{"id":5,"date":"2003-08-10T19:45:42","date_gmt":"2003-08-10T23:45:42","guid":{"rendered":"http:\/\/blogs.law.harvard.edu\/cull\/2003\/08\/10\/the-diamond-theorem\/"},"modified":"2003-08-10T19:45:42","modified_gmt":"2003-08-10T23:45:42","slug":"the-diamond-theorem","status":"publish","type":"post","link":"https:\/\/archive.blogs.harvard.edu\/m759\/2003\/08\/10\/the-diamond-theorem\/","title":{"rendered":"The Diamond Theorem"},"content":{"rendered":"

<\/a><\/p>\n

The Diamond Theorem<\/b><\/font>
(4×4 Case)<\/b><\/font>
\n
\n <\/b><\/big>
\n
\nby Steven H. Cullinane<\/b><\/font>\n<\/div>\n


\n
\n <\/b><\/big><\/p>\n\n\n\n
\n

\"dtheorem-AbstrD.jpg:<\/p>\n<\/td>\n

 We regard
\n the four-diamond figure D at left as a 4×4 array of two-color
\n diagonally-divided square tiles.<\/font><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Let G be the group of 322,560
\n permutations of these 16 tiles generated<\/a>
\n by arbitrarily<\/i> mixing
\n random<\/i>
\n permutations of rows and of columns with
\n permutations of the four 2×2 quadrants.<\/font><\/p>\n

<\/font><\/p>\n

THEOREM: Every G-image of D<\/a> (as at right, below) has some ordinary or
\n color-interchange symmetry.<\/font>\n <\/p>\n

Example:<\/font><\/b><\/big><\/p>\n

\"dtheorem-AbstrExample.jpg:<\/p>\n

where g, a permutation in G, is a product of two disjoint 7-cycles.  Note that Dg has
\n rotational color-interchange symmetry like that of the famed yin-yang
\n symbol.<\/font><\/p>\n
Remarks:<\/b><\/font>\n<\/div>\n<\/div>\n<\/div>\n

<\/p>\n


\nG is isomorphic to the affine group<\/a> A on the linear 4-space
\n over GF(2).  The 35 structures<\/a> of the 840 = 35 x 24 G-images of D<\/a> are
\n isomorphic to the 35 lines<\/a> in the 3-dimensional projective space over
\n GF(2). <\/p>\n

This can be seen by viewing the 35 structures<\/a> as three-sets of line diagrams,
\nbased on the three partitions of the four-set of square two-color tiles
\ninto two two-sets, and indicating the locations of these two-sets of
\ntiles within the 4×4 patterns.  The lines of the line diagrams may be
\nadded in a binary fashion (i.e., 1+1=0).  Each three-set of line diagrams
\nsums to zero — i.e., each diagram in a three-set is the binary sum of the
\nother two diagrams in the set.  Thus, the 35 three-sets of line diagrams
\ncorrespond to the 35 three-point lines<\/a> of the finite
\nprojective 3-space PG(3,2). <\/p>\n

<\/font><\/p>\n

For example, here are the line diagrams
for the figures above:
\n<\/font><\/div>\n
\n\"dtheorem-LineDiagrams.gif:<\/p>\n


\nShown below are the 15 possible line diagrams
\n<\/font>
\n
\nresulting from row\/column\/quadrant permutations.
\n<\/font>
\n
\nThese 15 diagrams may, as noted above, be regarded
\n<\/font>
\n
\nas the 15 points of the projective 3-space PG(3,2).<\/font><\/div>\n

<\/p>\n

\n \"dtheorem-ProjPoints.gif:<\/p>\n

\nThe symmetry of the line diagrams accounts for the symmetry
\nof the two-color patterns<\/a>. 
\n(A proof shows that a 2nx2n two-color triangular half-squares pattern
\nwith such line diagrams must have a 2×2 center with a symmetry, and
\nthat this symmetry must be shared by the entire pattern.)
\n<\/font>\n <\/div>\n<\/p><\/div>\n


Among the 35 structures of the
\n840 4×4 arrays of tiles, orthogonality<\/a> (in the sense of Latin-square
\northogonality) corresponds to skewness of lines in the finite
\nprojective space PG(3,2). 
\nThis was stated by the author in a 1978 note<\/a>.  (The note apparently had little effect. 
\nA quarter-century later, P. Govaerts, D. Jungnickel, L. Storme, and J.
\nA. Thas wrote that skew (i.e., nonintersecting) lines in a projective space seem “at first sight not at all related<\/a>” to orthogonal Latin squares.)<\/p>\n

We can define sums and products so that the G-images of D<\/a>  generate an ideal
\n (1024 patterns characterized by all horizontal or vertical “cuts” being
\n uninterrupted) of a ring of 4096 symmetric patterns. There is an infinite
\n family of such “diamond” rings, isomorphic to rings of matrices over
\n GF(4).<\/font><\/p>\n

The proof uses a decomposition technique<\/a> for functions into a finite field that might be of more general use.<\/p>\n

The underlying geometry<\/a>
\nof the 4×4 patterns is closely related to the Miracle Octad Generator<\/a> (pdf)
\nof R. T. Curtis– used in the construction of the Steiner system
\nS(5,8,24)<\/a>— and hence is also related to the Leech lattice<\/a>, which, as Walter Feit has remarked<\/a>, “is a blown up version of S(5,8,24).”<\/p>\n

For a movable JavaScript version of these 4×4 patterns, see  The Diamond 16 Puzzle<\/a>. <\/font><\/p><\/div>\n\n\n\n
\n

The above is an expanded version of Abstract 79T-A37, “Symmetry invariance
\nin a diamond ring,” by S. H. Cullinane,
\nNotices of the American Mathematical Society, February 1979,
\npages A-193,194.<\/font>\n<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

For a discussion of other cases of the theorem, click here<\/a>.<\/p>\n

Posted Sept. 22, 2005; replaces previous post.<\/font><\/font><\/p>\n","protected":false},"excerpt":{"rendered":"

The Diamond Theorem (4×4 Case) by Steven H. Cullinane We regard the four-diamond figure D at left as a 4×4 array of two-color diagonally-divided square tiles. Let G be the group of 322,560 permutations of these 16 tiles generated by arbitrarily mixing random permutations of rows and of columns with permutations of the four 2×2 […]<\/p>\n","protected":false},"author":1179,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1459],"tags":[],"class_list":["post-5","post","type-post","status-publish","format-standard","hentry","category-m759stories"],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/archive.blogs.harvard.edu\/m759\/wp-json\/wp\/v2\/posts\/5","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/archive.blogs.harvard.edu\/m759\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/archive.blogs.harvard.edu\/m759\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/m759\/wp-json\/wp\/v2\/users\/1179"}],"replies":[{"embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/m759\/wp-json\/wp\/v2\/comments?post=5"}],"version-history":[{"count":0,"href":"https:\/\/archive.blogs.harvard.edu\/m759\/wp-json\/wp\/v2\/posts\/5\/revisions"}],"wp:attachment":[{"href":"https:\/\/archive.blogs.harvard.edu\/m759\/wp-json\/wp\/v2\/media?parent=5"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/m759\/wp-json\/wp\/v2\/categories?post=5"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/m759\/wp-json\/wp\/v2\/tags?post=5"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}