In 2009, Klee founded Quantum Gravity Research. The institute’s mission is to discover the geometric first-principles unification of space, time, matter, energy, information and consciousness. The model, called *emergence theory*, is based on quasicrystalline mathematics.

## Research

### 2022

### Character Varieties and Algebraic Surfaces for the Topology of Quantum Computing

#### Michel Planat, Marcelo Amaral, Fang Fang, David Chester, Raymond Aschheim, Klee Irwin

It is shown that the representation theory of some finitely presented groups thanks to their $SL_2(\mathbb{C})$ character variety is related to algebraic surfaces. We make use of the Enriques-Kodaira classification of algebraic surfaces and the related topological tools to make such surfaces explicit. We study the connection of $SL_2(\mathbb{C})$ character varieties to topological quantum computing (TQC) as an alternative to the concept of anyons. The Hopf link $H$, whose character variety is a Del Pezzo surface $f_H$ (the trace of the commutator), is the kernel of our view of TQC. Qutrit and two-qubit magic state computing, derived from the trefoil knot in our previous work, may be seen as TQC from the Hopf link. The character variety of some two-generator Bianchi groups as well as that of the fundamental group for the singular fibers $\tilde{E}_6$ and $\tilde{D}_4$ contain $f_H$. A surface birationally equivalent to a $K_3$ surface is another compound of their character varieties.

### SYNTHESIS OF THE Ti-Zr-Ni ALLOYS BY THE “HYDRIDE CYCLE” METHOD

#### Oleksii Dmytrenko, Igor Kolodiy, T. Yanko, V M Borysenko, Klee Irwin, R L Vasilenko

A comprehensive study of the technical parameters and conditions for the synthesis of ternary alloys in the Ti-Zr-Ni system by the “hydride cycle” method was carried out. The influence on the synthesis process of such parameters as: temperature and annealing time, heating rate, cooling conditions, material composition, dispersion, hydrogen content in the hydrides used, the presence of impurities, mixing and pressing methods, and the degree of pressing of the starting components was determined. The alloys of the following compositions were synthesized and investigated: Ti40.5Zr31.9Ni27.6, Ti41.5Zr41.5Ni17, Ti40Zr40Ni20 , Ti44Zr40Ni16. The optimal technological parameters and conditions for the synthesis of ternary alloys are determined. It has been established that the key factors in the process of compound formation during hydride dissociation are the dispersion and homogeneity of the initial compacted components. It was found that the synthesis of ternary alloys in the Ti-Zr-Ni system occurs during a short-term exothermic reaction in the “thermal explosion” mode, which begins in the temperature region corresponding to the α↔β polymorphic transformation of zirconium and titanium.

### 2021

### On the Emergence of Spacetime and Matter from Model Sets

#### Marcelo Amaral, Fang Fang, Raymond Aschheim, Klee Irwin

Starting from first principles and the mathematics of model sets we propose a framework where emergence of spacetime and matter can be addressed.

### Group Theory of Syntactical Freedom in DNA Transcription and Genome Decoding

#### Michel Planat, Marcelo Amaral, Fang Fang, David Chester, Raymond Aschheim, Klee Irwin

Transcription factors (TFs) are proteins that recognize specific DNA fragments in order to decode the genome and ensure its optimal functioning. TFs work at the local and global scales by specifying cell type, cell growth and death, cell migration, organization and timely tasks. We investigate the structure of DNA-binding motifs with the theory of finitely generated groups. The DNA ‘word’ in the binding domain -the motif- may be seen as the generator of a finitely generated group Fdna on four letters, the bases A, T, G and C. It is shown that, most of the time, the DNA-binding motifs have subgroup structure close to free groups of rank three or less, a property that we call ‘syntactical freedom’. Such a property is associated to the aperiodicity of the motif when it is seen as a substitution sequence. Examples are provided for the major families of TFs such as leucine zipper factors, zinc finger factors, homeo-domain factors, etc. We also discuss the exceptions to the existence of such a DNA syntactical rule and their functional role. This includes the TATA box in the promoter region of some genes, the single nucleotide markers (SNP) and the motifs of some genes of ubiquitous role in transcription and regulation.

### Graph Coverings for Investigating Non Local Structures in Proteins, Music and Poems

#### Michel Planat, Raymond Aschheim, Marcelo M. Amaral, Fang Fang, Klee Irwin

We explore the structural similarities in three different languages, first in the protein language whose primary letters are the amino acids, second in the musical language whose primary letters are the notes, and third in the poetry language whose primary letters are the alphabet. For proteins, the non local (secondary) letters are the types of foldings in space (α-helices, β-sheets, etc.); for music, one is dealing with clear-cut repetition units called musical forms and for poems the structure consists of grammatical forms (names, verbs, etc.). We show in this paper that the mathematics of such secondary structures relies on finitely presented groups fp on r letters, where r counts the number of types of such secondary non local segments. The number of conjugacy classes of a given index (also the number of graph coverings over a base graph) of a group fp is found to be close to the number of conjugacy classes of the same index in the free group Fr−1 on r−1 generators. In a concrete way, we explore the group structure of a variant of the SARS-Cov-2 spike protein and the group structure of apolipoprotein-H, passing from the primary code with amino acids to the secondary structure organizing the foldings. Then, we look at the musical forms employed in the classical and contemporary periods. Finally, we investigate in much detail the group structure of a small poem in prose by Charles Baudelaire and that of the Bateau Ivre by Arthur Rimbaud.

### The Curled Up Dimension in Quasicrystals

#### Fang Fang, Richard Clawson, Klee Irwin

Most quasicrystals can be generated by the cut-and-project method from higher dimensional parent lattices. In doing so they lose the periodic order their parent lattice possess, replaced with aperiodic order, due to the irrationality of the projection. However, perfect periodic order is discovered in the perpendicular space when gluing the cut window boundaries together to form a curved loop. In the case of a 1D quasicrystal projected from a 2D lattice, the irrationally sloped cut region is bounded by two parallel lines. When it is extrinsically curved into a cylinder, a line defect is found on the cylinder. Resolving this geometrical frustration removes the line defect to preserve helical paths on the cylinder. The degree of frustration is determined by the thickness of the cut window or the selected pitch of the helical paths. The frustration can be resolved by applying a shear strain to the cut-region before curving into a cylinder. This demonstrates that resolving the geometrical frustration of a topological change to a cut window can lead to preserved periodic order.

