Volume Growth Entropy in Hilbert Geometry Matches Hyperbolic Space — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: The search results are a mix of competition-problem videos (windmill problem, "hardest easy geometry," USAMO problems) and one arXiv paper on Hilbert geometry asymptotics. The only substantive mathematical content is the Hilbert geometry paper; the rest are pedagogical artifacts without extractable equations.
MATH: From the Hilbert geometry paper (arXiv:0711.0446v1): - Volume growth entropy of metric balls in Hilbert geometry equals that of Lobachevsky space (hyperbolic space). - Asymptotic ratio: Vol(BR)/Area(SR) ∼ R/(n−1) (for dimension n), matching hyperbolic space behavior. - No explicit constants like 0.382, 0.618, 0.786, 1.618, 2.618 appear in the abstract.
CONNECTION: Hilbert geometry is a projective generalization of hyperbolic geometry. Its metric balls share the same exponential growth rate as Lobachevsky space — implying a deep projective invariance of the entropy. No direct link to golden-ratio or base-60 struc
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026volume,
title = {{Volume Growth Entropy in Hilbert Geometry Matches Hyperbolic Space — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23409563},
url = {https://zenodo.org/records/23409563},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Volume Growth Entropy in Hilbert Geometry Matches Hyperbolic Space — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/16
PB - Zenodo
DO - 10.5281/zenodo.23409563
UR - https://zenodo.org/records/23409563
AB - FINDING: The search results are a mix of competition-problem videos (windmill problem, "hardest easy geometry," USAMO problems) and one arXiv paper on Hilbert geometry asymptotics. The only substantive mathematical content is the Hilbert geometry paper; the rest are pedagogical artifacts without extractable equations. MATH: From the Hilbert geometry paper (arXiv:0711.0446v1): - Volume growth entropy of metric balls in Hilbert geometry equals that of Lobachevsky space (hyperbolic space). - Asymptotic ratio: Vol(B_R)/Area(S_R) ∼ R/(n−1) (for dimension n), matching hyperbolic space behavior. - No explicit constants like 0.382, 0.618, 0.786, 1.618, 2.618 appear in the abstract. CONNECTION: Hilbert geometry is a projective generalization of hyperbolic geometry. Its metric balls share the same exponential growth rate as Lobachevsky space — implying a deep projective invariance of the entropy. No direct link to golden-ratio or base-60 struc Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -