User:Tomruen/semiregular polytopes

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[edit] Semiregular figures (polytopes and honeycombs)

Extending the definition of semiregular to higher dimensional polytopes and tessellations, Thorold Gosset in 1900 published a complete list of regular and semiregular figures as defined by being vertex-transitive and constructed from convex regular facets and convex vertex figures.

His list included infinite tessellations (tilings and honeycombs) which he calls checks, and partial tilings and honeycombs which include infinite figures, which he calls semichecks.

Regular figures in n-dimensions:

1. n-ic pyramid - The Simplex family {3,3,3...} - ...
   • Includes equilateral triangle, tetrahedron, pentachoron ...
   • It is bounded by n+1 (n-1)-ic pyramids, n of which meet in each vertex;
   • and by (n+1)!/[(r+1)!(n-r)!]' r-ic pyramids, n!/[r!(n-r)!] of which meet at each vertex, where r=1 to (n-2).
   • It has 'n+1' vertices.
   • (n+t)!/[(r+t)!(n-r)!] r-ic pyramids meet in each t-ic pyramid.
2. n-ic double pyramid - the cross-polytope family {3,3,...4} - ...
   • Includes square, octahedron, 16-cell, ...
   • It is bounded by 2ⁿ (n-1)-ic pyramids, 2^n-1 of which meet in each vertex.
   • and by 2^r+1n!/[(r+1)!(n-r-1)!] r-ic pyramids, 2^r(n-1)!/(r!(n-r-1)!] r-ic pyramids meet in each t-ic pyramid.
3. n-ic cubic - The measure polytope family {4,3,3.....} - ...
   • Includes the square, cube, tesseract, ...
   • It is bounded by 2n (n-1)-ic cubes, n of which meet in each vertex; and by 2^n-r*n!/[r!(n-r)!] r-ic cubes, n~/[r!(n-r)!] of which meet at each vertex, where r=1 to (n-2).
   • It has 2ⁿ vertices.
   • (n-t)!/[(r-t)!(n-r)!] r-ic cubes meet in each t-ic cube.
4. (n-1)-ic Check - cubic {4,3.....3,4} honeycomb - ...
   • Includes the apeirogon, square tiling, cubic honeycomb, ...
   • Analogous to the pattern of a chess board. This figure is infinite.
   • It is bounded by ∞ (n-1)-ic cubes, 2^n-1 of which meet at each vertex;
   • and by (n-1)!/[(n-r-1)!r!] ∞ r-ic cubes, 2^r(n-t-1)!/[(r-t)!(n-r-1)!] of which meet at each vertex, where r=1 to (n-2).
   • It has ∞ vertices, 2^r-1(n-t-1)!/[(r-t)!(n-r-1)!] r-ic cubes meet in each t-ic cube.
   • It is self-dual.

Regular figures in 4-dimensions:

1. Octahedric - 24-cell {3,4,3} - =
2. Tetrahedric - 600-cell {3,3,5} -
3. Dodecahedric - 120-cell {5,3,3} -

Regular figures in 5-dimensions:

1. Octahedric check - {3,4,3,3}: honeycomb -
2. Double pyramidal check - {3,3,4,3} honeycomb - =

Semi-regular figures in n-dimensions:

1. (n-1)-ic semi-check (Partial-honeycomb). Analogous to the pattern of a single file of a chess board. (infinite)
   1. (infinite prism)
   2. (Infinite duoprism)
   3. (Infinite triprism)
   4. ...

Semi-regular figures in 4-dimensions:

1. Tetroctahedric - Rectified 5-cell -

1. Tetricosahedric - Snub 24-cell -
2. Octicosahedric - Rectified 600-cell -
3. Simple Tetroctahedric Check - Tetrahedral-octahedral honeycomb - or
4. Complex Tetroctahedric Check - Gyrated alternated cubic honeycomb (Non-Wythoffian)
5. Tetroctahedric Semi-check - partial honeycomb -

Semi-regular figures in 5-dimensions:

1. 5-ic semi-regular - Demipenteract - or

Semi-regular figures in 6-dimensions:

1. 6-ic semi-regular - E6 polytope -

Semi-regular figures in 7-dimensions:

1. 7-ic semi-regular - E7 polytope -

Semi-regular figures in 8-dimensions:

1. 8-ic semi-regular - E8 polytope -

Semi-regular figures in 9-dimensions:

1. 9-ic semi-check - T9 honeycomb -

[edit] References

• T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
• Alicia Boole Stott Geometrical deduction of semiregular from regular polytopes and space fillings, Verhandelingen of the Koninklijke academy van Wetenschappen width unit Amsterdam, Eerste Sectie 11,1, Amsterdam, 1910
  • Stott, A. B. "Geometrical Deduction of Semiregular from Regular Polytopes and Space Fillings." Verhandelingen der Koninklijke Akad. Wetenschappen Amsterdam 11, 3-24, 1910.
  • Alicia Boole Stott, "Geometrical deduction of semiregular from regular polytopes and space fillings," Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam, (eerste sectie), Vol. 11, No. 1, pp. 1-24 plus 3 plates, 1910.
  • Stott, A. B. 1910. "Geometrical Deduction of Semiregular from Regular Polytopes and Space Fillings." Verhandelingen der Koninklijke Akad. Wetenschappen Amsterdam
• Schoute, P. H., Analytical treatment of the polytopes regularly derived from the regular polytopes, Ver. der Koninklijke Akad. van Wetenschappen te Amsterdam (eerstie sectie), vol 11.5, 1913.
• H.S.M. Coxeter: Regular and Semi-Regular Polytopes, Part I, Mathematische Zeitschrift, Springer, Berlin, 1940
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
• H.S.M. Coxeter: Regular and Semi-Regular Polytopes, Part II, Mathematische Zeitschrift, Springer, Berlin, 1985
• H.S.M. Coxeter: Regular and Semi-Regular Polytopes, Part III, Mathematische Zeitschrift, Springer, Berlin, 1988
• G.Blind and R.Blind, "The semi-regular polyhedra", Commentari Mathematici Helvetici 66 (1991) 150--154