A338806 Number of polyhedra formed when an n-antiprism, formed from two n-sided regular polygons joined by 2n adjacent alternating triangles, is internally cut by all the planes defined by any three of its vertices.

8, 195, 834, 6365, 22770, 81769, 271702, 688793

Offset

3,1

Comments

For an n-antiprism, formed from two n-sided regular polygons joined by 2n adjacent alternating triangles, create all possible internal planes defined by connecting any three of its vertices. For example, in the case of a triangular 3-antiprism this results in 3 planes. Use all the resulting planes to cut the prism into individual smaller polyhedra. The sequence lists the number of resulting polyhedra for antiprisms with n>=3.

See A338808 for the number and images of the k-faced polyhedra in each antiprism dissection.

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

Links

Table of n, a(n) for n=3..10.

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

Scott R. Shannon, 4-antiprism, showing the 16 plane cuts on the external edges and faces.

Scott R. Shannon, 4-antiprism, showing the 195 polyhedra post-cutting. The 4-faced polyhedra are colored red, the 5-faced polyhedra are colored orange. The 6 and 8 faced polyhedra are not visible on the surface.

Scott R. Shannon, 4-antiprism, showing the 195 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. The 6 and 8 faced polyhedra are colored yellow and green respectively.

Scott R. Shannon, 7-antiprism, showing the 91 plane cuts on the external edges and faces.

Scott R. Shannon, 7-antiprism, showing the 22770 polyhedra post-cutting. The 4,5,6,7,8,9 faced polyhedra are shown as red, orange, yellow, green, blue, indigo respectively. The polyhedra with 10,11,12,14,21 faces are not visible on the surface.

Scott R. Shannon, 7-antiprism, showing the 22770 polyhedra post-cutting and exploded.

Scott R. Shannon, 10-antiprism, showing the 280 plane cuts on the external edges and faces.

Scott R. Shannon, 10-antiprism, showing the 688793 polyhedra post-cutting. The 4,5,6,7,8,9,10 faced polyhedra are colored red, orange, yellow, green, blue, indigo, violet respectively. The polyhedra with 11,12,20 faces are not visible on the surface.

Eric Weisstein's World of Mathematics, Antiprism.

Wikipedia, Antiprism.

Example

a(3) = 8. The 3-antiprism is cut with 3 internal planes resulting in 8 polyhedra, all 8 pieces having 4 faces.
a(4) = 195. The 4-antiprism is cut with 16 internal planes resulting in 195 polyhedra; 128 with 4 faces, 56 with 5 faces, 8 with 6 faces, and 3 with 8 faces. Note the number of 8-faced polyhedra is not a multiple of 4 - they lie directly along the z-axis so are not symmetric with respect to the number of edges forming the regular n-gons.

Crossrefs

Cf. A338808 (number of k-faced polyhedra), A338783 (regular prism), A338571 (Platonic solids), A333539 (n-dimensional cube).

Sequence in context: A172340 A103500 A232911 * A119299 A363986 A024287

Adjacent sequences: A338803 A338804 A338805 * A338807 A338808 A338809

Keyword

nonn,more

Author

Scott R. Shannon, Nov 10 2020

Status

approved