AXE, CLEO and FUJI: the Gap Pieces of the Icosahedron
- 1Independent artist, Osaka, Japan
Abstract
Regular icosahedra cannot fill space: two of them sharing an edge leave 83.62° around it. DICTO's thirteen-icosahedron cluster, the Lotus seed, closes every gap with three small pieces, AXE, CLEO and FUJI. With whole icosahedra they fill an icosahedron of edge φ² exactly, in 113 pieces, or in 63 pieces as DICTO first built it: a dodecahedron, 12 J11, 30 AXE and 20 FUJI. AXE and CLEO are two of the Mosseri–Sadoc tiles, scaled by 1/τ; here they arise as the gap pieces of a cluster of whole icosahedra. FUJI, a hexagonal pyramid, is new. It is not a union of Mosseri–Sadoc tiles at its own scale or larger (proofs by the Dehn invariant and by the angles of its base). The three pieces recur throughout DICTO's work: the Dodecahedral Chain (great stellated and excavated dodecahedra), the dodecahedron as a small icosahedron with 12 caps, 30 AXE and 20 CLEO, the Chain of Karma at every scale, and the Stella-Corona network. The names are DICTO's: AXE is shaped like an axe head; CLEO is named after Cleopatra's Needle in London; FUJI has the shape of Mount Fuji and the colour of the fuji flower, Japanese wisteria. Every volume, angle and count is recomputed from the study data (CHECKS.json). Printable A4 nets are included, so the Lotus seed can be built by hand. Files: the paper (PDF), the cut-out nets (PDF), and the data, scripts, figures and checks (zip). DICTO · dictospheres.com. Computation and checking with Claude (Anthropic).
BibTeX
@misc{dicto2026cleo,
title = {{AXE, CLEO and FUJI: the Gap Pieces of the Icosahedron}},
author = {DICTO},
howpublished = {Zenodo (CERN European Organization for Nuclear Research)},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23351291},
url = {https://doi.org/10.5281/zenodo.23351291},
note = {CC BY-SA},
}
RIS
TY - UNPB
TI - AXE, CLEO and FUJI: the Gap Pieces of the Icosahedron
AU - DICTO
PY - 2026
DA - 2026/10/14
PB - Zenodo (CERN European Organization for Nuclear Research)
DO - 10.5281/zenodo.23351291
UR - https://doi.org/10.5281/zenodo.23351291
AB - Regular icosahedra cannot fill space: two of them sharing an edge leave 83.62° around it. DICTO's thirteen-icosahedron cluster, the Lotus seed, closes every gap with three small pieces, AXE, CLEO and FUJI. With whole icosahedra they fill an icosahedron of edge φ² exactly, in 113 pieces, or in 63 pieces as DICTO first built it: a dodecahedron, 12 J11, 30 AXE and 20 FUJI. AXE and CLEO are two of the Mosseri–Sadoc tiles, scaled by 1/τ; here they arise as the gap pieces of a cluster of whole icosahedra. FUJI, a hexagonal pyramid, is new. It is not a union of Mosseri–Sadoc tiles at its own scale or larger (proofs by the Dehn invariant and by the angles of its base). The three pieces recur throughout DICTO's work: the Dodecahedral Chain (great stellated and excavated dodecahedra), the dodecahedron as a small icosahedron with 12 caps, 30 AXE and 20 CLEO, the Chain of Karma at every scale, and the Stella-Corona network. The names are DICTO's: AXE is shaped like an axe head; CLEO is named after Cleopatra's Needle in London; FUJI has the shape of Mount Fuji and the colour of the fuji flower, Japanese wisteria. Every volume, angle and count is recomputed from the study data (CHECKS.json). Printable A4 nets are included, so the Lotus seed can be built by hand. Files: the paper (PDF), the cut-out nets (PDF), and the data, scripts, figures and checks (zip). DICTO · dictospheres.com. Computation and checking with Claude (Anthropic).
ER -