The DICTO Truncated-Icosahedral Hecatodihedron
Abstract
We describe a convex polyhedron with 102 faces, 80 corners and 180 edges that arises as the seventh stage of our Spheres of Dharma, a cycle of nested icosahedral shells. It is the truncated icosahedron with a low hexagonal pyramid (a Shield) on each of its 20 hexagons, at the one height at which the two triangles of neighbouring Shields across every hexagon–hexagon edge lie in a single plane. Its faces are 12 regular pentagons, 30 rhombi and 60 isosceles triangles; it has only two edge angles. Combinatorially it is Conway's hexakis truncated icosahedron k6tI (132 faces) with its 60 coplanar triangle pairs merged into 30 rhombi. The rhombi and triangles all touch one sphere; the pentagons lie a little inside it, so the solid has no insphere. We name it the DICTO Truncated-Icosahedral Hecatodihedron (hecatodi-, one hundred and two). This corrects an earlier statement of ours that all 102 faces touch one sphere. DICTO · dictospheres.com. Computation and checking with Claude (Anthropic). Discovery #28 in DISCOVERIES.md of https://github.com/DICTOR-Master/kaleidohedra. Version 2. Version 2 corrects "60 pairs" to 30, and adds an exact relation found in a prior-art search: the Hecatodihedron's 20 Shield apices are the canonical pentakis dodecahedron's three-fold vertices, so it is the joined truncated icosahedron (jtI) with its 12 five-fold corners cut off at the pentagon planes, an unnamed solid derived from a known one; and the DICTO Hecatogdoecontahedron, its self-dual compound with the DICTO Ogdoecontahedron (182 faces), whose core and hull are those of the classical truncated icosahedron–pentakis dodecahedron compound. View in the app: https://kaleidohedra.dictospheres.com/?open=cycles
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BibTeX
@misc{dicto2026dicto,
title = {{The DICTO Truncated-Icosahedral Hecatodihedron}},
author = {DICTO},
howpublished = {Zenodo (CERN European Organization for Nuclear Research)},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23360857},
url = {https://doi.org/10.5281/zenodo.23360857},
note = {CC BY-SA},
}
RIS
TY - UNPB
TI - The DICTO Truncated-Icosahedral Hecatodihedron
AU - DICTO
PY - 2026
DA - 2026/10/14
PB - Zenodo (CERN European Organization for Nuclear Research)
DO - 10.5281/zenodo.23360857
UR - https://doi.org/10.5281/zenodo.23360857
AB - We describe a convex polyhedron with 102 faces, 80 corners and 180 edges that arises as the seventh stage of our Spheres of Dharma, a cycle of nested icosahedral shells. It is the truncated icosahedron with a low hexagonal pyramid (a Shield) on each of its 20 hexagons, at the one height at which the two triangles of neighbouring Shields across every hexagon–hexagon edge lie in a single plane. Its faces are 12 regular pentagons, 30 rhombi and 60 isosceles triangles; it has only two edge angles. Combinatorially it is Conway's hexakis truncated icosahedron k6tI (132 faces) with its 60 coplanar triangle pairs merged into 30 rhombi. The rhombi and triangles all touch one sphere; the pentagons lie a little inside it, so the solid has no insphere. We name it the DICTO Truncated-Icosahedral Hecatodihedron (hecatodi-, one hundred and two). This corrects an earlier statement of ours that all 102 faces touch one sphere. DICTO · dictospheres.com. Computation and checking with Claude (Anthropic). Discovery #28 in DISCOVERIES.md of https://github.com/DICTOR-Master/kaleidohedra. Version 2. Version 2 corrects "60 pairs" to 30, and adds an exact relation found in a prior-art search: the Hecatodihedron's 20 Shield apices are the canonical pentakis dodecahedron's three-fold vertices, so it is the joined truncated icosahedron (jtI) with its 12 five-fold corners cut off at the pentagon planes, an unnamed solid derived from a known one; and the DICTO Hecatogdoecontahedron, its self-dual compound with the DICTO Ogdoecontahedron (182 faces), whose core and hull are those of the classical truncated icosahedron–pentakis dodecahedron compound. View in the app: https://kaleidohedra.dictospheres.com/?open=cycles
ER -