Modelling quasicrystals at positive temperature (2009)
We consider a two-dimensional lattice model of equilibrium statistical mechanics, using nearest neighbor interactions based on the matching conditions for an aperiodic set of 16 Wang tiles. This...
AN ALGEBRAIC INVARIANT FOR SUBSTITUTION TILING SYSTEMS by (2009)
The Erwin, Schrödinger International Boltzmanngasse, Charles Radin, Lorenzo Sadun, Charles Radin, Lorenzo Sadun
We consider tilings of Euclidean spaces by polygons or polyhedra, in particular, tilings made by a substitution process, such as the Penrose tilings of the plane. We define an isomorphism invariant...
The characterization of ground states (2009)
Bellissard, Jean, Radin, Charles, Shlosman, Senya
We consider limits of equilibrium distributions as temperature approaches zero, for systems of infinitely many particles, and the characterization of the support of such limiting distributions. Such...
Argo IV, Theodore F., Guild, Matthew D., Wilson, Preston S., Schröter, Matthias, Radin, Charles, Swinney, Harry L.
Sound propagation in water-saturated granular sediments is known to depend on the sediment porosity, but few data in the literature address both the frequency and porosity dependency. To begin to...
Random Loose Packing in Granular Matter by (2009)
We introduce and simulate a two-dimensional Edwards-style model of granular matter at vanishing pressure. The model incorporates gravity and friction in a simplified manner, and exhibits a random...
Random Loose Packing in Granular Matter (2008)
Aristoff, David, Radin, Charles
We introduce and simulate a two dimensional probabilistic model of granular matter at vanishing pressure. The model exhibits a perfectly sharp random loose packing density, a phenomenon that should...
A“pinwheel ” tiling of the plane (as in Figure 1) does not have any translational or rotational symmetry in the usual sense. But in a tantalizing, unconventional way it is highly symmetric, and...
Charles Radin, Marjorie Senechal
$59.95 Hardcover Penrose tilings (Fig. 1) are beautiful. They also suggest significant new mathematics, so it is about time someone wrote a book about them which is readable (in fact, eminently...
Editor: R. de la Llave Existence of Ground State Configurations (2008)
Abstract. We prove the existence of ground state configurations for systems of infinitely many particles interacting, in d-dimensional Euclidean space, through many-body potentials with hard core....
Abstract. We propose a method to analyze the density of packings of spheres of fixed radius in the hyperbolic space of any dimension m ≥ 2, and prove that for all but countably many radii,...
TOPOLOGICAL CONJUGACY BETWEEN APERIODIC TILING DYNAMICAL SYSTEMS (2008)
We extend to certain aperiodic tiling dynamical systems associated with an underlying substitution a geometric invariant for topological conjugacy.
Invent. math. 132, 179±188 (1998) Quaquaversal tilings and rotations (2008)
Abstract. We construct a hierarchical tiling of 3 dimensional Euclidean space based on a triangular prism, using repeated rotations, about orthogonal axes, by angles 2p=m and 2p=n. To analyze the...
Orbits of Orbs: Sphere Packing Meets (2008)
Penrose Tilings, Charles Radin
arrange objects like balls or polyhedra in a given container. There is more scope for rich mathematics in having the “container ” be all of space, so there are no boundaries to spoil the...
Random Close Packing of Granular Matter by (2008)
We propose an interpretation of the random close packing of granular materials as a phase transition, and discuss the possibility of experimental verification.
TOPOLOGICAL CONJUGACY BETWEEN APERIODIC TILING DYNAMICAL SYSTEMS (2008)
We extend to certain aperiodic tiling dynamical systems associated with an underlying substitution a geometric invariant for topological conjugacy.
Editor: R. de la Llave UNIQUENESS AND SYMMETRY IN PROBLEMS OF OPTIMALLY DENSE PACKINGS (2008)
Lewis Bowen, Charles Holton, Charles Radin
Abstract. Part of Hilbert’s eighteenth problem is to classify the symmetries of the densest packings of bodies in Euclidean and hyperbolic spaces, for instance the densest packings of balls or...
