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Charles L Radin

Department of Mathematics

Mathematical Physics, Discrete Geometry


Phone: 512-471-0174

Office Location
PMA 12.114

Postal Address
AUSTIN, TX 78712

Ph.D., University of Rochester (1970)

Research Interests

Mathematical Physics, Discrete Geometry

Current research interest: Hilbert's eighteenth problem - understanding the symmetry of optimally dense packings, of spheres or polyhedra, in Euclidean and hyperbolic spaces, including aperiodic tilings such as the pinwheel, the quaquaversal, and the Penrose kites and darts. I'm also working on the closely related physics problem of the origin of structure in high density/low energy matter, using probabilistic models of sphere packings. In particular I'm interested in phase transitions in various exotic materials in and out of thermal equilibrium, for instance soft matter, such as sand piles or crumpled sheets, and quasicrystals. And although applications are not yet as compelling, I am also applying similar ideas to random networks.


C. Radin, Approach to equilibrium in a simple model, J. Math. Phys., 11(1970), 2945-2955. 

C. Radin, Noncommutative mean ergodic theory, Comm. Math. Phys., 21(1971), 291-302.

C. Radin, Gentle perturbations, Comm. Math. Phys., 23(1971), 189-198.

G. Emch and C. Radin, Relaxation of local thermal deviations from equilibrium, J. Math. Phys., 12(1971), 2043-2046. 

C. Radin, Ergodicity in von Neumann algebras, Pac. J. Math., 48(1973), 235-239. 

C. Radin, Automorphisms of von Neumann algebras as point transformations, Proc. Amer. Math. Soc., 39(1973), 343-346. 

C. Radin, Dynamics of limit models, Comm. Math. Phys., 33(1973), 283-292. 

C. Radin, A noncommutative L_1-mean ergodic theorem, Advances in Math., 21(1976), 110-111. 

C. Radin, Some remarks on the evolution of a Schrödinger particle in an attractive 1/r^2 potential, J. Math. Phys., 16(1975), 544-547. 

C. Radin, Particle lattice models and the dynamical instability of many-body systems, Comm. Math. Phys., 44(1975), 165-168. 

R.T. Powers and C. Radin, Average boundary conditions in Cauchy problems, J. Functional Analysis, 23(1976), 23-32. 

C. Radin, The dynamical instability of nonrelativistic many-body systems, Comm. Math. Phys., 54(1977), 69-80. 

C. Radin, Pointwise ergodic theory and operator algebras, J. Math. Phys. 19(1978), 1983-1985. 

C. Radin and B. Simon, Invariant domains for the time-dependent Schrödinger equation, J. Diff. Eqns. 29(1978), 289-296. 

J. A. Goldstein, C. Radin and R. E. Showalter, Convergence rates of ergodic limits for semigroups and cosine functions, Semigroup Forum, 16(1978), 89-95. 

C. Radin, Signal propagation in lattice models of quantum many body systems, Comm. Math. Phys., 62(1978), 159-166. 

C. S. Gardner and C. Radin, The infinite volume ground state of the Lennard-Jones potential, J. Stat. Phys., 20(1979), 719-724. 

G. Hamrick and C. Radin, The symmetry of ground states under perturbation, J. Stat. Phys., 21(1979), 601-607. 

R. Heitmann and C. Radin, The ground state for sticky disks, J. Stat. Phys., 22(1980), 281-287. 

B. Borden and C. Radin, The crystal structure of the noble gases, J. Chem. Phys., 75(1981), 2012-2013. 

C. Radin, The ground state for soft disks, J. Stat. Phys., 26(1981), 365-373.

C. Radin, Crystalline symmetry and surface tension, Physica, 113A(1982), 338-342. 

F. Nicolò and C. Radin, A first-order phase transition between crystal phases in the shift model, J. Stat. Phys., 28(1982), 473-478. 

