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

Author

Scott R. Shannon, Nov 08 2020

Keywords

• nonn
• more

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.

Examples

			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.
		

Links

• 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).

Crossrefs

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