Editor: R. de la Llave Existence of Ground State Configurations (2008)
Abstract. We prove the existence of ground state configurations for systems of infinitely many particles interacting, in d-dimensional Euclidean space, through many-body potentials with hard core....
Editor: R. de la Llave UNIQUENESS AND SYMMETRY IN PROBLEMS OF OPTIMALLY DENSE PACKINGS (2008)
Lewis Bowen, Charles Holton, Charles Radin
Abstract. Part of Hilbert’s eighteenth problem is to classify the symmetries of the densest packings of bodies in Euclidean and hyperbolic spaces, for instance the densest packings of balls or...
Editor: R. de la Llave Existence of Ground State Configurations (2008)
Abstract. We prove the existence of ground state configurations for systems of infinitely many particles interacting, in d-dimensional Euclidean space, through many-body potentials with hard core....
DENSEST PACKING OF EQUAL CIRCLES IN THE HYPERBOLIC PLANE (2007)
We propose a definition of density for packings of circles of fixed radius in the hyperbolic plane, and prove that for all but countably many radii, optimally dense packings must have low symmetry.
Aperiodic Tiliings In Higher (2007)
Dimensions By Charles, Charles Radin
We show that in dimensions d 3, aperiodic tilings can naturally avoid more symmetries than just translations. 1991 Classification: 58F11, 52C20, 47A35 * Research supported in part by NSF Grant No....
So Supported By, John H. Conway, Charles Radin, Lorenzo Sadun
We consider rotations A; B of finite order in SO(3), about axes separated by an angle of restricted type, and attempt to classify the possible group relations between A and B. We show that the...
Aubry--Mather Theory For Functions On Lattices. (2007)
Hans Koch Rafael, Charles Radin
. We generalize the Aubry-Mather theorem on the existence of quasiperiodic solutions of one dimensional difference equations to situations in which the independent variable ranges over more...
Optimally dense packings of hyperbolic space (2007)
In previous work a probabilistic approach to controlling di#culties of density in hyperbolic space led to a workable notion of optimal density for packings of bodies. In this paper we extend an...
This is a slightly expanded version of a talk given at the Janos Bolyai Conference
DENSEST PACKING OF EQUAL CIRCLES IN THE HYPERBOLIC PLANE (2007)
We propose a definition of density for packings of circles of fixed radius in the hyperbolic plane, and prove that for all but countably many radii, optimally dense packings must have low symmetry.
AN ALGEBRAIC INVARIANT FOR SUBSTITUTION TILING SYSTEMS by (2007)
The Erwin, Schrodinger International Boltzmanngasse, Charles Radin, Charles Radin, Lorenzo Sadun, Lorenzo Sadun
We consider tilings of Euclidean spaces by polygons or polyhedra, in particular, tilings made by a substitution process, such as the Penrose tilings of the plane. We define an isomorphism invariant...
Uniqueness and Symmetry in Problems of Optimally Dense Packings (2007)
Lewis Bowen, Charles Holton, Charles Radin, Lorenzo Sadun
We analyze the general problem of determining optimally dense packings, in a Euclidean or hyperbolic space, of congruent copies of some fixed finite set of bodies. We are strongly guided by examples...
Conjugacies For Tiling Dynamical Systems (2007)
Charles Holton Charles, Charles Radin, Lorenzo Sadun
We consider tiling dynamical systems and topological conjugacies between them. We prove that the criterion of being finite type is invariant under topological conjugacy. For substitution tiling...
The Symmetry of Optimally Dense Packings (2007)
This is a slightly expanded version of a talk given at the Janos Bolyai Conference on Hyperbolic Geometry, held in Budapest in July, 2002. The general subject of the talk was the densest packings of...
Random close packing of granular matter (2007)
We propose an interpretation of the random close packing of granular materials as a phase transition, and discuss the possibility of experimental verification.
