The FUJI Family: Pyramids on Flat-Ended Triangles
Year: 2026
Type: preprint
Abstract: FUJI is the hexagonal pyramid that closes the corners of our thirteen-icosahedron cluster, the Lotus seed [1]. Its base is a flat-ended triangle: a hexagon with all angles 120° and sides 1 and 1/φ alternating. The same shape kept turning up elsewhere, in our cuboctahedral PET-bottle spheres and in the colouring of the Lotus seed, so we asked whether FUJI belongs to a family. It does. Put an apex over a flat-ended triangle with sides 1 and r, with all six side edges of length 1. This gives a one-parameter family FUJI(r), running from the regular tetrahedron (r = 0) through FUJI (r = 1/φ) to the flat regular hexagon (r = 1). We derive the base angles and prove: The base angles (derived): cos L = (1 + 2r)/3 and cos² S = (2 + r)/(3(2 − r)). The antiprism rule: the member whose base continues the plane of the n-gonal antiprism's triangles is r = 2 cos(π/n) − 1, so that 1 + r is the short diagonal of the regular n-gon (for n = 3, its side). The wheel bound: flat wheels crossing along different spokes number at most 3 (square), 4 (hexagonal) or 6 (decagonal), and five octagonal wheels are impossible. The octahedron gives the tetrahedron, the square antiprism √2 − 1, and the pentagonal antiprism (the middle of the icosahedron) gives FUJI. The members turn up in several places: The golden hexagon of a construction after Odom is FUJI's base; the same construction with any extension gives every member with r < 1. The space-filling truncated tetrahedron of Damasceno, Engel and Glotzer has the r = 1/2 member's base as its faces. In a breathing pyrochlore the kagome hexagons are flat-ended triangles, with r equal to the breathing ratio; as a Wythoff construction it tiles space for every r. The equal-depth cube–octahedron solid has the cuboctahedral member's base, r = (√3 − 1)/2, as its eight triangles, and its cube corners are exactly that member's apexes; we prove the two conditions are the same. Cleo-Icos and Antonius: the icosahedron's central pyramid is the corner of the prolate golden rhombohedron, which splits into Cleo-Icos + Antonius + Cleo-Icos; so the two fill space, two to one, and build the rhombic hexecontahedron. DICTO · dictospheres.com. Computation, checking and drafting with Claude (Anthropic). Data, scripts, proofs and checks: studies/2026-10-14-fuji-family in https://github.com/DICTOR-Master/kaleidohedra. Version 2 adds Figure 11: Pacioli's golden rectangles and the 20 Cleo-Icos. Version 3 credits the Cleo-Icos as the unit tetrahedron of the Mackay icosahedron (Mackay 1962; Ino 1969; Marks 1984; Kabai 2014) and the golden rhombohedron as a distorted fcc cell (Prodan et al. 2017). Version 4 (2026-10-14): the Cleo-Icos in classical geometry, as the twin of the dodecahedron's central pyramid in Hypsicles' "Book XIV" of Euclid (c. 150 BC); the golden-hexagon construction credited to Sándor Kabai's Wolfram Demonstration "Hexagons and the Golden Ratio" (2008). No other changes. View in the app: https://kaleidohedra.dictospheres.com/?open=breathing
Author: DICTO
Language: English
Topic: Mathematics and Applications
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221788584