### Quantum Information in the Protein Codes, 3-manifolds and the Kummer Surface

#### Michel Planat, Raymond Aschheim, Marcelo Amaral, Fang Fang, Klee Irwin

Every protein consists of a linear sequence over an alphabet of 20 letters/amino acids. The sequence unfolds in the 3-dimensional space through secondary (local foldings), tertiary (bonds) and quaternary (disjoint multiple) structures. The mere existence of the genetic code for the 20 letters of the linear chain could be predicted with the (informationally complete) irreducible characters of the finite group Gn:=Zn⋊2O (with n=5 or 7 and 2O the binary octahedral group) in our previous two papers. It turns out that some quaternary structures of protein complexes display n-fold symmetries. We propose an approach of secondary structures based on free group theory. Our results are compared to other approaches of predicting secondary structures of proteins in terms of α helices, β sheets and coils, or more refined techniques. It is shown that the secondary structure of proteins shows similarities to the structure of some hyperbolic 3-manifolds. The hyperbolic 3-manifold of smallest volume –Gieseking manifold–, some other 3 manifolds and Grothendieck’s cartographic group are singled out as tentative models of such secondary structures. For the quaternary structure, there are links to the Kummer surface.

### Geometric State Sum Models from Quasicrystals

#### Marcelo Amaral, Fang Fang, Dugan Hammock, Klee Irwin

In light of the self-simulation hypothesis, a simple form implementation of the principle of efficient language is discussed in a self-referential geometric quasicrystalline state sum model in three dimensions. Emergence is discussed in context of geometric state sum models.

### Finite Groups for the Kummer Surface: the Genetic Code and a Quantum Gravity Analogy

#### Michel Planat, David Chester, Raymond Aschheim, Marcelo Amaral, Fang Fang, Klee Irwin

The Kummer surface was constructed in 1864. It corresponds to the desingularisation of the quotient of a 4-torus by 16 complex double points. Kummer surface is known to play a role in some models of quantum gravity. Following our recent model of the DNA genetic code based on the irreducible characters of the finite group G5:=(240,105)≅Z5⋊2O (with 2O the binary octahedral group), we now find that groups G6:=(288,69)≅Z6⋊2O and G7:=(336,118)≅Z7⋊2O can be used as models of the symmetries in hexamer and heptamer proteins playing a vital role for some biological functions. Groups G6 and G7 are found to involve the Kummer surface in the structure of their character table. An analogy between quantum gravity and DNA/RNA packings is suggested.

### 2020

### Space, Matter and Interactions in a Quantum Early Universe. Part I : Kac-Moody and Borcherds Algebras

#### Piero Truini, Alessio Marrani, Michael Rios, Klee Irwin

We introduce a quantum model for the Universe at its early stages, formulating a mechanism for the expansion of space and matter from a quantum initial condition, with particle interactions and creation driven by algebraic extensions of the Kac-Moody Lie algebra e9. We investigate Kac-Moody and Borcherds algebras, and we propose a generalization that meets further requirements that we regard as fundamental in quantum gravity.

### Space, Matter and Interactions in a Quantum Early Universe. Part II : Superalgebras and Vertex Algebras

#### Piero Truini, Alessio Marrani, Michael Rios, Klee Irwin

In our investigation on quantum gravity, we introduce an infinite dimensional complex Lie algebra 𝔤𝗎 that extends e9. It is defined through a symmetric Cartan matrix of a rank 12 Borcherds algebra. We turn 𝔤𝗎 into a Lie superalgebra 𝔰𝔤𝗎 with no superpartners, in order to comply with the Pauli exclusion principle. There is a natural action of the Poincaré group on 𝔰𝔤𝗎, which is an automorphism in the massive sector. We introduce a mechanism for scattering that includes decays as particular {\it resonant scattering}. Finally, we complete the model by merging the local 𝔰𝔤𝗎 into a vertex-type algebra.

### Complete Quantum Information in the DNA Genetic Code

#### Michel Planat, Raymond Aschheim, Marcelo M. Amaral, Fang Fang, Klee Irwin

We find that the degeneracies and many peculiarities of the DNA genetic code may be described thanks to two closely related (fivefold symmetric) finite groups. The first group has signature $G=\mathbb{Z}_5 \rtimes H$ where $H=\mathbb{Z}_2 . S_4\cong 2O$ is isomorphic to the binary octahedral group $2O$ and $S_4$ is the symmetric group on four letters/bases. The second group has signature $G=\mathbb{Z}_5 \rtimes GL(2,3)$ and points out a threefold symmetry of base pairings. For those groups, the representations for the $22$ conjugacy classes of $G$ are in one-to-one correspondence with the multiplets encoding the proteinogenic amino acids. Additionally, most of the $22$ characters of $G$ attached to those representations are informationally complete. The biological meaning of these coincidences is discussed.

### Informationally complete characters for quark and lepton mixings

#### Michel Planat, Raymond Aschheim, Marcelo M. Amaral, Klee Irwin

A popular account of the mixing patterns for the three generations of quarks and leptons is through the characters κ of a finite group G. Here we introduce a d-dimensional Hilbert space with d=cc(G), the number of conjugacy classes of G. Groups under consideration should follow two rules, (a) the character table contains both two- and three-dimensional representations with at least one of them faithful and (b) there are minimal informationally complete measurements under the action of a d-dimensional Pauli group over the characters of these representations. Groups with small d that satisfy these rules coincide in a large part with viable ones derived so far for reproducing simultaneously the CKM (quark) and PNMS (lepton) mixing matrices. Groups leading to physical CP violation are singled out.