Random Close Packing of Granular Matter by (2007)
We propose an interpretation of the random close packing of granular materials as a phase transition.
Phase transition in a static granular system (2006)
Schröter, Matthias, Nägle, Sibylle, Radin, Charles, Swinney, Harry L.
We find that a column of glass beads exhibits a well-defined transition between two phases that differ in their resistance to shear. Pulses of fluidization are used to prepare static states with...
A Solidification Phenomenon in Random Packings (2005)
Bowen, Lewis, Lyons, Russell, Radin, Charles, Winkler, Peter
We prove that uniformly random packings of copies of a certain simply-connected figure in the plane exhibit global connectedness at all sufficiently high densities, but not at low densities.
Fluid/solid transition in a hard-core system (2005)
Bowen, Lewis, Lyons, Russell, Radin, Charles, Winkler, Peter
We prove that a system of particles in the plane, interacting only with a certain hard-core constraint, undergoes a fluid/solid phase transition.
Most stable structure for hard spheres (2005)
Koch, Hans, Radin, Charles, Sadun, Lorenzo
The hard sphere model is known to show a liquid-solid phase transition, with the solid expected to be either face centered cubic or hexagonal close packed. The difference in free energy between the...
Uniqueness and symmetry in problems of optimally dense packings (2005)
Bowen, Lewis, Holton, Charles, Radin, Charles, Sadun, Lorenzo
Part of Hilbert's eighteenth problem is to classify the symmetries of the densest packings of bodies in Euclidean and hyperbolic spaces, for instance the densest packings of balls or simplices. We...
Uniqueness and symmetry in problems of optimally dense packings (2005)
Bowen, Lewis, Holton, Charles, Radin, Charles, Sadun, Lorenzo
Part of Hilbert's eighteenth problem is to classify the symmetries of the densest packings of bodies in Euclidean and hyperbolic spaces, for instance the densest packings of balls or simplices. We...
The structure of the hard sphere solid (2004)
Radin, Charles, Sadun, Lorenzo
We show that near densest-packing the perturbations of the HCP structure yield higher entropy than perturbations of any other densest packing. The difference between the various structures shows up...
Charles Holton, Charles Radin, Lorenzo Sadun, C. Holton, C. Radin, L. Sadun
Abstract: We consider tiling dynamical systems and topological conjugacies between them. We prove that the criterion of being of finite type is invariant under topological conjugacy. For substitution...
Charles Holton, Charles Radin, Lorenzo Sadun, C. Holton, C. Radin, L. Sadun
Abstract: We consider tiling dynamical systems and topological conjugacies between them. We prove that the criterion of being of finite type is invariant under topological conjugacy. For substitution...
Optimally Dense Packings of Hyperbolic Space (2004)
In previous work a probabilistic approach to controlling di#culties of density in hyperbolic space led to a workable notion of optimal density for packings of bodies. In this paper we extend an...
Conjugacies for Tiling Dynamical Systems (2003)
Holton, Charles, Radin, Charles, Sadun, Lorenzo
We consider tiling dynamical systems and topological conjugacies between them. We prove that the criterion of being finite type is invariant under topological conjugacy. For substitution tiling...
Uniqueness and Symmetry in Problems of Optimally Dense Packings (2003)
Bowen, Lewis, Holton, Charles, Radin, Charles, Sadun, Lorenzo
We analyze the general problem of determining optimally dense packings, in a Euclidean or hyperbolic space, of congruent copies of some fixed finite set of bodies. We are strongly guided by examples...
Optimally dense packings of hyperbolic space (2002)
In previous work a probabilistic approach to controlling difficulties of density in hyperbolic space led to a workable notion of optimal density for packings of bodies. In this paper we extend an...
A Nash-Moser Implicit Function Theorem with Whitney Regularity (2002)
Hans Koch, Lorenzo Sadun, Charles Radin, Philip Morrison, John Andrew Vano, ...
