A092538 -id:A092538 - cp's OEIS frontend

A092536 Sorted numbers of edges in the Archimedean polyhedra.

18, 24, 36, 36, 48, 60, 60, 72, 90, 90, 120, 150, 180

Offset: 1

Eric W. Weisstein, Feb 26 2004

Prisms and antiprisms are excluded.
Ordering is truncated tetrahedron, cuboctahedron, truncated cube = truncated octahedron, small rhombicuboctahedron, icosidodecahedron = snub cube, great rhombicuboctahedron, truncated dodecahedron = truncated icosahedron, small rhombicosidodecahedron, snub dodecahedron, great rhombicosidodecahedron.
Also the sorted number of edges of the Catalan polyhedra, the duals of the Archimedean polyhedra. - Felix Fröhlich, Mar 05 2018

Tom Gettys, Archimedean Polyhedra
George W. Hart, Archimedean Polyhedra
Ha Le, Archimedean Polyhedra
Eric Weisstein's World of Mathematics, Archimedean Solid
Wikipedia, Archimedean solid
Wikipedia, Catalan solid

Cf. A092537, A092538.

A092537 Sorted numbers of faces in the Archimedean polyhedra.

Original entry on oeis.org

8, 14, 14, 14, 26, 26, 32, 32, 32, 38, 62, 62, 92

Offset: 1

Eric W. Weisstein, Feb 26 2004

Prisms and antiprisms are excluded.
Ordering is truncated tetrahedron, cuboctahedron = truncated cube = truncated octahedron, great rhombicuboctahedron = small rhombicuboctahedron, icosidodecahedron = truncated dodecahedron = truncated icosahedron, snub cube, great rhombicosidodecahedron = small rhombicosidodecahedron, snub dodecahedron.
Also the sorted number of vertices of the Catalan polyhedra, the duals of the Archimedean polyhedra. - Felix Fröhlich, Mar 05 2018

Tom Gettys, Archimedean Polyhedra
George W. Hart, Archimedean Polyhedra
Ha Le, Archimedean Polyhedra
Eric Weisstein's World of Mathematics, Archimedean Solid
Wikipedia, Archimedean solid
Wikipedia, Catalan solid

Cf. A092536, A092538.

A299114 Number of sides of a face of an Archimedean solid.

Original entry on oeis.org

3, 4, 5, 6, 8, 10

Offset: 1

Jonathan Sondow, Feb 02 2018

Values of n for which the regular n-gon is a face of some Archimedean solid.

Remarkably, the same is true for Johnson solids. Indeed, before Johnson (1966) and Zalgaller (1967) classified the 92 Johnson solids, Grünbaum and Johnson (1965) proved that the only polygons that occur as faces of a non-uniform regular-faced convex polyhedron (i.e., a Johnson solid) are triangles, squares, pentagons, hexagons, octagons, and decagons.

Branko Grünbaum, Norman Johnson, The faces of a regular-faced polyhedron, J. Lond. Math. Soc. 40, 577-586 (1965).
Norman W. Johnson, Convex Polyhedra with Regular Faces, Canadian Journal of Mathematics, 18 (1966), 169-200.
Joseph Malkevitch, Regular-Faced Polyhedra: Remembering Norman Johnson, AMS Feature Column, Jan. 2018.
Eric Weisstein's World of Mathematics, Archimedean Solid
Wikipedia, List of Johnson solids
Victor A. Zalgaller, Convex Polyhedra with Regular Faces, Zap. Nauchn. Sem. LOMI, 1967, Volume 2. Pages 5-221 (Mi znsl1408).

Cf. A092536, A092537, A092538, A242731, A242732, A242733.

A325176 Numbers k such that an Archimedean 4-polytope with k vertices exists.

Original entry on oeis.org

5, 8, 10, 16, 20, 24, 30, 32, 48, 60, 64, 96, 100, 120, 144, 192, 288, 384, 576, 600, 720, 1152, 1200, 1440, 2400, 3600, 7200, 14400

Offset: 1

Felix Fröhlich, Sep 05 2019

Also, numbers n such that a Catalan 4-polytope (dual of an Archimedean 4-polytope) with n cells (3-D facets) exists.

			Vertices |  Example 4-polytope
         |  (Schläfli symbol)
----------------------------------------
      5  | {3,3,3}
      8  | {3,3,4}
     10  | r{3,3,3}
     16  | {4,3,3}
     20  | t{3,3,3}
     24  | {3,4,3}
     30  | rr{3,3,3}
     32  | r{4,3,3}
     48  | t{3,3,4}
     60  | tr{3,3,3}
     64  | t{4,3,3}
     96  | rr{4,3,3}
    100  | grand antiprism
    120  | {3,3,5}
    144  | t_0,3{3,4,3}
    192  | t{3,4,3}
    288  | rr{3,4,3}
    384  | t_0,1,2,3{3,3,4}
    576  | tr{3,4,3}
    600  | {5,3,3}
    720  | r{3,3,5}
   1152  | t_0,1,2,3{3,4,3}
   1200  | r{5,3,3}
   1440  | t{3,3,5}
   2400  | t{5,3,3}
   3600  | rr{5,3,3}
   7200  | tr{5,3,3}
  14400  | t_0,1,2,3{5,3,3}

J. H. Conway, H. Burgiel and Chaim Goodman-Strauss, The Symmetries of Things, A K Peters, Ltd., 2008, pp. 389-403, ISBN 978-1-56881-220-5.

Marco Möller, 4-dimensionale Archimedische Polytope, Results in Mathematics, Vol. 46, No. 3-4 (2004), 271-360 (in German) [Subscription required].
Marco Möller, 4-dimensionale Archimedische Polytope, Dissertation, Universität Hamburg, 2004 (in German).
Wikipedia, Schläfli symbol - Polychora and honeycombs
Wikipedia, Uniform 4-polytope

Cf. A092538.

cp's OEIS Frontend

A092536 Sorted numbers of edges in the Archimedean polyhedra.

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A092537 Sorted numbers of faces in the Archimedean polyhedra.

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A299114 Number of sides of a face of an Archimedean solid.

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A325176 Numbers k such that an Archimedean 4-polytope with k vertices exists.

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