### Aspects of aperiodicity and randomness in theoretical physics

#### Leonardo Ortiz, Marcelo Amaral, Klee Irwin

In this work we explore how the heat kernel, which gives the solution to the diffusion equation and the Brownian motion, would change when we introduce quasiperiodicity in the scenario. We also study the random walk in the Fibonacci sequence. We discuss how these ideas would change the discrete approaches to quantum gravity and the construction of quantum geometry.

### Symmetry transformation in Pd quasicrystals upon heating and hydrogenation

#### Vladimir Dubinko, Denys Laptiev, Dmitry Terentyev, Sergey Dmitriev, Klee Irwin

In this work, the structural transformation from a crystalline to quasicrystalline symmetry in palladium (Pd) and palladium-hydrogen (Pd-H) atomic clusters upon thermal annealing and hydrogenation has been addressed by means of atomistic simulations. A structural analysis of the clusters was performed during the heating up to the melting point to identify the temperature for the phase transformation. It has been demonstrated that nanometric pure Pd clusters transform from cuboctahedral to icosahedral structures under heating. This transformation is thermally activated process and the activation barrier depends on the cluster size. The activation energy of the cubo-ico symmetry transformation was measured using the variable heating rate method and was found to increase with the cluster size from 0.05 eV for 55 atomic cluster up to 0.66 eV for 147 atomic cluster. Hydrogenation of the nanometric Pd clusters yields to the modification of the transformation barrier in a non-monotonic form. At low H concentration, the transformation barrier decreases, while by increasing H concentration above a certain threshold, the barrier grows again thus making a minimum around a specific hydrogen concentration. This behaviour was rationalized as a competition between two processes, namely: the structure symmetry breaking at low H concentrations and stabilization of cuboctahedral phase of the clusters at high H concentration. The obtained results provide an estimation of the temperature range at which the symmetry transformation should occur under thermal annealing with experimentally achievable heating rates.

### Synthesis of hydrogen storage materials in a Ti-Zr-Ni system using the hydride cycle technology during dehydrogenation by an electron beam in a vacuum

#### Oleksii Dmytrenko, V. I. Dubinko, Valeriy Borysenko, Klee Irwin

The synthesis of intermetallic material was carried out by means of dehydrogenating annealing of a (TiH 2) 30 Zr 45 Ni 25 sample in vacuum by an electron beam. The properties of the obtained material were studied for establishing the structural phase composition by scanning electron microscopy and X-ray structural analysis. It was found that prolonged exposure of an electron beam to a sample containing titanium hydride leads to a number of structural transformations in the material, accompanied by a redistribution of hydrogen from titanium to zirconium and culminating in the synthesis of a ternary alloy with characteristic growth structures. The processes of hydrogen sorption-desorption by a synthesized sample were studied, the temperature ranges of these processes and the absorption capacity of the obtained material were established. It was shown that the structure of the sample formed upon heating by an electron beam promotes the absorption of hydrogen at room temperature up to 1.41 wt.%.

### Quantum computation and measurements from an exotic space-time R4

#### Michel Planat, Raymond Aschheim, Marcelo. M. Amaral, Klee Irwin

The authors previously found a model of universal quantum computation by making use of the coset structure of subgroups of a free group G with relations. A valid subgroup H of index d in G leads to a ‘magic’ state $||ψ⟩$ in d-dimensional Hilbert space that encodes a minimal informationally complete quantum measurement (or MIC), possibly carrying a finite ‘contextual’ geometry. In the present work, we choose G as the fundamental group $π1(V)$ of an exotic 4-manifold V, more precisely a ‘small exotic’ (space-time) $R^4$ (that is homeomorphic and isometric, but not diffeomorphic to the Euclidean $R^4$). Our selected example, due to to S. Akbulut and R.~E. Gompf, has two remarkable properties: (i) it shows the occurence of standard contextual geometries such as the Fano plane (at index 7), Mermin’s pentagram (at index 10), the two-qubit commutation picture $GQ(2,2)$ (at index 15) as well as the combinatorial Grassmannian $Gr(2,8)$(at index 28) , (ii) it allows the interpretation of MICs measurements as arising from such exotic (space-time) $R^4$’s. Our new picture relating a topological quantum computing and exotic space-time is also intended to become an approach of ‘quantum gravity’.

### The Self–Simulation Hypothesis Interpretation of Quantum Mechanics

#### Klee Irwin, Marcelo Amaral, and David Chester

We modify the simulation hypothesis to a self–simulation hypothesis, where the physical universe, as a strange loop, is a mental self–simulation that might exist as one of a broad class of possible code theoretic quantum gravity models of reality obeying the principle of efficient language axiom. This leads to ontological interpretations about quantum mechanics. We also discuss some implications of the self–simulation hypothesis such as an informational arrow of time.

### 2019

### Empires: The Nonlocal Properties of Quasicrystals

#### Fang Fang, Sinziana Paduroiu, Dugan Hammock and Klee Irwin

In quasicrystals, any given local patch—called an emperor—forces at all distances the existence of accompanying tiles—called the empire—revealing thus their inherent nonlocality. In this chapter, we review and compare the methods currently used for generating the empires, with a focus on the cut-and-project method, which can be generalized to calculate empires for any quasicrystals that are projections of cubic lattices. Projections of non-cubic lattices are more restrictive and some modifications to the cut-and-project method must be made in order to correctly compute the tilings and their empires. Interactions between empires have been modeled in a game-of-life approach governed by nonlocal rules and will be discussed in 2D and 3D quasicrystals. These nonlocal properties and the consequent dynamical evolution have many applications in quasicrystals research, and we will explore the connections with current material science experimental research.

### Holographic Code Rate

#### Noah Bray-Ali, David Chester, Dugan Hammock, Marcelo M. Amaral, Klee Irwin, Michael F. Rios

Holographic codes grown with perfect tensors on regular hyperbolic tessellations using an inflation rule protect quantum information stored in the bulk from errors on the boundary provided the code rate is less than one. Hyperbolic geometry bounds the holographic code rate and guarantees quantum error correction for codes grown with any inflation rule on all regular hyperbolic tessellations in a class whose size grows exponentially with the rank of the perfect tensors for rank five and higher. For the tile completion inflation rule, holographic triangle codes have code rate more than one but all others perform quantum error correction.

