This study examines n-balls, n-simplices, and n-orthoplices in real dimensions using novel recurrence relations that remove the indefiniteness present in known formulas. They show that in the negative, integer dimensions, the volumes of n-balls are zero if n is even, positive if n = −4k − 1, and negative if n = −4k − 3, for natural k. The volumes and surfaces of n-cubes inscribed in n-balls in negative dimensions are complex, wherein for negative, integer dimensions they are associated with integral powers of the imaginary unit. The relations are continuous for n ∈ ℝ and show that the constant of π is absent for 0 ≤ n < 2. For n < −1, self-dual n-simplices are undefined in the negative, integer dimensions, and their volumes and surfaces are imaginary in the negative, fractional ones and divergent with decreasing n. In the negative, integer dimensions, n-orthoplices reduce to the empty set, and their real volumes and imaginary surfaces are divergent in negative, fractional ones with decreasing n. Out of three regular, convex polytopes present in all natural dimensions, only n-orthoplices and n-cubes (and n-balls) are defined in the negative, integer dimensions.

This study argues that a biological cell, a dissipative structure, is the smallest agent capable of processing quantum information through its triangulated, holographic sphere of perception, where this mechanism has been extended by natural evolution to endo and exosemiosis in multicellular organisms and further to the language of Homo sapiens. Thus, life explains the measurement problem of quantum theory within the framework of the holographic principle, emergent gravity, and emergent dimensionality. Each Planck triangle on a black hole surface corresponds to a qubit in an equal superposition, attaining known bounds on the products of its energies and orthogonalization interval. Black holes generate entropy variation shells through the solid-angle correspondence. The entropic work introduces the bounds on the number of active Planck triangles dependent on the information capacity of the black hole generator. The velocity and dissipativity bounds and the bounds on the theoretical probabilities for active, energy-carrying Planck triangles were derived. In particular, this study shows that black holes, Turing machines, and viruses cannot assume the role of an observer. The entropy variation shells and black-body objects may hint at solutions to ball lightning and sonoluminescence unexplained physical spherical phenomena. “It is also possible that we learned that the principal problem is no longer the fight with the adversities of nature but the difficulty of understanding ourselves if we want to survive” [1].

The study shows that the volumes and surfaces of n-balls, n-simplices, and n-orthoplices are holomorphic functions of n, which makes those objects omnidimensional, that is well defined in any complex dimension. Applications of these formulas to the omnidimensional polytopes inscribed in and circumscribed about n-balls reveal previously unknown properties of these geometric objects. In particular, for $0<n<1$, the volumes of the omnidimensional polytopes are larger than those of circumscribing n-balls, and both their volumes and surfaces are smaller than those of inscribed n-balls. The surface of an n-simplex circumscribing a unit diameter n-ball is spirally convergent to zero with real n approaching negative infinity but first has a local maximum at $n=-3.5$. The surface of an n-orthoplex circumscribing a unit diameter n-ball is spirally divergent with real n approaching negative infinity but first has a local minimum at $n=-1.5$, where its real and imaginary parts are equal to each other; similarly, is its volume, where the similar local minimum occurs at $n=-3.5$. Reflection functions for volumes and surfaces of these polytopes inscribed in and circumscribed about n-balls are proposed. Symmetries of products and quotients of the volumes in complex dimensions n and $-n$ and of the surfaces in complex dimensions n and $2-n$ are shown to be independent of the metric factor and the gamma function. Specific symmetries also hold between the volumes and surfaces in dimensions $n=-1/2$ and $n=1/2$.

Using assembly theory, we investigate the assembly pathways of binary strings (bitstrings) of length N formed by joining bits present in the assembly pool and the bitstrings that entered the pool as a result of previous joining operations. We show that the bitstring assembly index is bounded from below by the shortest addition chain for N, and we conjecture about the form of the upper bound. We define the degree of causation for the minimum assembly index and show that, for certain N values, it has regularities that can be used to determine the length of the shortest addition chain for N. We show that a bitstring with the smallest assembly index for N can be assembled via a binary program of a length equal to this index if the length of this bitstring is expressible as a product of Fibonacci numbers. Knowing that the problem of determining the assembly index is at least NP-complete, we conjecture that this problem is NP-complete, while the problem of creating the bitstring so that it would have a predetermined largest assembly index is NP-hard. The proof of this conjecture would imply P ≠ NP since every computable problem and every computable solution can be encoded as a finite bitstring. The lower bound on the bitstring assembly index implies a creative path and an optimization path of the evolution of information, where only the latter is available to Turing machines (artificial intelligence). Furthermore, the upper bound hints at the role of dissipative structures and collective, in particular human, intelligence in this evolution.

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.