A339538 - OEIS

%I #20 Dec 16 2020 19:29:59

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

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

%C For an elongated n-bipyramid with faces that are squares and equilateral triangles, formed by joining the two halves of an n-gonal bipyramid by an n-prism, 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 k-faced polyhedra, where k>=4, for elongated n-bipyramids 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.

%H Hyung Taek Ahn and Mikhail Shashkov, <a href="https://cnls.lanl.gov/~shashkov/papers/ahn_geometry.pdf">Geometric Algorithms for 3D Interface Reconstruction</a>.

%H Scott R. Shannon, <a href="/A339538/a339538.jpg">Elongated 3-bypyramid, showing the 29 plane cuts on the external edges and faces</a>.

%H Scott R. Shannon, <a href="/A339538/a339538_11.jpg">Elongated 3-bypyramid, showing the 258 4-faced polyhedra</a>.

%H Scott R. Shannon, <a href="/A339538/a339538_3.jpg">Elongated 3-bypyramid, showing all 654 polyhedra post cutting</a>. The polyhedra are colored red,orange,yellow for face counts 4 to 6 respectively. No 6-faced polyhedra are visible on the surface.

%H Scott R. Shannon, <a href="/A339538/a339538_4.jpg">Elongated 3-bypyramid, showing all 654 polyhedra post cutting and exploded</a>. 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-faced polyhedra can now be seen.

%H Scott R. Shannon, <a href="/A339538/a339538_1.jpg">Elongated 4-bypyramid, showing the 40 plane cuts on the external edges and faces</a>.

%H Scott R. Shannon, <a href="/A339538/a339538_6.jpg">Elongated 4-bypyramid, showing the 424 4-faced polyhedra</a>.

%H Scott R. Shannon, <a href="/A339538/a339538_8.jpg">Elongated 4-bypyramid, showing the 48 7-faced polyhedra</a>.

%H Scott R. Shannon, <a href="/A339538/a339538_5.jpg">Elongated 4-bypyramid, showing all 1272 polyhedra post cutting</a>. The polyhedra are colored red,orange,yellow,green,blue for face counts 4 to 8 respectively.

%H Scott R. Shannon, <a href="/A339538/a339538_7.jpg">Elongated 4-bypyramid, showing all 1272 polyhedra post cutting and exploded</a>.

%H Scott R. Shannon, <a href="/A339538/a339538_2.jpg">Elongated 5-bypyramid, showing the 112 plane cuts on the external edges and faces</a>.

%H Scott R. Shannon, <a href="/A339538/a339538_12.jpg">Elongated 5-bypyramid, showing the 8830 4-faced polyhedra</a>. This contains very small polyhedra near the peaks of the pyramids due to the convergence of the cutting lines near these points.

%H Scott R. Shannon, <a href="/A339538/a339538_13.jpg">Elongated 5-bypyramid, showing the 8830 4-faced polyhedra viewed from above</a>.

%H Scott R. Shannon, <a href="/A339538/a339538_14.jpg">Elongated 5-bypyramid, showing the 1210 8-faced polyhedra viewed from above</a>.

%H Scott R. Shannon, <a href="/A339538/a339538_9.jpg">Elongated 5-bypyramid, showing all 43730 polyhedra post cutting</a>. The polyhedra are colored red,orange,yellow,green,blue.indigo,violet,light-blue for face counts 4 to 11 respectively.

%H Scott R. Shannon, <a href="/A339538/a339538_10.jpg">Elongated 5-bypyramid, showing all 43730 polyhedra post cutting and exploded</a>.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElongatedSquareDipyramid.html">Elongated Square Dipyramid</a>.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JohnsonSolid.html">Johnson Solid</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Elongated_bipyramid">Elongated bipyramid</a>.

%e The elongated 5-bipyramid 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.

%e 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.

%e No polyhedra with 12 or more faces are created.

%e The table is:

%e 258, 336, 60;

%e 424, 584, 208, 48, 8;

%e 8830, 16090, 12210, 5040, 1210, 260, 80, 10;

%Y Cf. A338825, A339528, A339348, A339349, A338622, A338801, A338808.

%K nonn,fini,full,tabf

%O 3,1

%A _Scott R. Shannon_, Dec 08 2020