“Dimension” Is Not One Thing: Mandelbrot’s Fractal Dimension and Not Conflating the Corpus’s Special Dimensions (From the Roughness of Markets).
The word “dimension” recurs in the corpus (n=2 forcing inverse-square in 3D, Paper 01; special dimensions over 3/4/5, Paper 64; why 3+1, Paper 69). But is “dimension” one concept? Benoît Mandelbrot (1924-2010) showed it is not: the FRACTAL dimension. Self-similar figures (Koch curve, coastlines, clouds) carry a non-integer dimension; for a self-similar set of N copies scaled by r, D = log N / log(1/r), and the Koch curve gives D = log4/log3 ≈ 1.26, between a line (1) and a plane (2). This is not “the number of independent directions (topological dimension = integer)” but a scaling exponent of how a set fills space — its roughness. This paper uses Mandelbrot’s viewpoint to sort the corpus’s “dimensions”: topological dimension (integer, direction count), Hausdorff/fractal dimension (non-integer, scaling), and the corpus’s special integer dimensions (3+1; 3/4/5 = Papers 69/64) are distinct roots merely sharing the word “dimension” (Papers 58/48). IN PARTICULAR the non-integer 1.26 must NOT be linked to the corpus’s special numbers (n=2, 3, 4pi) — numerology forbidden. Mandelbrot also pioneered the fractal analysis of financial price series (fat tails), intersecting the author’s origin (markets, quant). IMPORTANT (scope): a Layer-2 structural interpretation of standard fractal geometry; no new mathematics and no new prediction; it does NOT predict markets (Mandelbrot’s work is descriptive/critical). The Universe OS is a Layer-3 non-science lens.
On the making of this work: The ideas and content of this work stem from the author’s own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author’s. Feedback and corrections are sincerely appreciated.
Reference
Yamagishi, Y. (2026). “Dimension” Is Not One Thing [Paper 94]. Zenodo.
DOI: 10.5281/zenodo.21758581
ORCID: 0009-0006-8587-3450