Year: 2010
Type: preprint
Abstract: In Universal PiTet a single rule T builds each solid from the last: every corner v of a solid S gives the face plane v·x = v·v, through v and at right angles to the line from the centre, and T(S) is the solid these planes bound. Started at the icosidodecahedron it closes into a four-step cycle. DICTO proposed running it from the DICTO Ogdoecontahedron (OG), and also running it the other way. The reverse step P takes as new corners the foot points of S's face planes, the points of those planes nearest the centre. From the OG, four steps of T give four solids with 92, 180, 212 and 270 corners, the DICTO KRE, PHE, TPP and DPHP, and four steps of P give four with 80, 132, 140 and 162 corners, the DICTO TPH, TQH, TQP and TQHD. All eight are proved exactly in ℚ(√5), each with the full icosahedral symmetry, and each step of P is undone exactly by T. Several have regular faces: 12 regular pentagons (PHE, TPH, TQP), 20 regular hexagons (PHE, DPHP, TQH, TQHD); the KRE has 30 rhombi and 60 kites. The family does not cycle. It nests as shells for three steps up, OG ⊂ KRE ⊂ PHE ⊂ TPP, and two steps down, TQH ⊂ TPH ⊂ OG, and each of these shells, and also the shell TQHD ⊂ TQP, is filled exactly by one piece per face of the inner solid. Both directions then break in the same way: 60 corners of the TPP stand outside the DPHP, and 60 corners of the TQP outside the TQH. In the plane, T and P are the classical antipedal and pedal constructions; the polyhedral pedal construction was posed as an open problem in 2010. We found the eight solids in no source. Companion to The DICTO Ogdoecontahedron (#31), DICTO DOI 10.5281/zenodo.23404688, and Universal PiTet: The DICTO PiTet Proposition (#38), DICTO DOI 10.5281/zenodo.23463305. Finding #39 on dictospheres.com. Scripts, exact checks and data: https://github.com/DICTOR-Master/kaleidohedra/tree/058c2fb/studies/2026-10-19-five-fold-octet, also attached here (og-family-study-058c2fb.zip).
Source: arXiv (Cornell University)
Author: Smarandache, Florentin
Language: English
Topic: Mathematics and Applications
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W1641991810