### Group geometrical axioms for magic states of quantum computing

#### Michel Planat, Raymond Aschheim, Marcelo M. Amaral, Klee Irwin

Let H be a non trivial subgroup of index d of a free group G and N the normal closure of H in G. The coset organization in a subgroup H of G provides a group P of permutation gates whose common eigenstates are either stabilizer states of the Pauli group or magic states for universal quantum computing. A subset of magic states consists of MIC states associated to minimal informationally complete measurements. It is shown that, in most cases, the existence of a MIC state entails that the two conditions (i) $N=G$ and (ii) no geometry (a triple of cosets cannot produce equal pairwise stabilizer subgroups), or that these conditions are both not satisfied. Our claim is verified by defining the low dimensional MIC states from subgroups of the fundamental group $G=π1(M)$ of some manifolds encountered in our recent papers, e.g. the 3-manifolds attached to the trefoil knot and the figure-eight knot, and the 4-manifolds defined by 0-surgery of them. Exceptions to the aforementioned rule are classified in terms of geometric contextuality (which occurs when cosets on a line of the geometry do not all mutually commute).

### Quantum Gravity at the Fifth Root of Unity

#### Marcelo M. Amaral, Raymond Aschheim, Klee Irwin

We consider quantum transition amplitudes, partition functions and observables for 3D spin foam models within $SU(2)$ quantum group deformation symmetry, where the deformation parameter is a complex fifth root of unity. By considering fermionic cycles through the foam we couple this $SU(2)$ quantum group with the same deformation of $SU(3)$, so that we have quantum numbers linked with spacetime symmetry and charge gauge symmetry in the computation of observables. The generalization to higher-dimensional Lie groups $SU(N)$, $G2$ and $E8$ is suggested. On this basis we discuss a unifying framework for quantum gravity. Inside the transition amplitude or partition function for geometries, we have the quantum numbers of particles and fields interacting in the form of a spin foam network − in the framework of state sum models, we have a sum over quantum computations driven by the interplay between aperiodic order and topological order.

### Constructing numbers in quantum gravity: infinions

#### Raymond Aschheim & Klee Irwin

Based on the Cayley-Dickson process, a sequence of multidimensional structured natural numbers (infinions) creates a path from quantum information to quantum gravity. Octonionic structure, exceptional Jordan algebra, and *E* _{8} Lie algebra are encoded on a graph with *E* _{9} connectivity, decorated by integral matrices. With the magic star, a toy model for a quantum gravity is presented with its naturally emergent quasicrystalline projective compactification.

### Quantum computing, Seifert surfaces and singular fibers

#### Michel Planat, Raymond Aschheim, Marcelo M. Amaral, Klee Irwin

The fundamental group $π1(L)$ of a knot or link $L$ may be used to generate magic states appropriate for performing universal quantum computation and simultaneously for retrieving complete information about the processed quantum states. In this paper, one defines braids whose closure is the $L$ of such a quantum computer model and computes their Seifert surfaces and the corresponding Alexander polynomial. In particular, some d-fold coverings of the trefoil knot, with $d=3$, 4, 6 or 12, define appropriate links $L$ and the latter two cases connect to the Dynkin diagrams of $E6$ and $D4$, respectively. In this new context, one finds that this correspondence continues with the Kodaira’s classification of elliptic singular fibers. The Seifert fibered toroidal manifold $Σ′$, at the boundary of the singular fiber $E8˜$, allows possible models of quantum computing.

### Emergence Theory Conceptual Overview

#### Klee Irwin

Emergence theory is a code-theoretic first-principles based discretized quantum field theoretic approach to quantum gravity and particle physics. This overview covers the primary set of ideas being assembled by Quantum Gravity Research.

### Assessment of discrete breather in the metallic hydrides

#### Vladimir Dubinko, Denis Laptev, Dmitry Terentyev, Sergey V. Dmitriev, Klee Irwin

Computational assessment of the discrete breathers (also known as intrinsic localised modes) is performed in nickel and palladium hydrides with an even stoichiometry by means of molecular dynamics simulations. The breathers consisting of hydrogen and metallic atoms were excited following the experience obtained earlier by modelling the breathers in pure metallic systems. Stable breathers were only found in the nickel hydride system and only for the hydrogen atoms oscillating along 〈1 0 0〉 and 〈1 1 1〉 polarization axes. At this, two types of the stable breathers involving single oscillating hydrogen and a pair of hydrogen atoms beating in antiphase mode were discovered. Analysis of the breather characteristics reveals that its frequency is located in the phonon gap or lying in the optical phonon band of phonon spectrum near the upper boundary. Analysis of the movement of atoms constituting the breather was performed to understand the mechanism that enables the breather stabilization and long-term oscillation without dissipation its energy to the surrounding atoms. It has been demonstrated that, while in palladium hydride, the dissipation of the intrinsic breather energy due to hydrogen-hydrogen attractive interaction occurs, the stable oscillation in the nickel hydride system is ensured by the negligibly weak hydrogen-hydrogen interaction acting within a distance of the breather oscillation amplitude. Thus, our analysis provides an explanation for the existence of the long-living stable breathers in metallic hydride systems. Finally, the high energy oscillating states of hydrogen atoms have been observed for the NiH and PdH lattices at finite temperatures which can be interpreted as a fingerprint of the finite-temperature analogues of the discrete breathers.

### 2018

### Non-Local Game of Life in 2D Quasicrystals

#### Fang Fang, Sinziana Paduroiu, Dugan Hammock, Klee Irwin

On a two-dimensional quasicrystal, a Penrose tiling, we simulate for the first time a game of life dynamics governed by non-local rules. Quasicrystals have inherently non-local order since any local patch, the emperor, forces the existence of a large number of tiles at all distances, the empires. Considering the emperor and its local patch as a quasiparticle, in this case a glider, its empire represents its field and the interaction between quasiparticles can be modeled as the interaction between their empires. Following a set of rules, we model the walk of life in different setups and we present examples of self-interaction and two-particle interactions in several scenarios. This dynamic is influenced by both higher dimensional representations and local choice of hinge variables. We discuss our results in the broader context of particle physics and quantum field theory, as a first step in building a geometrical model that bridges together higher dimensional representations, quasicrystals and fundamental particles interactions.

