Four Cubes: On the Necessity of Four-Dimensional Perception

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 from0 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.