C. Radin and L. Schulman, Periodicity of classical ground states, Phys. Rev. Letts., 51(1983), 621-622. 

C. Radin, Classical ground states in one dimension, J. Stat. Phys., 35(1984), 109-117. 

C. Radin, Tiling, periodicity, and crystals, J. Math. Phys. 26(1985), 1342-1344. 

C. Radin, Crystals and quasicrystals: a lattice gas model, Phys. Lett., 114A(1986), 381-383. 

C. Radin, Crystals and quasicrystals: a continuum model, Comm. Math. Phys., 105(1986), 385-390.

C. Radin, Correlations in classical ground states, J. Stat. Phys., 43(1986), 707-712.  

J. Miekisz and C. Radin, The unstable chemical structure of quasicrystalline alloys, Phys. Lett., 119A(1986), 133-134.

J. Miekisz and C. Radin, The third law of thermodynamics, Mod. Phys. Lett., B1(1987), 61-66. 

C. Radin, Low temperature and the origin of crystalline symmetry, Intl. J. Mod. Phys., B1(1987), 1157-1191.

C. Radin, Symmetry: Unifying Human Understanding, Amer. Scientist, 76(1988), 392-393. 

J. Miekisz and C. Radin, Why solids are not really crystalline, Phys. Rev., B39(1989), 1950-1952. 

C. Radin, Ordering in lattice gases at low temperature, J. Phys., A22(1989), 317-319. 

C. Gardner, J. Miekisz, C. Radin and A. Van Enter) Fractal ground state for an Ising model, J. Phys. A22(1989), L1019-L1023. 

C. Radin, Disordered Ground States of Classical Lattice Models, Revs. Math. Phys. 3(1991), 125-135. 

C. Radin, Global Order from Local Sources, Review-Expository Paper, Bull. Amer. Math. Soc. 25(1991), 335-364. 

C. Radin and M. Wolff, Space tilings and local isomorphism, Geometriae Dedicata 42(1992), 355-360.

C. Radin, Z^n versus Z actions for systems of finite type, Contemporary Mathematics 135(1992), 339-342. 

D. Berend and C. Radin, Are there chaotic tilings?, Comm. Math. Phys. 152(1993), 215-219. 

C. Radin, Symmetry of tilings of the plane, Bull. Amer. Math. Soc. 29(1993), 213-217. 

C. Radin, The pinwheel tilings of the plane, Annals of Math. 139(1994), 661-702. 

C. Radin, Symmetry and tilings, Notices Amer. Math. Soc. 42(1995), 26-31.

C. Radin, Space tilings and substitutions, Geometriae Dedicata 55(1995), 257-264. 

C. Radin, Aperiodic tilings in higher dimensions, Proc. Amer. Math. Soc. 123(1995), 3543-3548. 

C. Radin, Quasicrystals and geometry, Notices Amer. Math. Soc. 43(1996), 416-421. 

C. Radin, Miles of Tiles, Ergodic theory of Z^d-actions, London Math. Soc. Lecture Notes Ser. 228(1996), Cambridge Univ Press, 237-258.

C. Radin and L. Sadun, The isoperimetric problem for pinwheel tilings, Comm. Math. Phys. 177(1996), 255-263.  

H. Koch, R. de la Llave and C. Radin, Aubry-Mather theory for functions on lattices, Discrete Contin. Dynam. Systems 3(1997), 135-151. 

C. Radin, Aperiodic tilings, ergodic theory and rotations, The Mathematics of Long-Range Aperiodic Order, Nato Adv. Sci. Institutes Ser. 489(1997), Kluwer, 499-519. 

J.H. Conway and C. Radin, Quaquaversal tilings and rotations, Inventiones math. 132(1998), 179-188. 

C. Radin and L. Sadun, Subgroups of SO(3) associated with tilings, J. Algebra 202(1998), 611-633. 

C. Radin, Statistical Symmetry, chapter in "Geometry at Work", MAA Notes volume 53 (Math. Assoc. Amer., Washington, DC, 2000). 

C. Radin and L. Sadun, An algebraic invariant for substitution tiling systems, Geometriae Dedicata, 73(1998), 21-37. 

C. Radin, Rashomon: pavages et rotations, Gaz. Soc. Math. de France 79(1999), 49-57. 

C. Radin and L. Sadun, On 2-generator subgroups of SO(3), Trans. Amer. Math. Soc. 351(1999), 4469-4480.

J.H. Conway, C. Radin and L. Sadun, On angles whose squared trigonometric functions are rational, Discrete Comput. Geom. 22(1999), 321-332. 

C. Radin, Symmetries of quasicrystals, J. Stat. Phys. 95(1999), 827-833. 

J.H. Conway, C. Radin and L. Sadun, Relations in SO(3) supported by geodetic angles, Discrete Comput. Geom. 23(2000), 453-463. 

C. Radin and L. Sadun, Isomorphism of hierarchical structures, Ergodic Theory Dynam. Systems 21(2001), 1239-1248. 

N. Ormes, C. Radin and L. Sadun, A homeomorphism invariant for substitution tiling spaces, Geometriae Dedicata 90(2002), 153-182. 