To my parents Who taught me the value of understanding To my wife Who makes sure I eat and sleep somewhat regularly And to all my cats Who occasionally inspire mathematical insight Acknowledgments I...
A homeomorphism invariant for substitution tiling spaces, Geom. Dedicata 90 (2002)
Nicholas Ormes, Charles Radin, Lorenzo Sadun
Abstract. We derive a homeomorphism invariant for those tiling spaces which are made by rather general substitution rules on polygonal tiles, including those tilings, like the pinwheel, which contain...
Abstract. In previous work, a probabilistic approach to controlling difficulties of density in hyperbolic space led to a workable notion of optimal density for packings of bodies. In this paper we...
A homeomorphism invariant for substitution tiling spaces, Geom. Dedicata 90 (2002)
Nicholas Ormes, Charles Radin, Lorenzo Sadun
We derive a homeomorphism invariant for those tiling spaces which are made by rather general substitution rules on polygonal tiles, including those tilings, like the pinwheel, which contain tiles in...
A homeomorphism invariant for substitution tiling spaces, Geom. Dedicata 90 (2002)
Nicholas Ormes, Charles Radin, Lorenzo Sadun
Abstract. We derive a homeomorphism invariant for those tiling spaces which are made by rather general substitution rules on polygonal tiles, including those tilings, like the pinwheel, which contain...
Abstract. In previous work, a probabilistic approach to controlling difficulties of density in hyperbolic space led to a workable notion of optimal density for packings of bodies. In this paper we...
A Homeomorphism Invariant for Substitution Tiling Spaces (2000)
Ormes, Nic, Radin, Charles, Sadun, Lorenzo
We derive a homeomorphism invariant for those tiling spaces which are made by rather general substitution rules on polygonal tiles, including those tilings, like the pinwheel, which contain tiles in...
A Homeomorphism Invariant For Substitution Tiling Spaces (2000)
Nicholas Ormes, Charles Radin, Lorenzo Sadun
We derive a homeomorphism invariant for those tiling spaces which are made by rather general substitution rules on polygonal tiles, including those tilings, like the pinwheel, which contain tiles in...
Abstract. We consider hierarchical structures such as Fibonacci sequences and Penrose tilings, and examine the consequences of different choices for the definition of isomorphism. In particular, we...
Libro de geometría en el que se abordan figuras simétricas o grupos de figuras (tiling). Contenido: Teoría ergódica; Física para matemáticos; Orden; Simetría.
Miles of Tiles / C. Radin. (1999)
Libro de geometría en el que se abordan figuras simétricas o grupos de figuras (tiling). Contenido: Teoría ergódica; Física para matemáticos; Orden; Simetría.
Tiles By Charles, Charles Radin
This article corresponds closely to four lectures given in conjunction with the July 1994 workshop at Warwick; the informal format of those lectures is maintained. Our main goal is to discuss the...
On 2-generator subgroups of SO(3 (1999)
Abstract. We classify all subgroups of SO(3) that are generated by two elements, each a rotation of finite order, about axes separated by an angle that is a rational multiple of π. In all cases we...
Isomorphism of Hierarchical Structures (1998)
Radin, Charles, Sadun, Lorenzo
We consider hierarchical structures such as Fibonacci sequences and Penrose tilings, and examine the consequences of different choices for the definition of isomorphism. In particular we discuss the...
On Angles Whose Squared Trigonometric Functions are Rational (1998)
Conway, John H., Radin, Charles, Sadun, Lorenzo
We consider the rational linear relations between real numbers whose squared trigonometric functions have rational values, angles we call ``geodetic''. We construct a convenient basis for the vector...
Subgroups Of So(3) Associated With Tilings (1998)
By Charles Radin, Charles Radin, Lorenzo Sadun
We give a thorough analysis of those subgroups of SO(3) generated by rotations about perpendicular axes by 2ß=p and 2ß=q. A corollary is that such a group is the free product of the cyclic groups...
