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 (i.e. checkerboards), and partial tilings and honeycombs which include infinite figures, which he calls semichecks.

The regular forms can be labeled by Schläfli symbols, and nearly all have Coxeter-Dynkin diagrams.

[edit] Regular figures in n-dimensions


4-simplex
4-orthoplex
4-cube
  1. n-ic pyramid - The simplex family {3,3,3...} - Image:CD ring.pngImage:CD 3b.pngImage:CD dot.pngImage:CD 3b.png...Image:CD 3b.pngImage:CD dot.pngImage:CD 3b.pngImage:CD dot.png
    • 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 orthoplex family {3,3,...4} - Image:CD dot.pngImage:CD 4.pngImage:CD dot.pngImage:CD 3b.png...Image:CD 3b.pngImage:CD dot.pngImage:CD 3b.pngImage:CD ring.png
    • Includes square, octahedron, 16-cell, ...
    • It is bounded by 2n (n-1)-ic pyramids, 2n-1 of which meet in each vertex.
    • and by 2r+1n!/[(r+1)!(n-r-1)!] r-ic pyramids, 2r(n-1)!/(r!(n-r-1)!] r-ic pyramids meet in each t-ic pyramid.
  3. n-ic cubic - The hypercube family {4,3,3.....} - Image:CD ring.pngImage:CD 4.pngImage:CD dot.pngImage:CD 3b.png...Image:CD 3b.pngImage:CD dot.pngImage:CD 3b.pngImage:CD dot.png
    • Includes the square, cube, tesseract, ...
    • It is bounded by 2n (n-1)-ic cubes, n of which meet in each vertex; and by 2n-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.

square tiling
cubic honeycomb
  1. (n-1)-ic Check - cubic {4,3.....3,4} honeycomb - Image:CD ring.pngImage:CD 4.pngImage:CD dot.pngImage:CD 3b.png...Image:CD 3b.pngImage:CD dot.pngImage:CD 4.pngImage:CD dot.png
    • Includes the apeirogon, square tiling, cubic honeycomb, Tesseractic tetracomb, ...
    • Analogous to the pattern of a chess board. This figure is infinite.
    • It is bounded by ∞ (n-1)-ic cubes, 2n-1 of which meet at each vertex;
    • and by (n-1)!/[(n-r-1)!r!] ∞ r-ic cubes, 2r(n-t-1)!/[(r-t)!(n-r-1)!] of which meet at each vertex, where r=1 to (n-2).
    • It has ∞ vertices, 2r-1(n-t-1)!/[(r-t)!(n-r-1)!] r-ic cubes meet in each t-ic cube.
    • It is self-dual.

[edit] Regular figures in 4-dimensions


24-cell
600-cell
120-cell
  1. Octahedric - 24-cell {3,4,3} - Image:CD ring.pngImage:CD 3b.pngImage:CD dot.pngImage:CD 4.pngImage:CD dot.pngImage:CD 3b.pngImage:CD dot.png = Image:CDW_dot.pngImage:CDW_4.pngImage:CDW_dot.pngImage:CDW_3.pngImage:CDW_ring.pngImage:CDW_3.pngImage:CDW_dot.png - Facets {3,4}
  2. Tetrahedric - 600-cell {3,3,5} - Image:CD dot.pngImage:CD 5.pngImage:CD dot.pngImage:CD 3b.pngImage:CD dot.pngImage:CD 3b.pngImage:CD ring.png - Facets {3,3}
  3. Dodecahedric - 120-cell {5,3,3} - Image:CD ring.pngImage:CD 5.pngImage:CD dot.pngImage:CD 3b.pngImage:CD dot.pngImage:CD 3b.pngImage:CD dot.png - Facets {5,3}

[edit] Regular figures in 5-dimensions

(No image)
{3,4,3,3}
{3,3,4,3}
  1. Octahedric check - {3,4,3,3}: honeycomb - Image:CD ring.pngImage:CD 3b.pngImage:CD dot.pngImage:CD 4.pngImage:CD dot.pngImage:CD 3b.pngImage:CD dot.pngImage:CD 3b.pngImage:CD dot.png - Facets {3,4,3}
  2. Double pyramidal check - {3,3,4,3} honeycomb - Image:CD dot.pngImage:CD 3b.pngImage:CD dot.pngImage:CD 4.pngImage:CD dot.pngImage:CD 3b.pngImage:CD dot.pngImage:CD 3b.pngImage:CD ring.png = Image:CDW_hole.pngImage:CDW_4.pngImage:CDW_dot.pngImage:CDW_3.pngImage:CDW_dot.pngImage:CDW_3.pngImage:CDW_dot.pngImage:CDW_4.pngImage:CDW_dot.png - Facets {3,3,4}

[edit] Semi-regular figures in n-dimensions


Infinite prism
Partial square tiling
Infinite duoprism
Partial cubic honeycomb
  1. (n-1)-ic semi-check (Partial-honeycomb). Analogous to the pattern of a single file of a chess board. (infinite)
    1. Image:CDW ring.pngImage:CDW infin.pngImage:CDW dot.pngImage:CDW 2.pngImage:CDW ring.png (infinite prism) - facets {4} and {∞}
    2. Image:CDW ring.pngImage:CDW infin.pngImage:CDW dot.pngImage:CDW 2.pngImage:CDW ring.pngImage:CDW infin.pngImage:CDW dot.pngImage:CDW 2.pngImage:CDW ring.png (Infinite duoprism) - facets {4,3} and {4,4}
    3. Image:CDW ring.pngImage:CDW infin.pngImage:CDW dot.pngImage:CDW 2.pngImage:CDW ring.pngImage:CDW infin.pngImage:CDW dot.pngImage:CDW 2.pngImage:CDW ring.pngImage:CDW infin.pngImage:CDW dot.pngImage:CDW 2.pngImage:CDW ring.png (Infinite triprism) - facets {4,3,3} and {4,3,4}
    4. ...

