Four Cubes: On the Necessity of Four-Dimensional Perception

Szymon Łukaszyk

doi:10.59973/ipil.242

Authors

Szymon Łukaszyk Łukaszyk Patent Attorneys, Głowackiego 8, 40-052 Katowice, Poland

DOI:

https://doi.org/10.59973/ipil.242

Keywords:

Boolean space, Spectral graph theory, Sparse distributed memory, Activation functions, Neural architectures, Geometric deep learning, Cotan Laplacian, Exotic R4, Vector equilibrium, Ramanujan graphs, Topological data analysis, Emergent dimensionality

Abstract

The study aimed to demonstrate that the perceived (3+0i)-dimensional space is necessary for biological evolution due to the exotic R⁴ property of such a space, which ensures variations of traits between individuals perceiving the same differentiable structures. Properties of graphs constructed in Boolean spaces {0, 1}ⁿ were researched. The cotan Laplacian of 2-face triangulated n-cube was shown to have a spectrum corresponding to the Hamming distance distribution of Boolean space, and its regular version was shown to be a Ramanujan graph for 2 ≤ n ≤ 5 with the smallest integral Ramanujan bound for n = 4. The spectrum of the distance matrix on the graph comprising 2ⁿ n-cubes sharing a common origin was shown to be bounded by irrational eigenvalues, and if its 2-faces are triangulated, the spectrum of the cotan Laplacian includes all integers from
0 to 3n without the eigenvalue of 3n − 1. The relations of these graphs with Buckminster Fuller’s vector equilibrium were discussed. Based on Watanabe’s ugly duckling theorem, we defined a trainable activation function of an artificial neuron in a sparse distributed memory model.

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