Symmetries of Quasicrystals (1998)
We consider tiling models of "round quasicrystals" which would have diffraction patterns which are fully rotation invariant -- rings instead of Bragg peaks. They can be distinguished from...
Isomorphism of Hierarchical Structures (1998)
We consider hierarchical structures such as Fibonacci sequences and Penrose tilings, and examine the consequences of different choices for the definition of isomorphism. In particular we discuss the...
RASHOMON (pavages et rotations) (1998)
this article there can be a significant advantage in considering various points of view of a complicated phenomenon, and it is not surprising that the further separated the worlds from which the...
An algebraic invariant for substitution tiling systems, Geometriae Dedicata 73 (1998)
Abstract. We consider tilings of Euclidean spaces by polygons or polyhedra, in particular, tilings made by a substitution process, such as the Penrose tilings of the plane. We define an isomorphism...
An Algebraic Invariant for Substitution Tiling Systems (1997)
Radin, Charles, Sadun, Lorenzo
We consider tilings of Euclidean spaces by polygons or polyhedra, in particular, tilings made by a substitution process, such as the Penrose tilings of the plane. We define an isomorphism invariant...
Radin, Charles, Sadun, Lorenzo
In 3-dimensional Euclidean space, let $A$ be a rotation by $2 \pi/p$ about a fixed axis, and let $B$ be a rotation by $2 \pi/q$ about a second axis that makes an angle of $2 \pi n/m$ with the first,...
On 2-Generator Subgroups of SO(3) (1997)
We classify all subgroups of SO(3) that are generated by two elements, each a rotation of finite order, about axes separated by an angle that is a rational multiple of ß. In all cases we give a...
Subgroups of SO(3) Associated with Tilings (1996)
Radin, Charles, Sadun, Lorenzo
We give a thorough analysis of those subgroups of SO(3) generated by rotations about perpendicular axes by 2\pi/p and 2\pi/q. A corollary is that such a group is the free product of the cyclic groups...
An Algebraic Invariant For Substitution Tiling Systems (1996)
We consider tilings of Euclidean spaces by polygons or polyhedra, in particular, tilings made by a substitution process, such as the Penrose tilings of the plane. We define an isomorphism invariant...
The Isoperimetric Problem For Pinwheel Tilings (1996)
By Charles Radin, Charles Radin, Lorenzo Sadun
In aperiodic "pinwheel" tilings of the plane there exist unions of tiles with ratio (area)/(perimeter) 2 arbitrarily close to that of a circle. Such approximate circles can be constructed...
Subgroups of SO 3 Associated with Tilings (1996)
We give a thorough analysis of those subgroups of SO 3 generated by rotations about perpendicular axes by 2��p and 2��q. A corollary is that such a group is the free product of the cyclic...
Subgroups of SO 3 Associated with Tilings (1996)
We give a thorough analysis of those subgroups of SO 3 generated by rotations about perpendicular axes by 2��p and 2��q. A corollary is that such a group is the free product of the cyclic...
Quaquaversal Tilings And Rotations (1995)
This paper is concerned with certain patterns in 3 dimensional Euclidean space, and their symmetries. One step in analyzing these symmetries involves determining the group G(m;n) generated by a pair...
Space Tilings And Substitutions (1995)
We generalize the study of symbolic dynamical systems of finite type and ZZ 2 action, and the associated use of symbolic substitution dynamical systems, to dynamical systems with IR 2 action. The new...
Quaquaversal Tilings And Rotations (1995)
This paper is concerned with certain patterns in 3 dimensional Euclidean space, and their symmetries. One step in analyzing these symmetries involves determining the group G(m;n) generated by a pair...
This article corresponds closely to four lectures given in conjunction with the July 1994 workshop at Warwick; the informal format of those lectures is maintained. Our main goal is to discuss the...
The Isoperimetric Problem For Pinwheel Tilings (1994)
In aperiodic "pinwheel" tilings of the plane there exist unions of tiles with ratio (area)/(perimeter) 2 arbitrarily close to that of a circle. Such approximate circles can be constructed...
