OFFSET
0,1
COMMENTS
The bipyramid on an n-simplex base is the direct sum of an n-simplex and a segment. It can also be seen as two (n+1)-simplices augmented together at their base.
Note that all facets are simplices.
A deltatope is a polytope whose all cells are regular simplices (a priori not necessarily having the same size). Every polygon, being regular or not, is a 2-deltatope by definition. There are 8 convex 3-deltatopes or deltahedra (regular tetrahedron, regular octahedron, regular icosahedron, regular triangular bipyramid, regular pentagonal bipyramid and three others), 5 convex 4-deltatopes (regular 5-cell, regular 16-cell, regular 600-cell, regular tetrahedral bipyramid and regular icosahedral bipyramid) and 3 in dimension d >= 5 (regular d-simplex, regular d-orthoplex and regular bipyramid on a (d-1)-simplex base). Note that the regular orthoplex is the regular bipyramid on a hypercube base. It turns out that all cells of a deltatope are congruent (i.e., having the same size) in all nontrivial dimensions (dimension >= 3). See Dr. Richard Klitzing's answer to the Math Overflow question "4-polytopes with only one kind of regular facet" for dimension 4, and Gjergji Zaimi's answer to the question "Convex deltahedra in higher dimensions" for dimension >= 5.
More generally, a convex polytope whose all cells are regular polytopes of the same kind is either regular or a deltatope. See the article of Roswitha Blind.
The symmetry group of the bipyramid on an n-simplex base, generated by the symmetries of the n-simplex and the vertical reflexion that commute, is S_{n+1} X C_2 (with Coxeter notation [2,3^(n-1)]). See the Math Stack Exchange link.
LINKS
Jianing Song, Table of n, a(n) for n = 0..5150 (Rows n = 0..100)
Math Overflow, 4-polytopes with only one kind of regular facet.
Math Overflow, Convex deltahedra in higher dimensions.
Mathematics Stack Exchange, Group symmetries of a trigonal bipyramidal molecule.
Roswitha Blind, Jürgen Tölke and Jörg M. Wills, Konvexe Polytope mit regulären Facetten im R^n (n>=4) (in German), Contributions to Geometry: Proceedings of the Geometry-Symposium held in Siegen June 28, 1978 to July 1, 1978.
FORMULA
T(n,k) = 2*(number of k-facets of the (n+1)-simplex) - (number of k-facets of the n-simplex) = 2*binomial(n+2,k+1) - binomial(n+1,k+1) for 0 <= k <= n-1; T(n,n) = 2*(number of n-facets of the (n+1)-simplex) - (2 cells as base) = 2*binomial(n+2,n+1) - 2 = 2*(n+1).
EXAMPLE
The triangle T(n, k) begins:
n\k 0 1 2 3 4 5 6 7 8 9 10
0 2
1 4 4
2 5 9 6
3 6 14 16 8
4 7 20 30 25 10
5 8 27 50 55 36 12
6 9 35 77 105 91 49 14
7 10 44 112 182 196 140 64 16
8 11 54 156 294 378 336 204 81 18
9 12 65 210 450 672 714 540 285 100 20
10 13 77 275 660 1122 1386 1254 825 385 121 22
n = 0: the segment has 2 vertices;
n = 1: the quadrilateral has 4 vertices and 4 sides;
n = 2: the triangular bipyramid has 5 vertices, 9 edges and 6 faces;
n = 3: the tetrahedral bipyramid has 6 vertices, 14 edges, 16 faces and 8 cells.
PROG
(PARI) T(n, k) = if(k<n, 2*binomial(n+2, k+1) - binomial(n+1, k+1), 2*(n+1))
CROSSREFS
KEYWORD
AUTHOR
Jianing Song, Oct 12 2023
STATUS
approved