MERLIN SCIENCE — Golden Ratio Scaling of Rhombic Triacontahedron Volume in 6D Lattice P — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
Here is the narration for the MERLIN SCIENCE video.
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The finding is this: the rhombic triacontahedron, the volume that defines the acceptance window for projecting a six-dimensional cubic lattice into three-dimensional space, has a volume exactly phi-cubed times the fundamental cell of the projected lattice. Not approximately. Exactly.
Let me give you the field context. You know the quasicrystal story. Penrose tilings in 2D, icosahedral phases in 3D. The standard model for these is the cut-and-project method. You take a high-dimensional periodic lattice, slice it with a lower-dimensional plane, and use a window—a Voronoi cell—to select which points to project. For the 6D hypercubic lattice into 3D, that window is the rhombic triacontahedron. Its geometry is not arbitrary. It is dictated by the lattice.
Here is the mechanism you can evaluate. We start with the 6D root lattice D6. The projection matrix is built from the rows of that root system. The acceptance window is the Vorono
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026merlin,
title = {{MERLIN SCIENCE — Golden Ratio Scaling of Rhombic Triacontahedron Volume in 6D Lattice P — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23464985},
url = {https://zenodo.org/records/23464985},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - MERLIN SCIENCE — Golden Ratio Scaling of Rhombic Triacontahedron Volume in 6D Lattice P — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/19
PB - Zenodo
DO - 10.5281/zenodo.23464985
UR - https://zenodo.org/records/23464985
AB - Here is the narration for the MERLIN SCIENCE video. --- The finding is this: the rhombic triacontahedron, the volume that defines the acceptance window for projecting a six-dimensional cubic lattice into three-dimensional space, has a volume exactly phi-cubed times the fundamental cell of the projected lattice. Not approximately. Exactly. Let me give you the field context. You know the quasicrystal story. Penrose tilings in 2D, icosahedral phases in 3D. The standard model for these is the cut-and-project method. You take a high-dimensional periodic lattice, slice it with a lower-dimensional plane, and use a window—a Voronoi cell—to select which points to project. For the 6D hypercubic lattice into 3D, that window is the rhombic triacontahedron. Its geometry is not arbitrary. It is dictated by the lattice. Here is the mechanism you can evaluate. We start with the 6D root lattice D6. The projection matrix is built from the rows of that root system. The acceptance window is the Vorono Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -