

A339538


Irregular table read by rows: The number of kfaced polyhedra, where k>=4, created when an elongated nbipyramid, with faces that are squares and equilateral triangles, is internally cut by all the planes defined by any three of its vertices.


1



258, 336, 60, 424, 584, 208, 48, 8, 8830, 16090, 12210, 5040, 1210, 260, 80, 10
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OFFSET

3,1


COMMENTS

For an elongated nbipyramid with faces that are squares and equilateral triangles, formed by joining the two halves of an ngonal bipyramid by an nprism, create all possible internal planes defined by connecting any three of its vertices. Use all the resulting planes to cut the polyhedron into individual smaller polyhedra. The sequence lists the number of resulting kfaced polyhedra, where k>=4, for elongated nbipyramids where 3 <= n <= 5. These three elongated bipyramids are the only possible elongated bipyramids that are Johnson solids, i.e., their faces are all regular polygons.


LINKS

Table of n, a(n) for n=3..18.
Hyung Taek Ahn and Mikhail Shashkov, Geometric Algorithms for 3D Interface Reconstruction.
Scott R. Shannon, Elongated 3bypyramid, showing the 29 plane cuts on the external edges and faces.
Scott R. Shannon, Elongated 3bypyramid, showing the 258 4faced polyhedra.
Scott R. Shannon, Elongated 3bypyramid, showing all 654 polyhedra post cutting. The polyhedra are colored red,orange,yellow for face counts 4 to 6 respectively. No 6faced polyhedra are visible on the surface.
Scott R. Shannon, Elongated 3bypyramid, showing all 654 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 6faced polyhedra can now be seen.
Scott R. Shannon, Elongated 4bypyramid, showing the 40 plane cuts on the external edges and faces.
Scott R. Shannon, Elongated 4bypyramid, showing the 424 4faced polyhedra.
Scott R. Shannon, Elongated 4bypyramid, showing the 48 7faced polyhedra.
Scott R. Shannon, Elongated 4bypyramid, showing all 1272 polyhedra post cutting. The polyhedra are colored red,orange,yellow,green,blue for face counts 4 to 8 respectively.
Scott R. Shannon, Elongated 4bypyramid, showing all 1272 polyhedra post cutting and exploded.
Scott R. Shannon, Elongated 5bypyramid, showing the 112 plane cuts on the external edges and faces.
Scott R. Shannon, Elongated 5bypyramid, showing the 8830 4faced polyhedra. This contains very small polyhedra near the peaks of the pyramids due to the convergence of the cutting lines near these points.
Scott R. Shannon, Elongated 5bypyramid, showing the 8830 4faced polyhedra viewed from above.
Scott R. Shannon, Elongated 5bypyramid, showing the 1210 8faced polyhedra viewed from above.
Scott R. Shannon, Elongated 5bypyramid, showing all 43730 polyhedra post cutting. The polyhedra are colored red,orange,yellow,green,blue.indigo,violet,lightblue for face counts 4 to 11 respectively.
Scott R. Shannon, Elongated 5bypyramid, showing all 43730 polyhedra post cutting and exploded.
Eric Weisstein's World of Mathematics, Elongated Square Dipyramid.
Eric Weisstein's World of Mathematics, Johnson Solid.
Wikipedia, Elongated bipyramid.


EXAMPLE

The elongated 5bipyramid has 12 vertices, 25 edges and 15 faces (5 squares and 10 equilateral triangles). It is cut by 112 internal planes defined by any three of its vertices, resulting in the creation of 43730 polyhedra.
The 11 faced polyhedra are unusual in that all 10 are visible on the surface; most polyhedra cut with their own planes have the resulting polyhedra with the most faces near the center of the original polyhedra and are thus not visible on its surface.
No polyhedra with 12 or more faces are created.
The table is:
258, 336, 60;
424, 584, 208, 48, 8;
8830, 16090, 12210, 5040, 1210, 260, 80, 10;


CROSSREFS

Cf. A338825, A339528, A339348, A339349, A338622, A338801, A338808.
Sequence in context: A250741 A031514 A173981 * A252264 A202892 A158230
Adjacent sequences: A339535 A339536 A339537 * A339539 A339540 A339541


KEYWORD

nonn,fini,full,tabf


AUTHOR

Scott R. Shannon, Dec 08 2020


STATUS

approved