### On the Poincaré Group at the 5th Root of Unity A code theoretic particle physics model from lower dimensional representations of Lie groups

#### Marcelo M. Amaral, Klee Irwin

Considering the predictions from the standard model of particle physics coupled with experimental results from particle accelerators, we discuss a scenario in which from the infinite possibilities in the Lie groups we use to describe particle physics, nature needs only the lower dimensional representations − an important phenomenology that we argue indicates nature is code theoretic. We show that the “quantum” deformation of the SU (2) Lie group at the 5th root of unity can be used to address the quantum Lorentz group and gives the right low dimensional physical realistic spin quantum numbers confirmed by experiments. In this manner, we can describe the spacetime symmetry content of relativistic quantum fields in accordance with the well known Wigner classification. Further connections of the 5th root of unity quantization with the mass quantum number associated with the Poincaré Group and the SU(N) charge quantum numbers are discussed as well as their implication for quantum gravity.

### Quasicrystal Tilings in Three Dimensions and Their Empires

#### Dugan Hammock, Fang Fang, Klee Irwin

The projection method for constructing quasiperiodic tilings from a higher dimensional lattice provides a useful context for computing a quasicrystal’s vertex configurations, frequencies, and empires (forced tiles). We review the projection method within the framework of the dual relationship between the Delaunay and Voronoi cell complexes of the lattice being projected. We describe a new method for calculating empires (forced tiles) which also borrows from the dualization formalism and which generalizes to tilings generated projections of non-cubic lattices. These techniques were used to compute the vertex configurations, frequencies and empires of icosahedral quasicrystals obtained as projections of the D6 and Z6 lattices to R3 and we present our analyses. We discuss the implications of this new generalization.

### Quantum Computing with Bianchi Groups

#### Michel Planat, Raymond Aschheim, Marcelo M. Amaral, Klee Irwin

It has been shown that non-stabilizer eigenstates of permutation gates are appropriate for allowing d-dimensional universal quantum computing (uqc) based on minimal informationally complete POVMs. The relevant quantum gates may be built from subgroups of finite index of the modular group Γ = PSL(2,Z) [M. Planat, Entropy 20, 16 (2018)] or more generally from subgroups of fundamental groups of 3- manifolds [M. Planat, R. Aschheim, M. M. Amaral and K. Irwin, arXiv 1802.04196(quant-ph)]. In this paper, previous work is encompassed by the use of torsion-free subgroups of Bianchi groups for deriving the quantum gate generators of uqc. A special role is played by a chain of Bianchi congruence n-cusped links starting with Thurston’s link.

### Closing Gaps in Geometrically Frustrated Symmetric Clusters: Local Equivalence between Discrete Curvature and Twist Transformations

#### Fang Fang, Richard Clawson, Klee Irwin

In geometrically frustrated clusters of polyhedra, gaps between faces can be closed without distorting the polyhedra by the long established method of discrete curvature, which consists of curving the space into a fourth dimension, resulting in a dihedral angle at the joint between polyhedra in 4D. An alternative method—the twist method—has been recently suggested for a particular case, whereby the gaps are closed by twisting the cluster in 3D, resulting in an angular offset of the faces at the joint between adjacent polyhedral. In this paper, we show the general applicability of the twist method, for local clusters, and present the surprising result that both the required angle of the twist transformation and the consequent angle at the joint are the same, respectively, as the angle of bending to 4D in the discrete curvature and its resulting dihedral angle. The twist is therefore not only isomorphic, but isogonic (in terms of the rotation angles) to discrete curvature. Our results apply to local clusters, but in the discussion we offer some justification for the conjecture that the isomorphism between twist and discrete curvature can be extended globally. Furthermore, we present examples for tetrahedral clusters with three-, four-, and fivefold symmetry.

### Universal Quantum Computing and Three-Manifolds

#### Michel Planat, Raymond Aschheim, Marcelo M. Amaral, Klee Irwin

A single qubit may be represented on the Bloch sphere or similarly on the 3-sphere $S3$. Our goal is to dress this correspondence by converting the language of universal quantum computing (UQC) to that of 3-manifolds. A magic state and the Pauli group acting on it define a model of UQC as a POVM that one recognizes to be a 3-manifold $M^3$. E. g., the d-dimensional POVMs defined from subgroups of finite index of the modular group $PSL(2,\mathbb{Z})$ correspond to d-fold $M^3$ – coverings over the trefoil knot. In this paper, one also investigates quantum information on a few \lq universal’ knots and links such as the figure-of-eight knot, the Whitehead link and Borromean rings, making use of the catalog of platonic manifolds available on SnapPy. Further connections between POVMs based uqc and $M^3$’s obtained from Dehn fillings are explored.

### 2017

### Methods for Calculating Empires in Quasicrystals

#### Fang Fang, Dugan Hammock, Klee Irwin

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 lattice (the mother lattice) that can be cut-and-projected into the lower dimensional quasicrystal. We compare our method to the two existing methods, namely one method that uses the algorithm of the Fibonacci chain to force the Ammann bars in order to find the forced tiles of an empire and the method that follows the work of N.G. de Bruijn on constructing a Penrose tiling as the dual to a pentagrid. This new method is not only conceptually simple and clear, but it also allows us to calculate the empires of the vertex configurations in a defected quasicrystal by reversing the configuration of the quasicrystal to its higher dimensional lattice, where we then apply the restrictions. These advantages may provide a key guiding principle for phason dynamics and an important tool for self error-correction in quasicrystal growth.

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

#### Klee Irwin, Fang Fang, Marcelo Amaral, Raymond Aschheim

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 ratio in fundamental physics by addressing charge and length quantization and by analyzing the ratios of fundamental constants−the limits of nature. The limit of minimal length and volume arising in quantum gravity theory indicates an underlying principle that we develop herein.