L. Bowen and C. Radin, Densest packing of equal spheres in hyperbolic space, Discrete Comput. Geom. 29(2003), 23-39. 

L. Bowen and C. Radin, Optimally dense packings of hyperbolic space, Geometriae Dedicata 104(2004), 37-59. 

C. Radin, The symmetry of optimally dense packings, in Non-Euclidean Geometries: János Bolyai Memorial Volume, Mathematics and Its Applications, Vol. 581, Springer, 2006, pp. 197-207.

L. Bowen, C. Holton, C. Radin and L. Sadun, Uniqueness and symmetry in problems of optimally dense packings, Math. Phys. Electron. J. 11(2005), Paper 1, 34pp. (electronic) 

C. Radin, Orbits of orbs: sphere packing meets Penrose tilings, Amer. Math. Monthly 111(2004), 137-149.

C. Holton, C. Radin and L. Sadun, Conjugacies for tiling dynamical systems, Comm. Math. Phys. 209(2005), 343-359. 

C. Radin, Existence of ground state configurations, Math. Phys. Electron. J. 10(2004), Paper 6, 7pp. (electronic). 

C. Holton and C. Radin, Topological conjugacy between aperiodic tiling dynamical systems. 

C. Radin and L. Sadun, Structure of the hard sphere solid, Phys. Rev. Lett. 94(2005), 015502. 

C. Radin, A review of two books on sphere packing, Amer. Math. Monthly 113(2006), 87-90. 

H. Koch, C. Radin and L. Sadun, Most stable structure for hard spheres, Phys. Rev. E 72(2005), 016708. 

L. Bowen, R. Lyons, C. Radin and P. Winkler, Fluid-solid transition in a hard-core system, Phys. Rev. Lett. 96(2006), 025701. 

L. Bowen, R. Lyons, C. Radin and P. Winkler, A solidification phenomenon in random packings, SIAM J. Math. Anal. 38(2006), 1075-1089. 
M. Schröter, S. Nägle, C. Radin and H.L. Swinney, Phase transition in a static granular system, EPL 78(2007), 44004. 
C. Radin, Conference report II, Appl. Rheol. 17:3(2007), 166.
C. Radin, Random close packing of granular matter, J. Stat. Phys. 131(2008), 567-573. 

D. Aristoff and C. Radin, Random loose packing in granular matter, J. Stat. Phys., 135(2009), 1-23. 

J. Bellissard, C. Radin and S. Shlosman, The characterization of ground states, J. Phys. A: Math. Theor. 43(2010), 305001. 

D. Aristoff and C. Radin, Random close packing in a granular model, J. Math. Phys. 51(2010), 113302. 

H. Koch and C. Radin, Modelling quasicrystals at positive temperature, J. Stat. Phys., 138(2010), 465-475.

D. Aristoff and C. Radin, Layering in crumpled sheets, EPL 91(2010), 56003. 

D. Aristoff and C. Radin, Dilatancy transition in a granular model, J. Stat. Phys. 143(2011), 215-225. 

T. Argo, M. Guild, P. Wilson, M. Schröter, C. Radin and H. Swinney, Sound speed in water-saturated glass beads as a function of frequency and porosity, J. Acoust. Soc. Am. 129(2011) EL101-EL107. 

J.-F. Métayer, D.J. Suntrup III, C. Radin, H.L. Swinney and M. Schröter, Shearing of frictional sphere packings, EPL 93(2011), 64003

D. Aristoff and C. Radin, First order phase transition of a long polymer chain, J. Phys. A: Math. Theor. 44(2011), 065004.

D. Aristoff and C. Radin, First order phase transition in a model of quasicrystals, J. Phys. A: Math. Theor. 44(2011), 255001.

D. Aristoff and C. Radin, Rigidity in solids, J. Stat. Phys. 144(2011) 1247-1255. 

C. Radin and M. Yin, Phase transitions in exponential random graphs, Ann. Appl. Probab. (to appear). arXiv:1108.0649 

D. Aristoff and C. Radin, Emergent structures in large networks, J. Appl. Probab. (to appear). arXiv:1110.1912 

C. Radin, A revolutionary material, Notices Amer. Math. Soc. 60(2013), 310-315. 

C. Radin and L. Sadun, Phase transitions in a complex network, J. Phys. A: Math. Theor. 46 (2013) 305002 print.pdf; The previous version, with title `A mean field analysis of the fluid/solid phase transition', is here

C. Radin and L. Sadun, Singularities in the entropy of asymptotically large simple graphs, arXiv:1302.3531


C. Radin, Miles of Tiles, Student Mathematical Library, Vol 1 (Amer. Math. Soc., Providence, 1999).