W e present an icosahedral quasicrystal, a Fibonacci icosagrid, obtained by spacing the parallel ... more W e present an icosahedral quasicrystal, a Fibonacci icosagrid, obtained by spacing the parallel planes in an icosagrid with the Fibonacci sequence. This quasicrystal can also be thought 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 Fibonacci icosagrid, and they can be enriched to form the Fibonacci icosagrid. This creates a mapping between the Fibonacci icosagrid and the E−8 lattice. It is known that the combined structure and dynamics of all gravitational and Standard Model particle fields, including fermions, are part of the E8 Lie algebra. Because of this, the Fibonacci icosagrid is a good candidates, for representing states and interactions between particles and fields in quantum mech...
We explore the structural similarities in three different languages, first in the protein languag... more 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 concre...
Most quasicrystals can be generated by the cut-and-project method from higher dimensional parent ... more 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 geometrica...
Every protein consists of a linear sequence over an alphabet of 20 letters/amino acids. The seque... more 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 qua-ternary (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...
The Kummer surface was constructed in 1864. It corresponds to the desingularisation of 1 the quot... more The Kummer surface was constructed in 1864. It corresponds to the desingularisation of 1 the quotient of a 4-torus by 16 complex double points. Kummer surface is known to play a role in 2 some models of quantum gravity. Following our recent model of the DNA genetic code based on the 3 irreducible characters of the finite group G 5 := (240, 105) ∼ = Z 5 2O (with 2O the binary octahedral 4 group), we now find that groups G 6 := (288, 69) ∼ = Z 6 2O and G 7 := (336, 118) ∼ = Z 7 2O can be 5 used as models of the symmetries in hexamer and heptamer proteins playing a vital role for some 6 biological functions. Groups G 6 and G 7 are found to involve the Kummer surface in the structure of 7 their character table. An analogy between quantum gravity and DNA/RNA packings is suggested. 8
In quasicrystals, any given local patch-called an emperor-forces at all distances the existence o... more 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.
The soybean lecithin–gallic acid complex ameliorates hepatic damage and iron-overload induced by ... more The soybean lecithin–gallic acid complex ameliorates hepatic damage and iron-overload induced by alcohol and exerts hepatoprotective effects.
The projection method for constructing quasiperiodic tilings from a higher dimensional lattice pr... more 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 dualisation 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 a projections of the D 6 and Z 6 lattices to R 3 and we present our analyses. We discuss the implications of this new generalization.
We present an icosahedral quasicrystal as a modification of the icosagrid, a multigrid with 10 pla... more We present an icosahedral quasicrystal as a modification of the icosagrid, a multigrid with 10 plane sets that are arranged with icosahedral symmetry. We use the Fibonacci chain to space the planes, thereby obtaining a quasicrystal with icosahedral symmetry. It has a surprising correlation to the Elser-Sloane quasicrystal1, a 4D cut-and-project of the E8 lattice. We call this quasicrystal the Fibonacci modified icosagrid quasicrystal. We found that this structure totally embeds another quasicrystal that is a compound of 20 3D slices of the Elser-Sloane quasicrystal. The slices, which contain only regular tetrahedra, are put together by a certain golden ratio based rotation2. Interesting 20-tetrahedron clusters arranged with the golden ratio based rotation appear repetitively in the structure. They are arranged with icosahedral symmetry. It turns out that this rotation is the dihedral angle of the 600-cell (the super-cell of the Elser-Sloane quasicrystal) and the angle between the tetr...
Quasicrystals are fractal because they are scale invariant and self similar. In this paper, a new... more 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 = φ. The corresponding pointwise dimension is 0.7. Various variations such as truncation from the head or tail, scrambling the orders of the sequence, changeing 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.
In light of the self-simulation hypothesis, a simple form implementation of the principle of effi... more 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.
On a two-dimensional quasicrystal, a Penrose tiling, we simulate for the first time a game of lif... more 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.
In geometrically frustrated clusters of polyhedra, gaps between faces can be closed without disto... more 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.
This paper reviews the empire problem for quasiperiodic tilings and the existing methods for gene... more 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.
Since antiquity, the packing of convex shapes has been of great interest to many scientists and m... more 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 semi-quasicrystalline 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.
In this paper we present the construction of several aggregates of tetrahedra. Each construction ... more 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 interesting properties.
We show that when projecting an edge-transitive N-dimensional polytope onto an M-dimensional subs... more We show that when projecting an edge-transitive N-dimensional polytope onto an M-dimensional subspace of R^N, the sums of the squares of the original and projected edges are in the ratio N=M.
We present the construction of a dense, quasicrystalline packing of regular tetrahedra with icosa... more 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. This small number of plane classes was achieved by a certain "golden r...
The Boerdijk-Coxeter helix is a helical structure of tetrahedra which possesses no non-trivial tr... more 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".
In this paper the experimental results of spectral modulation of a self-guided laser pulse in an ... more In this paper the experimental results of spectral modulation of a self-guided laser pulse in an underdense plasma will be presented. Experiments were conducted using an ultrashort laser pulse ($\ sim $50 fs) generated from the UCLA Ti: Sapphire laser system capable of ...