### Toward a Unification of Physics and Numbers Theory

#### Klee Irwin

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 divides $(^p_k)$ for all $0<k<p$. Our geometric form provokes a novel hypothesis about the distribution of prime-simplexes that, if solved, may lead to a proof of the Riemann hypothesis. Specifically, if a geometric algorithm predicting the number of prime simplexes within any bound n-simplexes or associated $A_n$ lattices is discovered, a deep understanding of the error factor of the prime number theorem would be realized – the error factor corresponding to the distribution of the non-trivial zeta zeros. In Part III, we discuss the mysterious link between physics and the Riemann hypothesis. We suggest how quantum gravity and particle physicists might benefit from a simplex-integer based quasicrystal code formalism. An argument is put forth that the unifying idea between number theory and physics is code theory, where reality is information theoretic and 3-simplex integers form physically realistic aperiodic dynamic patterns from which space, time and particles emerge from the evolution of the code syntax. Finally, an appendix provides an overview of the conceptual framework of emergence theory, an approach to unification physics based on the quasicrystalline spin network.

### The Code Theoretic Axiom: The Third Ontology

#### Klee Irwin

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 language of space-time form the code theoretic substrate upon which particle and gravitational physics emerge.

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

#### Amrik Sen, Raymond Aschheim, Klee Irwin

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 to its icosahedral symmetry, bears the signature of a 6D lattice that manifests in the Dirichlet integer representation. We present an interpretation whereby the three-dimensional 20G can be regarded as the core substratum from which the higher dimensional lattices emerge. This emergent geometry is based on an induction principle supported by the Clifford multi-vector formalism of three-dimensional (3D) Euclidean space. This lays a geometric framework for understanding several physics theories related to $SU(5)$, $E_{6}$, $E_8$ Lie algebras and their composition with the algebra associated with the even unimodular lattice in $R^{3,1}$. The construction presented here is inspired by Penrose’s three world mode.

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

#### Marcelo Amaral, Raymond Aschheim, Laurentiu Bubuianu, Klee Irwin, Sergiu Vacaru, Daniel Woolridge

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 from loop quantum gravity, LQG. We apply the anholonomic frame deformation method, AFDM, in order to decouple the (modified) gravitational and matter field equations in general form. This allows us to find integral varieties of cosmological solutions determined by generating functions, effective sources, integration functions and constants. The coefficients of metrics and connections for such cosmological configurations depend, in general, on all spacetime coordinates and can be chosen to generate observable (quasi)-periodic/ aperiodic/ fractal / stochastic / (super) cluster / filament / polymer like (continuous, stochastic, fractal and/or discrete structures) in MGTs and/or GR. In this work, we study new classes of solutions for anamorphic cosmology with LQG holonomy corrections. Such solutions are characterized by nonlinear symmetries of generating functions for generic off–diagonal cosmological metrics and generalized connections, with possible nonholonomic constraints to Levi-Civita configurations and diagonalizable metrics depending only on a timelike coordinate. We argue that anamorphic quasiperiodic cosmological models integrate the concept of quantum discrete spacetime, with certain gravitational QC-like vacuum and nonvacuum structures. And, that of a contracting universe that homogenizes, isotropizes and flattens without introducing initial conditions or multiverse problems.

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

#### Raymond Aschheim, Carlos Castro Perelman, Klee Irwin

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 imaginary parts of the nontrivial zeta zeros in the critical line : *sn = 1/2 + i$\lambda_n$* . The latter results are consistent with the validity of the Bohr-Sommerfeld semi-classical quantization condition. It is shown how one may modify the parameters which define the potential, and fine tune its values, such that the energy spectrum of the (modified) Hamiltonian matches not only the first two zeroes but the other consecutive zeroes. The highly non-trivial functional form of the potential is found via the Bohr-Sommerfeld quantization formula using the full-fledged Riemann-von Mangoldt counting formula ($without$ any truncations) for the number *N (E)* of zeroes in the critical strip with imaginary part greater than *0* and less than or equal to *E*.

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

#### Sergiu I. Vacaru, Klee Irwin

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 in very general forms are considered. This allows us to construct a variety of exact solutions when the coefficients of fundamental geometric/physical objects depend on all higher dimensional spacetime coordinates via corresponding classes of generating and integration functions, generalized effective sources and integration constants. Such generalized solutions are determined by generic off-diagonal metrics and nonlinear and/or linear connections. In particular, as configurations which are warped/compactified to lower dimensions and for Levi-Civita connections. The corresponding metrics can have (non) Killing and/or Lie algebra symmetries and/or describe (1+2)-d and/or (1+3)-d domain wall configurations, with possible warping nearly almost-Kähler manifolds, with gravitational and gauge instantons for nonlinear vacuum configurations and effective polarizations of cosmological and interaction constants encoding string gravity effects. A series of examples of exact solutions describing generic off-diagonal supergravity modifications to black hole/ ellipsoid and solitonic configurations are provided and analyzed. We prove that it is possible to reproduce the Kerr and other type black solutions in general relativity (with certain types of string corrections) in 4D and to generalize the solutions to non-vacuum configurations in (super) gravity/string theories.

### 2016

### Heterotic Supergravity with Internal Almost-Kahler Configurations and Gauge SO(32), or E8xE8, Instantons

#### Laurentiu Bubuianu, Klee Irwin, Sergiu Vacaru

Heterotic supergravity with (1+3)-dimensional domain wall configurations and (warped) internal, 6-dimensional, almost-K\”{a}hler manifolds 6X are studied. Considering on 10-dimensional spacetime, nonholonomic distributions with conventional double fibrations, 2+2+…=2+2+3+3, and associated SU(3) structures on internal space, we generalize for real, internal, almost symplectic gravitational structures the constructions with gravitational and gauge instantons of tanh-kink type. They include the first α′ corrections to the heterotic supergravity action, parameterized in a form to imply nonholonomic deformations of the Yang-Mills sector and corresponding Bianchi identities. We show how it is possible to construct a variety of solutions, depending on the type of nonholonomic distributions and deformations of ‘prime’ instanton configurations characterized by two real supercharges. This corresponds to N=1/2 supersymmetric, nonholonomic manifolds from the 4-dimensional point of view. Our method provides a unified description of embedding nonholonomically deformed tanh-kink-type instantons into half-BPS solutions of heterotic supergravity. This allows us to elaborate new geometric methods of constructing exact solutions of motion equations, with first order α′ corrections to the heterotic supergravity. Such a formalism is applied for general and/or warped almost-K% \”{a}hler configurations, which allows us to generate nontrivial (1+3)-d domain walls. This formalism is utilized in our associated publication \cite{partner} in order to construct and study generic off-diagonal nonholonomic deformations of the Kerr metric, encoding contributions from heterotic supergravity.

