A347753 Number of polyhedra formed when a row of n adjacent cubes are internally cut by all the planes defined by any three of their vertices.

96, 2968, 42384, 319416

Offset: 1

Author

Scott R. Shannon, Sep 12 2021

Keywords

• nonn
• more

Comments

For a row of n adjacent cubes create all possible planes defined by connecting any three of their vertices. For example, in the case of a single cube this results in fourteen planes; six planes between the pairs of parallel edges connected to each end of the face diagonals, and eight planes from connecting the three vertices adjacent to each corner vertex. Use all the resulting planes to cut the entire solid into individual smaller polyhedra. The sequence lists the numbers of resulting polyhedra for n adjacent cubes.

See A347918 for the number of k-faced polyhedra for each value of n.

Examples

			a(1) = 96. A single cube, with eight vertices, has 14 internal cutting planes resulting in 96 polyhedra. See A333539 and A338571.
a(2) = 2968. Two adjacent cubes, with twelve vertices, have 51 internal cutting planes resulting in 2968 polyhedra.
a(3) = 42384. Three adjacent cubes, with sixteen vertices, have 124 internal cutting planes resulting in 42384 polyhedra.
a(4) = 319416. Four adjacent cubes, with twenty vertices, have 245 internal cutting planes resulting in 319416 polyhedra.
		

Links

• Scott R. Shannon, The 245 cutting planes on the surface of 4 adjacent cubes.
• Scott R. Shannon, The surface of the 4 adjacent cubes after cutting. The 4-, 5-, 6-, 7-, 8-, and 9-faced polyhedra created by the planes are colored red, orange, yellow, green, blue, and indigo, respectively. The 10-, 11-, and 12-faced polyhedra are not visible on the surface. See also A347918.
• Scott R. Shannon, The 4 adjacent cubes after cutting exploded. Each of the 319416 polyhedra is moved away from the center of the solid a distance proportional to the average distance of its vertices from the center.

Crossrefs

Cf. A347918 (number of k-faced polyhedra), A333539 (n-dimensional cube), A338571 (Platonic solids), A338783 (n-prism), A338809 (n-bipyramid), A007588.

Formula

a(1) = A333539(3).

Conjectured formula for the number of internal cutting planes for n adjacent cubes is A007588(n+1).