A338801 Irregular table read by rows: The number of k-faced polyhedra, where k>=4, created when an n-prism, formed from two n-sided regular polygons joined by n adjacent rectangles, is internally cut by all the planes defined by any three of its vertices.

17, 0, 1, 72, 24, 575, 450, 232, 60, 15, 0, 3, 1728, 1668, 948, 144, 24, 12, 8799, 10080, 6321, 3052, 898, 490, 161, 14, 35, 14, 7, 22688, 24080, 12784, 4160, 1248, 272, 80, 32, 78327, 101142, 70254, 39708, 19584, 6894, 2369, 1062, 351, 54, 27, 18, 27, 36, 11, 165500, 203220, 134860, 62520, 21240, 5720, 1080, 300, 100, 20

Offset

3,1

Comments

See A338783 for further details and images for this sequence.

The author thanks Zach J. Shannon for assistance in producing the images for this sequence.

Links

Table of n, a(n) for n=3..64.

Hyung Taek Ahn and Mikhail Shashkov, Geometric Algorithms for 3D Interface Reconstruction.

Scott R. Shannon, 3-prism, showing the 18 polyhedra post-cutting and exploded. Each piece has been moved away from the origin by a distance proportional to the average distance of its vertices from the origin. Red shows the seventeen 4-faced polyhedra, orange the single 6-faced polyhedron.

Scott R. Shannon, 7-prism, showing the 8799 4-faced polyhedra.

Scott R. Shannon, 7-prism, showing the 10080 5-faced polyhedra.

Scott R. Shannon, 7-prism, showing the 6321 6-faced polyhedra.

Scott R. Shannon, 7-prism, showing the 3052 7-faced polyhedra.

Scott R. Shannon, 7-prism, showing the 898 8-faced polyhedra.

Scott R. Shannon, 7-prism, showing the 490 9-faced polyhedra.

Scott R. Shannon, 7-prism, showing the 161 10-faced polyhedra.

Scott R. Shannon, 7-prism, showing the 14 11-faced, 35 12-faced, 14 13-faced, 7 14-faced polyhedra. These are colored white, black, yellow, red respectively. None of these are visible on the surface.

Scott R. Shannon, 7-prism, showing all 29871 polyhedra. The 4,5,6,7,8,9,10 faced polyhedra are colored red, orange, yellow, green, blue, indigo, violet respectively. The 11,12,13,14 faced polyhedra are not visible on the surface.

Scott R. Shannon, 10-prism, showing all 594560 polyhedra.

Eric Weisstein's World of Mathematics, Prism.

Wikipedia, Prism (geometry).

Formula

Sum of row n = A338783(n).

Example

The triangular 3-prism is cut with 6 internal planes defined by all 3-vertex combinations of its 6 vertices. This leads to the creation of seventeen 4-faced polyhedra and one 6-faced polyhedra, eighteen pieces in all. The single 6-faced polyhedra lies at the very center of the original 3-prism.
The 9-prism is cut with 207 internal planes leading to the creation of 319864 pieces. It is noteworthy in creating all k-faced polyhedra from k=4 to k=18.
The table begins:
17,0,1;
72,24;
575,450,232,60,15,0,3;
1728,1668,948,144,24,12;
8799,10080,6321,3052,898,490,161,14,35,14,7;
22688,24080,12784,4160,1248,272,80,32;
78327,101142,70254,39708,19584,6894,2369,1062,351,54,27,18,27,36,11;
165500,203220,134860,62520,21240,5720,1080,300,100,20;

Crossrefs

Cf. A338783 (number of polyhedra), A338808 (antiprism), A338622 (Platonic solids), A333543 (n-dimensional cube).

Sequence in context: A341690 A341689 A243776 * A198631 A185685 A144692

Adjacent sequences: A338798 A338799 A338800 * A338802 A338803 A338804

Keyword

nonn,tabf

Author

Scott R. Shannon, Nov 10 2020

Status

approved