### Starobinsky Inflation and Dark Energy and Dark Matter Effects from Quasicrystal Like Spacetime Structures

#### Raymond Aschheim, Laurenµiu Bubuianu, Fang Fang, Klee Irwin, Vyacheslav Ruchin, Sergiu I. Vacaru

The goal of this work on mathematical cosmology and geometric methods in modified gravity theories, MGTs, is to investigate Starobinsky-like inflation scenarios determined by gravitational and scalar field configurations mimicking quasicrystal, QC, like structures. Such spacetime aperiodic QCs are different from those discovered and studied in solid state physics but described by similar geometric methods. We prove that an inhomogeneous and locally anisotropic gravitational and matter field effective QC mixed continuous and discrete “aether” can be modeled by exact cosmological solutions in MGTs and Einstein gravity. The coefficients of corresponding generic off-diagonal metrics and generalized connections depend (in general) on all spacetime coordinates via generating and integration functions and certain smooth and discrete parameters. Imposing additional nonholonomic constraints, prescribing symmetries for generating functions and solving the boundary conditions for integration functions and constants, we can model various nontrivial torsion QC structures or extract cosmological Levi–Civita configurations with diagonal metrics reproducing de Sitter (inflationary) like and other types homogeneous inflation and acceleration phases. Finally, we speculate how various dark energy and dark matter effects can be modeled by off-diagonal interactions and deformations of a nontrivial QC like gravitational vacuum structure and analogous scalar matter fields.

### Catalytic Mechanism of LENR in Quasicrystals Based on Localized Anharmonic Vibrations and Phasons

#### Volodymyr Dubinko, Denis Laptev, Klee Irwin

Quasicrystals (QCs) are a novel form of matter, which are neither crystalline nor amorphous. Among many surprising properties of QCs is their high catalytic activity. We propose a mechanism explaining this peculiarity based on unusual dynamics of atoms at special sites in QCs, namely, localized anharmonic vibrations (LAVs) and phasons. In the former case, one deals with a large amplitude (~ fractions of an angstrom) time-periodic oscillations of a small group of atoms around their stable positions in the lattice, known also as discrete breathers, which can be excited in regular crystals as well as in QCs. On the other hand, phasons are a specific property of QCs, which are represented by very large amplitude (~angstrom) oscillations of atoms be-tween two quasi-stable positions determined by the geometry of a QC. Large amplitude atomic motion in LAVs and phasons results in time-periodic driving of adjacent potential wells occupied by hydrogen ions (protons or deuterons) in case of hydrogenated QCs. This driving may result in the increase of amplitude and energy of zero-point vibrations (ZPV). Based on that, we demonstrate a drastic increase of the D-D or D-H fusion rate with increasing number of modulation periods evaluated in the framework of Schwinger model, which takes into account suppression of the Coulomb barrier due to lattice vibrations. In this context, we present numerical solution of Schrodinger equation for a particle in a non-stationary double well potential, which is driven time-periodically imitating the action of a LAV or phason. We show that the rate of tunneling of the particle through the potential barrier separating the wells is enhanced drastically by the driving, and it increases strongly with increasing amplitude of the driving. These results support the concept of nuclear catalysis in QCs that can take place at special sites provided by their inherent topology.

### Golden, Quasicrystalline, Chiral Packings of Tetrahedra

#### Fang Fang, Garrett Sadler, Julio Kovacs, Klee Irwin (written 2012, published 2016)

Since antiquity, the packing of convex shapes has been of great interest to many scientists and mathematicians [1-7]. Recently, particular interest has been given to packings of three-dimensional tetrahedra [8-20]. Dense packings of both crystalline [8, 10, 15, 17, 19] and semi-quasicrystalline [14] have been reported. It is interesting that a semiquasicrystalline packing of tetrahedra can emerge naturally within a thermodynamic simulation approach [14]. However, this packing is not perfectly quasicrystalline and the packing density, while dense, is not maximal. Here we suggest that a “golden rotation” between tetrahedral facial junctions can arrange tetrahedra into a perfect quasicrystalline packing. Using this golden rotation, tetrahedra can be organized into “triangular”, “pentagonal”, and “spherical” locally dense aggregates. Additionally, the aperiodic Boerdijk-Coxeter helix [23, 24] (tetrahelix) is transformed into a structure of 3- or 5-fold periodicity—depending on the relative chiralities of the helix and rotation—herein referred to as the “philix”. Further, using this same rotation, we build (1) a shell structure which resembles a Penrose tiling upon projection into two dimensions, and (2) a “tetragrid” structure assembled of golden rhombohedral unit cells. Our results indicate that this rotation is closely associated with Fuller’s “jitterbug transformation” [21] and that the total number of face-plane classes (defined below) is significantly reduced in comparison with general tetrahedral aggregations, suggesting a quasicrystalline packing of tetrahedra which is both dynamic and dense. The golden rotation that we report presents a novel tool for arranging tetrahedra into perfect quasicrystalline, dense packings.

### The Unexpected Fractal Signatures in Fibonacci Chains

#### Fang Fang and Klee Irwin

Quasicrystals are fractal due to their self similar property. In this paper, a new cycloidal fractal signature possessing the cardioid shape in the Mandelbrot set is presented in the Fourier space of a Fibonacci chain with two lengths, $L$ and $S$, where $L/S = \phi$. The corresponding pointwise dimension is 0.7. Various modifications, such as truncation from the head or tail, scrambling the orders of the sequence, and changing the ratio of the L and S, are done on the Fibonacci chain. The resulting patterns in the Fourier space show that that the fractal signature is very sensitive to changes in the Fibonacci order but not to the L/S ratio.