[edit] Semi-regular figures in 4-dimensions


Rectified 5-cell
Snub 24-cell
Rectified 600-cell
  1. Tetroctahedric - Rectified 5-cell - Image:CD dot.pngImage:CD 3b.pngImage:CD ring.pngImage:CD 3b.pngImage:CD dot.pngImage:CD 3b.pngImage:CD dot.png - t1{3,3,3} - Facets {3,3} and {3,4}
  2. Tetricosahedric - Snub 24-cell - Image:CD_hole.pngImage:CD_3.pngImage:CD_downbranch-snub.pngImage:CD_3.pngImage:CD_hole.png - s{31,1,1} - Facets {3,3} and {3,5}
  3. Octicosahedric - Rectified 600-cell - Image:CD dot.pngImage:CD 5.pngImage:CD dot.pngImage:CD 3b.pngImage:CD ring.pngImage:CD 3b.pngImage:CD dot.png - t1{3,3,5} - Facets {3,4} and {3,5}

tet-oct
Gyro tet-oct
tet-oct slab
  1. Simple Tetroctahedric Check - Tetrahedral-octahedral honeycomb - Image:CD_ring.pngImage:CD_3b.pngImage:CD_downbranch-00.pngImage:CD_3b.pngImage:CD_4.pngImage:CD_dot.png or Image:CDW_hole.pngImage:CDW_4.pngImage:CDW_dot.pngImage:CDW_3.pngImage:CDW_dot.pngImage:CDW_4.pngImage:CDW_dot.png - Facets {3,3} and {3,4}
  2. Complex Tetroctahedric Check - Gyrated alternated cubic honeycomb (Non-Wythoffian) - Facets {3,3} and {3,4}
  3. Tetroctahedric Semi-check - partial honeycomb - Image:CDW_hole.pngImage:CDW_6.pngImage:CDW_dot.pngImage:CDW_3.pngImage:CDW_dot.pngImage:CDW_2b.pngImage:CDW_hole.png - Facets {3,3}, {3,4}, and {3,6}

Semi-regular figures in 5 or more dimensions


Demipenteract
E6
E7
E8
  1. 5-ic semi-regular - Demipenteract - Image:CDW_hole.pngImage:CDW_4.pngImage:CDW_dot.pngImage:CDW_3b.pngImage:CDW_dot.pngImage:CDW_3b.pngImage:CDW_dot.pngImage:CDW_3b.pngImage:CDW_dot.png or Image:CD dot.pngImage:CD 3b.pngImage:CD dot.pngImage:CD 3b.pngImage:CD_downbranch-00.pngImage:CD 3b.pngImage:CD ring.png - 11,2 - facets {3,3,3} and {3,3,4}
  2. 6-ic semi-regular - E6 polytope - Image:CD dot.pngImage:CD 3b.pngImage:CD dot.pngImage:CD 3b.pngImage:CD_downbranch-00.pngImage:CD 3b.pngImage:CD dot.pngImage:CD 3b.pngImage:CD ring.png - 22,1 - facets {3,3,3,3} and {3,3,3,4}
  3. 7-ic semi-regular - E7 polytope - Image:CD dot.pngImage:CD 3b.pngImage:CD dot.pngImage:CD 3b.pngImage:CD_downbranch-00.pngImage:CD 3b.pngImage:CD dot.pngImage:CD 3b.pngImage:CD dot.pngImage:CD 3b.pngImage:CD ring.png - 32,1 - facets {3,3,3,3,3} and {3,3,3,3,4}
  4. 8-ic semi-regular - E8 polytope - Image:CD dot.pngImage:CD 3b.pngImage:CD dot.pngImage:CD 3b.pngImage:CD_downbranch-00.pngImage:CD 3b.pngImage:CD dot.pngImage:CD 3b.pngImage:CD dot.pngImage:CD 3b.pngImage:CD dot.pngImage:CD 3b.pngImage:CD ring.png - 42,1 - facets {3,3,3,3,3,3} and {3,3,3,3,3,4}
  5. 9-ic check - E8 lattice - Image:CD dot.pngImage:CD 3b.pngImage:CD dot.pngImage:CD 3b.pngImage:CD_downbranch-00.pngImage:CD 3b.pngImage:CD dot.pngImage:CD 3b.pngImage:CD dot.pngImage:CD 3b.pngImage:CD dot.pngImage:CD 3b.pngImage:CD dot.pngImage:CD 3b.pngImage:CD ring.png - 52,1 - facets {3,3,3,3,3,3} and {3,3,3,3,3,4}

[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