In 2009, Irwin founded Quantum Gravity Research, a non-profit organization whose mission is to discover the geometric first-principles unification of space, time, matter, energy, information, and consciousness. Emergence theory aims to produce a unified first- principles theory of everything founded upon the mathematics of quasicrystals.

Irwin’s research has principally focused on the organization of dense aggregates of tetrahedra in three- and four-dimensional space. Klee is the author and co-author of numerous papers and presentations, and emergence theory’s framework is described in the new film “What Is Reality?”.

## Research

### 2018

### 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 S*^3*. 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,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.

### Encoding Geometric Frustration In Tetrahedral Packing With Gaps, Discrete Curvature, Distortion or Twisting

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

This paper presents various ways of encoding geometric frustration in tetrahedral packings, by introducing small gaps to form quasicrystalline order, by curving to the 4th dimension with discrete curvatures, by distortion of tetrahedral edges and by twisting the edge-sharing and/or vertex-sharing local tetrahedral clusters. The key to these methods is to encode the deficit of the tetrahedral dihedral angle in closing a circle which is the cause of the geometric frustration. A surprising connection between the discrete curvature method and the twisting method is that both the transformation angle and the joint angle are the same in the one case as in the other. This connection leads to a way of encoding discrete curvature with twisting, which may help to model spacetime based on a quasicrystalline network that serves as a discrete version of a pseudo-Riemannian space.

### 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 \binom{p}{k}=(p(p-1)…(p-k+1))/(k(k-1)…1) 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 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 3-dimensional icosahedral seed, a 6-dimensional 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 3-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 multivector formalism of 3D Euclidean space. This lays a geometric framework for understanding several physics theories related to SU(5), E6, E8 Lie algebras and their composition with the algebra associated with the even unimodular lattice in R3,1. The construction presented here is inspired by Penrose’s “three world” model.

### Amorphic 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 Hypoth-esis 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 zetazeroes only in the asymptotic (very large N ) region. The ordinates λ n (positive imaginary parts) of the non-trivial zeta zeros in the critical line: sn = 12 ± iλn 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 tuneits values, such that the energy spectrum of the (modiﬁed) Hamiltonianmatches 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 because they are scale invariant and self-similar. 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 = Ø. The corresponding pointwise dimension is 0.7. Various variations such as truncation from the head or tail, scrambling the orders of the sequence, 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 the change 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.