### An Icosohedral Quasicrystal and E8 Derived Quasicrystals

#### Fang Fang, Klee Irwin

We present the construction of an icosahedral quasicrystal, a quasicrystalline spin network, obtained by spacing the parallel planes in an icosagrid with the Fibonacci sequence. This quasicrystal can also be thought of as a golden composition of five sets of Fibonacci tetragrids. We found that this quasicrystal embeds the quasicrystals that are golden compositions of the three-dimensional tetrahedral cross-sections of the Elser-Sloane quasicrystal, which is a four-dimensional cut-and-project of the E8 lattice. These compound quasicrystals are subsets of the quasicrystalline spin network, and the former can be enriched to form the later. This creates a mapping between the quasicrystalline spin network and the E8 lattice.

### 2014

### A New Approach to the Hard Problem of Consciousness: A Quasicrystalline Language of “Primitive Units of Consciousness” in Quantized Spacetime

#### Klee Irwin

The hard problem of consciousness must be approached through the ontological lens of 20th-century physics, which tells us that reality is information theoretic and quantized at the level of Planck scale spacetime. Through careful deduction, it becomes clear that information cannot exist without consciousness – the awareness of things. And to be aware is to hold the meaning of relationships of objects within consciousness – perceiving abstract objects while enjoying degrees of freedom within the structuring of those relationships. This defines consciousness as language – (1) a set of objects and (2) an ordering scheme with (3) degrees of freedom used for (4) expressing meaning. And since even information at the Planck scale cannot exist without consciousness, we propose an entity called a “primitive unit of consciousness”, which acts as a mathematical operator in a quantized spacetime language. Quasicrystal mathematics based on E8 geometry seems to be a candidate for the language of reality, possessing several qualities corresponding to recent physical discoveries and various physically realistic unification models.

### Eight Things a First Principles Theory of Everything Should Possess

#### Klee Irwin

A first principles theory of everything has never been achieved. An E8 derived code of quantized spacetime could meet the following suggested requirements: (1) First principles explanation of time dilation, inertia, the magnitude of the Planck constant and the speed of light (2) First principles explanation of conservation laws and gauge transformation symmetry. (3) Must be fundamentally relativistic with nothing that is invariant being absolute. (4) Pursuant to the deduction that reality is fundamentally information-theoretic, all information must be generated by observation/measurement at the simplest Planck scale of the code/language. (5) Must be non-deterministic. (6) Must be computationally efficient. (7) Must be a code describing “jagged” (quantized) waveform – a waveform language. (8) Must have a first principles explanation for preferred chirality in nature.

### An Icosohedral Quasicrystal as a Packing of Regular Tetrahedra

#### Fang Fang, Julio Kovacs, Garrett Sadler, Klee Irwin

We present the construction of a dense, quasicrystalline packing of regular tetrahedra with icosahedral symmetry. This quasicrystalline packing was achieved through two independent approaches. The first approach originates in the Elser- Sloane 4D quasicrystal. A 3D slice of the quasicrystal contains a few types of prototiles. An initial structure is obtained by decorating these prototiles with tetrahedra. This initial structure is then modified using the Elser- Sloane quasicrystal itself as a guide. The second approach proceeds by decorating the prolate and oblate rhombohedra in a 3-dimensional Ammann tiling. The resulting quasicrystal has a packing density of 59.783%. We also show a variant of the quasicrystal that has just 10 plane classes (compared with the 190 of the original), defined as the total number of distinct orientations of the planes in which the faces of the tetrahedra are contained. The small number of plane classes was achieved by a certain “golden rotation” of the tetrahedra.

### 2013

### Periodic Modification of the Boerdijk-Coxeter Helix (Tetrahelix)

#### Garrett Sadler, Fang Fang, Julio Kovacs, Klee Irwin

The Boerdijk-Coxeter helix is a helical structure of tetrahedra which possesses no non-trivial translational or rotational symmetries. In this document, we develop a procedure by which this structure is modiﬁed to obtain both translational and rotational (upon projection) symmetries along/about its central axis. We report the ﬁnding of several, distinct periodic structures, and focus on two particular forms related to the pentagonal and icosahedral aggregates of tetrahedra as well as Buckminster Fuller’s “jitterbug transformation”.

### Cabinet of Curiosities: The Interesting Geometry of the Angle β = arccos((3φ – 1)/4)

#### Fang Fang, Klee Irwin, Julio Kovacs, Garrett Sadler

In this paper, we present the construction of several aggregates of tetrahedra. Each construction is obtained by performing rotations on an initial set of tetrahedra that either (1) contains gaps between adjacent tetrahedra, or (2) exhibits an aperiodic nature. Following this rotation, gaps of the former case are “closed” (in the sense that faces of adjacent tetrahedra are brought into contact to form a “face junction”) while translational and rotational symmetries are obtained in the latter case. In all cases, an angular displacement of {\beta} = arccos((3{\phi} – 1)/4) (or a closely related angle), where {\phi} is the golden ratio, is observed between faces of a junction. Additionally, the overall number of plane classes, defined as the number of distinct facial orientations in the collection of tetrahedra, is reduced following the transformation. Finally, we present several “curiosities” involving the structures discussed here with the goal of inspiring the reader’s interest in constructions of this nature and their attending, interesting properties.

### Law of Sums of the Squares of Areas, Volumes and Hyper Volumes of Regular Polytopes from Clifford Polyvectors

#### Carlos Perelman, Fang Fang, Garrett Sadler, Klee Irwin

Inspired by the recent sums of the squares law obtained by we derive the law of the sums of the squares of the areas, volumes and hyper-volumes associated with the faces, cells and hyper-cells of regular polytopes in diverse dimensions after using Clifford algebraic methods.

### 2012

### The Sum of Squares Law

#### Julio Kovacs, Fang Fang, Garrett Sadler, Klee Irwin

We show that when projecting an edge-transitive N-dimensional polytope onto an M-dimensional subspace of RN, the sums of the squares of the original and projected edges are in the ratio N/M.