A338783 Number of polyhedra formed 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.

18, 96, 1335, 4524, 29871, 65344, 319864, 594560

Offset

3,1

Comments

For an n-prism, formed from two n-sided regular polygons joined by n adjacent rectangles, create all possible internal planes defined by connecting any three of its vertices. For example, in the case of a triangular prism this results in 6 planes. Use all the resulting planes to cut the prism into individual smaller polyhedra. The sequence lists the number of resulting polyhedra for prisms with n>=3.

See A338801 for the number and images of the k-faced polyhedra in each prism 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, 3-prism, showing the 6 plane cuts on the external edges and faces.

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 4-faced polyhedra, orange the single 6-faced polyhedron.

Scott R. Shannon, 7-prism, showing the 98 plane cuts on the external edges and faces.

Scott R. Shannon, 7-prism, showing the 29871 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 11,12,13,14 faces are not visible on the surface.

Scott R. Shannon, 7-prism, showing the 29871 polyhedra post-cutting and exploded.

Scott R. Shannon, 10-prism, showing the 275 plane cuts on the external edges and faces

Scott R. Shannon, 10-prism, showing the 594560 polyhedra post-cutting. The 4,5,6,7,8,9 faced polyhedra are colored red, orange, yellow, green, blue, indigo respectively. The polyhedra with 10,11,12,13 faces are not visible on the surface.

Eric Weisstein's World of Mathematics, Prism.

Wikipedia, Prism (geometry).

Example

a(3) = 18. The triangular 3-prism has 6 internal cutting planes resulting in 18 polyhedra; seventeen 4-faced polyhedra and one 6-faced polyhedron.
a(4) = 96. The square 4-prism (a cuboid) has 14 internal cutting planes resulting in 96 polyhedra; seventy-two 4-faced polyhedra and twenty-four 5-faced polyhedra. See A338622.

Crossrefs

Cf. A338801 (number of k-faced polyhedra), A338806 (antiprism), A338571 (Platonic solids), A338622 (k-faced polyhedra in Platonic solids), A333539 (n-dimensional cube).

Sequence in context: A047929 A243995 A264202 * A324304 A118864 A118606

Adjacent sequences: A338780 A338781 A338782 * A338784 A338785 A338786

Keyword

nonn,more

Author

Scott R. Shannon, Nov 08 2020

Status

approved