Constructions of Polyhedrons with Equal Polyspheres


2004/08/24


There are some constructions with equal polyspheres, which look like polyhedrons with regular faces. In the picture one of the used pieces is blue colored. The table below shows the properties of those figures and the number of spheres needed.

Name Made of Faces Volume n=3
Tetrahedron triangle(n) +..+ triangle(1) 4 triangles n(n+1)(n+2)/6
Square Pyramid square(n) +..+ square(1) 4 triangles,
1 square		 n(n+1)(2n+1)/6
Octahedron square pyramid(n) +
square pyramid(n-1)		 8 triangles n(2n^2+1)/3
Cubeoctahedron octahedron(2n-1) -
6*square pyramid(n-1)		 8 triangles,
6 squares		 (2n-1)(5n^2-5n+3)/3
Truncated Tetrahedron tetrahedron(3n-2) -
4*tetrahedron(n-1)		 4 triangles,
4 hexagons		 n(23n^2-27n+10)/6
Truncated Octahedron octahedron(3n-2) -
6*square pyramid(n-1)		 6 squares,
8 hexagons		 16n^3-33n^2+24n-6

With multiple copies of some polyspheres it is often possible to construct these polyhedrons. For the following pieces some solutions are shown. Torsten Sillke has checked some more pieces on his site.

Click on the numbers to see the layers and some pictures of the constructions.

Name/Size 2 3 4 5 6 7 8 9 10
Tetrahedron 4 10 20 35 56 84 120 165 220
Y4, D5,... D5 Y4, J4 Y4 C3, D3, Y4, D5 D5 Y4, D5
Square Pyramid 5 14 30 55 91 140 204 285 385
L3 J4, Y4, D5 C3, L3, Y4
Octahedron 6 19 44 85 146 231 344 489 670
L3 J4, Y4 D5
Cubeoctahedron 13 55 147 309 561 923 1415 2057 2869
D5 C3, L3
Truncated Tetrahedron 16 68 180 375 676 1106 1688 2445 3400
Y4 C4, J4, Y4 C3, D3, I3, C4, J4, Y4, D5
Truncated Octahedron 38 201 586 1289 2406 4033 6266 9201 12934
C3, L3


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