## Methods for Calculating Empires in Quasicrystals

This paper reviews the empire problem for quasiperiodic tilings and the existing methods for generating the empires of the vertex configurations in quasicrystals, while introducing a new and more efficient method based on the cut-and-project technique. Using Penrose tiling as an example, this method finds the forced tiles with the restrictions in the high dimensional […]

## Quantum Walk on a Spin Network and the Golden Ratio as the Fundamental Constant of Nature

We apply a discrete quantum walk from a quantum particle on a discrete quantum spacetime from loop quantum gravity and show that the related entanglement entropy drives an entropic force. We apply these concepts to a model where walker positions are topologically encoded on a spin network. Then, we discuss the role of the golden […]

## Toward a Unification of Physics and Numbers Theory

In Part I, we introduce the notion of simplex-integers and show how, in contrast to digital numbers, they are the most powerful numerical symbols that implicitly express the information of an integer and its set theoretic substructure. In Part II, we introduce a geometric analogue to the primality test that when $p$ is prime, it […]

## The Code Theoretic Axiom: The Third Ontology

A logical-physical ontology is code theory, wherein reality is neither deterministic nor random. In light of Conway and Kochens free will theorem and strong free will theorem, we discuss the plausibility of a third axiomatic option – geometric language; the code theoretic axiom. We suggest freewill choices at the syntactically free steps of a geometric […]

## Emergence of an Aperiodic Dirichlet Space from the Tetrahedral Units of an Icosahedral Internal Space

We present the emergence of a root system in six dimensions from the tetrahedra of an icosahedral core known as the 20-group (20G) within the framework of Clifford’s geometric algebra. Consequently, we establish a connection between a three-dimensional icosahedral seed, a six-dimensional (6D) Dirichlet quantized host and a higher dimensional lattice structure. The 20G, owing […]

## Anamorphic Quasiperiodic Universes in Modified and Einstein Gravity with Loop Quantum Gravity Corrections

The goal of this work is to elaborate on new geometric methods of constructing exact and parametric quasiperiodic solutions for anamorphic cosmology models in modified gravity theories, MGTs, and general relativity, GR. There exist previously studied generic off-diagonal and diagonalizable cosmological metrics encoding gravitational and matter fields with quasicrystal like structures, QC, and holonomy corrections […]

## The Search for a Hamiltonian whose Energy Spectrum coincides with the Riemann Zeta Zeroes

Inspired by the Hilbert-Polya proposal to prove the Riemann Hypothesis we have studied the Schroedinger QM equation involving a highly non-trivial potential, and whose self-adjoint Hamiltonian operator has for its energy spectrum one which approaches the imaginary parts of the zeta zeroes only in the asymptotic (very large N) region. The ordinates $\lambda_n$ are the positive […]

## Off-Diagonal Deformations of Kerr Metrics and Black Ellipsoids in Heterotic Supergravity

Geometric methods for constructing exact solutions of motion equations with first order α′ corrections to the heterotic supergravity action implying a non-trivial Yang-Mills sector and six dimensional, 6-d, almost-Kähler internal spaces are studied. In 10-d spacetimes, general parameterizations for generic off-diagonal metrics, nonlinear and linear connections and matter sources, when the equations of motion decouple […]