174 geometry/cycle.polynomial.p
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This article is from the Puzzles FAQ, by Chris Cole chris@questrel.questrel.com and Matthew Daly mwdaly@pobox.com with numerous contributions by others.
174 geometry/cycle.polynomial.p
What are the cycle polynomials for the Platonic solids?
geometry/cycle.polynomial.s
For future reference, here are the cycle polynomials for the five platonic
solids (and I threw in the tesseract for good measure). Most combinatoric
coloring problems are simple plug-ins to these polynomials. For details,
see any book on combinatorics that presents Polya counting theory.
tetrahedron: (x1^4+3x2^2+8x1*x3)/12
cube: (x1^6+6x2^3+3x1^2*x2^2+8x3^2+6x1^2*x4)/24
octahedron: (x1^8+9x2^4+8x1^2*x3^2+6x4^2)/24
dodecahedron: (x1^12+15x2^6+20x3^4+24x1^2*x5^2)/60
icosahedron: (x1^20+15x2^10+20x1^2*x3^6+24x5^4)/60
tesseract: (32x6^4+x2^12+48x8^3+x1^24+24x1^2*x2^11+12x2^2*x4^5+32x3^8+12x4^6
+18x1^4*x2^10+12x1^4*x4^5)/192
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