The Pinwheel Tilings of the Plane (1994)
this paper, and many long hours of discussions with Dani Berend. 2 CHARLES RADIN means "disordered" in the probabilistic sense used to study patterns in nonlinear dynamics. Within the field...
Symmetry Of Tilings Of The Plane (1994)
. We discuss two new results on tilings of the plane. In the first, we give sufficient conditions for the tilings associated with an inflation rule to be uniquely ergodic under translations, the...
Symmetry of tilings of the plane (1993)
We discuss two new results on tilings of the plane. In the first, we give sufficient conditions for the tilings associated with an inflation rule to be uniquely ergodic under translations, the...
Are there chaotic tilings (1993)
We develop a class of examples in the form of tiling dynamical systems for use as toy models in statistical mechanics, to analyze the possible existence of disordered crystals. We give the first such...
Aperiodic Tiliings In Higher Dimensions (1993)
We show that in dimensions d 3, aperiodic tilings can naturally avoid more symmetries than just translations. 1991 Classification: 58F11, 52C20, 47A35 * Research supported in part by NSF Grant No....
Symmetry of Tilings of the Plane (1993)
We discuss two new results on tilings of the plane. In the first, we give sufficient conditions for the tilings associated with an inflation rule to be uniquely ergodic under translations, the...
Are There Chaotic Tilings? (1993)
We modify a method of S. Mozes to produce interesting tiling dynamical systems with an R n -action in place of Z n -action. An example is given which is weakly mixing.
Space Tilings And Local Isomorphism (1992)
We prove that for every finite tile set, isometric copies of which can tile euclidean n-space, there is a tiling with the local isomorphism property. * Research supported in part by NSF Grant No....
Disordered ground states of classical lattice models, Revs (1991)
We use strictly ergodic dynamical systems to describe two methods for constructing short range interactions of classical statistical mechanics models with unique ground states and unusual properties...
Global Order From Local Sources (1991)
This article contains introductions to three open problems of significant research interest, taken from number theory, logic, and condensed matter physics. All three problems will be shown to have at...
Z n Versus Z Actions for Systems of Finite Type (1991)
We consider dynamical systems of finite type with Z n actions, and discuss the differences between the cases n = 1 and n 2. For the latter we examine the degree of "order" which is possible...
Disordered Ground States Of Classical Lattice Models (1991)
We use strictly ergodic dynamical systems to describe two methods for constructing short range interactions of classical statistical mechanics models with unique ground states and unusual properties...
Ordering in lattice gases at low temperature (1989)
Abstract. We show that for generic short-range classical lattice gas models the third law of thermodynamics holds-entropy goes to zero with temperature. The most fundamental aspect of low-temperature...
CRYSTALS AND QUASICRYSTALS: A LATI'ICE GAS MODEL:~ (1986)
We give an example of a lattice gas model with a completely symmetric, short-range, two-body interaction which has quasiperiodic, but no periodic, ground states. The discovery of quasicrystalline...
The unstable chemical structure of the quasicrystalline alloys, Phys (1986)
We show that a typical toy model of an ordered quasicrystalline alloy has a range of stoichiometries. We consider the low temperature ordered phases discrete model with nondegenerate periodic ground...
CRYSTALLINE SYMMETRY AND SURFACE TENSION (1981)
We analyze a model of interacting ions and show that surface tension breaks the crystalline symmetry of the ground state. 1.
An Algebraic Invariant For Substitution Tiling Systems
We consider tilings of Euclidean spaces by polygons or polyhedra, in particular, tilings made by a substitution process, such as the Penrose tilings of the plane. We define an isomorphism invariant...
On Angles Whose Squared Trigonometric Functions Are Rational
John H. Conway, Charles Radin, Lorenzo Sadun
We consider the rational linear relations between real numbers whose squared trigonometric values are rational, angles we call "geodetic". We construct a convenient basis for the vector...