbol. It is so called because it resembles a conventional image of a star (heraldry), heraldic star. Computer scientists and Mathematici ...

and blank space, respectively. Each column and the top row have a property known as the ''count'', which is the number of asterisks in that particular line. One of the criteria for a set of 24 coordinates to be a codeword in the binary Golay code is for all seven counts to be of the same parity. The other restriction is that the ''scores'' of each column form a word in the hexacode. The score of a column can be either 0, 1, ω, or ω-bar, depending on its contents. The score of a column is evaluated by the following rules: * If a column contains exactly one asterisk, it has a score of 0 if it resides in the top row, 1 if it is in the second row, ω for the third row, and ω-bar for the bottom row. * Simultaneously complementing every bit in a column does not affect its score. * Complementing the bit in the top row does not affect its score, either. A codeword can be derived from just its top row and score, which proves that there are exactly 4096 codewords in the binary Golay code.

MiniMOG

John Horton Conway John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician. He was active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many b ...

developed a 4 × 3 array known as the MiniMOG. The MiniMOG provides the same function for the Mathieu group M₁₂ and

ternary Golay code In coding theory, the ternary Golay codes are two closely related error-correcting codes. The code generally known simply as the ternary Golay code is an 1, 6, 53-code, that is, it is a linear code over a ternary alphabet; the relative distan ...

as the Miracle Octad Generator does for M₂₄ and binary Golay code, respectively. Instead of using a quaternary hexacode, the MiniMOG uses a ternary tetracode.

Cullinane diamond theorem

The Cullinane diamond theorem is a theorem about the Galois geometry underlying the MOG.Cullinane diamond theorem
at the Encyclopedia of Mathematics The theorem also explains symmetry properties of the sort of chevron or diamond designs often found on quilts. It was first published in the journal Computer Graphics and Art, Vol. 2, No. 1, in February 1977. An updated version was published in Notices of the American Mathematical Society, Issue 192, Vol. 26, No. 2, February 1979, as Abstract 79T-A37, on pages A-193-194. Background on the theorem and its author is at https://bio.site/cullinane. The theorem is available as
The Square Model of Fano's 1892 Finite 3-Space
at Harvard's DASH repository. Some historical background for the theorem: Ibáñez, Raúl,
The Truchet Tiles and the Diamond Puzzle
" In April 2024, Ibáñez described the theore
in UNIÓN
a periodical for mathematics educators throughout Latin America. For a cryptographic application, se
a 2016 paper from SASTRA University
—

Notes

References

* * *V. Harish, N. R. Kumar and N. R. Raajan, "New visual secret sharing scheme for gray-level images using diamond theorem correlation pattern structure," 2016 International Conference on Circuit, Power and Computing Technologies (ICCPCT), Nagercoil, India, 2016, pp. 1-5, doi: 10.1109/ICCPCT.2016.7530155.

External links

* {{planetmath, urlname=miracleoctadgenerator Sporadic groups