Crystallographic Groups, Strictly Tessellating Polytopes, and Analytic Eigenfunctions

Abstract The mathematics of crystalline structures connects analysis, geometry, algebra, and number theory. The planar crystallographic groups were classified in the late 19th century. One hundred years later, Bérard proved that the fundamental domains of all such groups satisfy a very special analytic property: the Dirichlet eigenfunctions for the Laplace eigenvalue equation are all trigonometric functions. In 2008, McCartin proved that in two dimensions, this special analytic property has both an equivalent algebraic formulation, as well as an equivalent geometric formulation. Here we generalize the results of Bérard and McCartin to all dimensions. We prove that the following are equivalent: the first Dirichlet eigenfunction for the Laplace eigenvalue equation on a polytope is real analytic, the polytope strictly tessellates space, and the polytope is the fundamental domain of a crystallographic Coxeter group. Moreover, we prove that under any of these equivalent conditions, all of the eigenfunctions are trigonometric functions. To conclude, we connect these topics to the Fuglede and Goldbach conjectures and give a purely geometric formulation of Goldbach’s conjecture.

[1]  Wilhelm Killing,et al.  Die Zusammensetzung der stetigen endlichen Transformationsgruppen , 1888 .

[2]  E. V. Chuprunov,et al.  n-dimensional space groups and regular point systems , 1988 .

[3]  Zhen Zhen,et al.  Graphene: Fundamental research and potential applications , 2017 .

[4]  A. Wiles,et al.  Ring-Theoretic Properties of Certain Hecke Algebras , 1995 .

[5]  B. Farkas,et al.  On Fuglede's Conjecture and the Existence of Universal Spectra , 2006, math/0612016.

[6]  R. Courant,et al.  Methoden der mathematischen Physik , .

[7]  Nicolas Bourbaki,et al.  Groupes et algèbres de Lie , 1971 .

[8]  J. E. Chisholm,et al.  International Tables for Crystallography, Vol. A: Space-group Symmetry , 1984, Mineralogical Magazine.

[9]  Andre K. Geim,et al.  Electric Field Effect in Atomically Thin Carbon Films , 2004, Science.

[10]  K. Novoselov,et al.  A roadmap for graphene , 2012, Nature.

[11]  H. Brown,et al.  Crystallographic Groups of Four-Dimensional Space , 1978 .

[12]  A. Schönflies.  Ueber Gruppen von Bewegungen , 1887 .

[13]  J. Kappraff.  Symmetry of the Plane , 2001 .

[14]  R. Young,et al.  Mechanical properties of graphene and graphene-based nanocomposites , 2017 .

[15]  L. Bieberbach,et al.  Über die Bewegungsgruppen der Euklidischen Räume (Zweite Abhandlung.) Die Gruppen mit einem endlichen Fundamentalbereich , 1912 .

[16]  Nicolas Bourbaki,et al.  Eléments de Mathématique , 1964 .

[17]  Gebräuchliche Fertigarzneimittel,et al.  V , 1893, Therapielexikon Neurologie.

[18]  R. Sarpong,et al.  Bio-inspired synthesis of xishacorenes A, B, and C, and a new congener from fuscol† †Electronic supplementary information (ESI) available. See DOI: 10.1039/c9sc02572c , 2019, Chemical science.

[19]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[20]  L. Bieberbach,et al.  Über die Bewegungsgruppen der Euklidischen Räume , 1911 .

[21]  Owen Jones,et al.  The Grammar of Ornament , 1988 .

[22]  Mihail N. Kolountzakis,et al.  Complex Hadamard matrices and the spectral set conjecture , 2004 .

[23]  Bent Fuglede,et al.  Commuting self-adjoint partial differential operators and a group theoretic problem , 1974 .

[24]  Harold R. Parks,et al.  A Primer of Real Analytic Functions , 1992 .

[25]  Terence Tao,et al.  Fuglede's conjecture is false in 5 and higher dimensions , 2003, math/0306134.

[26]  M. Matolcsi.  Fuglede’s conjecture fails in dimension 4 , 2005, math/0611936.

[27]  Spectres et groupes cristallographiques I: Domaines euclidiens , 1980 .

[28]  John Bamberg,et al.  The Crystallographic Restriction, Permutations, and Goldbach's Conjecture , 2003, Am. Math. Mon..

[29]  Safaa A. Elsherif,et al.  Revolution of Graphene for different applications: State-of-the-art , 2017 .

[30]  The Fuglede conjecture for convex domains is true in all dimensions , 2019, 1904.12262.

[31]  Nir Lev,et al.  Fuglede's spectral set conjecture for convex polytopes , 2016, 1602.08854.

[32]  M. Matolcsi,et al.  Tiles with no spectra , 2004, math/0406127.

[33]  A. Wiles.  Modular Elliptic Curves and Fermat′s Last Theorem(抜粋) (フェルマ-予想がついに解けた!?) , 1995 .

[34]  J. Bamberg,et al.  The Crystallographic Restriction, Permutations, and Goldbach's Conjecture , 2003 .

[35]  M BrianJ.,et al.  On Polygonal Domains with Trigonometric Eigenfunctions of the Laplacian under Dirichlet or Neumann Boundary Conditions , 2008 .

[36]  G. Folland.  Fourier analysis and its applications , 1992 .

[37]  T. Tao,et al.  The Fuglede spectral conjecture holds for convex planar domains , 2003 .