W e present an icosahedral quasicrystal, a Fibonacci icosagrid, obtained by spacing the parallel ... more W e present an icosahedral quasicrystal, a Fibonacci icosagrid, obtained by spacing the parallel planes in an icosagrid with the Fibonacci sequence. This quasicrystal can also be thought 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 Fibonacci icosagrid, and they can be enriched to form the Fibonacci icosagrid. This creates a mapping between the Fibonacci icosagrid and the E−8 lattice. It is known that the combined structure and dynamics of all gravitational and Standard Model particle fields, including fermions, are part of the E8 Lie algebra. Because of this, the Fibonacci icosagrid is a good candidates, for representing states and interactions between particles and fields in quantum mech...
We explore the structural similarities in three different languages, first in the protein languag... more 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 concre...
Most quasicrystals can be generated by the cut-and-project method from higher dimensional parent ... more 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 geometrica...
Every protein consists of a linear sequence over an alphabet of 20 letters/amino acids. The seque... more 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 qua-ternary (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...
The Kummer surface was constructed in 1864. It corresponds to the desingularisation of 1 the quot... more The Kummer surface was constructed in 1864. It corresponds to the desingularisation of 1 the quotient of a 4-torus by 16 complex double points. Kummer surface is known to play a role in 2 some models of quantum gravity. Following our recent model of the DNA genetic code based on the 3 irreducible characters of the finite group G 5 := (240, 105) ∼ = Z 5 2O (with 2O the binary octahedral 4 group), we now find that groups G 6 := (288, 69) ∼ = Z 6 2O and G 7 := (336, 118) ∼ = Z 7 2O can be 5 used as models of the symmetries in hexamer and heptamer proteins playing a vital role for some 6 biological functions. Groups G 6 and G 7 are found to involve the Kummer surface in the structure of 7 their character table. An analogy between quantum gravity and DNA/RNA packings is suggested. 8
In quasicrystals, any given local patch-called an emperor-forces at all distances the existence o... more 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.
The soybean lecithin–gallic acid complex ameliorates hepatic damage and iron-overload induced by ... more The soybean lecithin–gallic acid complex ameliorates hepatic damage and iron-overload induced by alcohol and exerts hepatoprotective effects.
The projection method for constructing quasiperiodic tilings from a higher dimensional lattice pr... more 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 dualisation 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 a projections of the D 6 and Z 6 lattices to R 3 and we present our analyses. We discuss the implications of this new generalization.
We present an icosahedral quasicrystal as a modification of the icosagrid, a multigrid with 10 pla... more We present an icosahedral quasicrystal as a modification of the icosagrid, a multigrid with 10 plane sets that are arranged with icosahedral symmetry. We use the Fibonacci chain to space the planes, thereby obtaining a quasicrystal with icosahedral symmetry. It has a surprising correlation to the Elser-Sloane quasicrystal1, a 4D cut-and-project of the E8 lattice. We call this quasicrystal the Fibonacci modified icosagrid quasicrystal. We found that this structure totally embeds another quasicrystal that is a compound of 20 3D slices of the Elser-Sloane quasicrystal. The slices, which contain only regular tetrahedra, are put together by a certain golden ratio based rotation2. Interesting 20-tetrahedron clusters arranged with the golden ratio based rotation appear repetitively in the structure. They are arranged with icosahedral symmetry. It turns out that this rotation is the dihedral angle of the 600-cell (the super-cell of the Elser-Sloane quasicrystal) and the angle between the tetr...
Quasicrystals are fractal because they are scale invariant and self similar. In this paper, a new... more 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 = φ. The corresponding pointwise dimension is 0.7. Various variations such as truncation from the head or tail, scrambling the orders of the sequence, changeing 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.
In light of the self-simulation hypothesis, a simple form implementation of the principle of effi... more 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.
On a two-dimensional quasicrystal, a Penrose tiling, we simulate for the first time a game of lif... more 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.
In geometrically frustrated clusters of polyhedra, gaps between faces can be closed without disto... more 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.
This paper reviews the empire problem for quasiperiodic tilings and the existing methods for gene... more 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.
Since antiquity, the packing of convex shapes has been of great interest to many scientists and m... more 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 semi-quasicrystalline 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.
In this paper we present the construction of several aggregates of tetrahedra. Each construction ... more 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 interesting properties.
We show that when projecting an edge-transitive N-dimensional polytope onto an M-dimensional subs... more We show that when projecting an edge-transitive N-dimensional polytope onto an M-dimensional subspace of R^N, the sums of the squares of the original and projected edges are in the ratio N=M.
We present the construction of a dense, quasicrystalline packing of regular tetrahedra with icosa... more 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. This small number of plane classes was achieved by a certain "golden r...
The Boerdijk-Coxeter helix is a helical structure of tetrahedra which possesses no non-trivial tr... more 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".
In this paper the experimental results of spectral modulation of a self-guided laser pulse in an ... more In this paper the experimental results of spectral modulation of a self-guided laser pulse in an underdense plasma will be presented. Experiments were conducted using an ultrashort laser pulse ($\ sim $50 fs) generated from the UCLA Ti: Sapphire laser system capable of ...
Uploads
Papers by Fang Fang