Classification of spherical monohedral polygonal tilings
The following three motivations come from the work of great mathematicians David Hilbert (January 23, 1862 Königsberg - February 14, 1943) and Branko Grünbaum ( 2 October 1929 Croatia - present. A fellow of American mathematical society), as well as from recent study of supramolecules.
Hilbert's eighteenth problem, question No.2 asked whether there is an anisohedral polyhedron of euclidean spaces. An anisohedral tile, by definition, admits a monohedral tiling but no isohedral tiling. We say a tiling is isohedral, if the symmetry group acts transitively on the tiles. Karl Reinhardt answered Question No.2 of Hilbert's eighteenth problem in 1928 by finding examples of such polyhedra. Heesch then gave an example of an anisohedral tile in the plane in 1935. By using the terminology of Archimedean duals given in Chapter 21 of (J. H. Conway, H. Burgiel, and C. Goodman-Strauss. The Symmetries of Things. A K Peters Ltd., Wellesley, MA, 2008.), [Sakano-Akama15] classified all spherical monohedral (kite/dart/rhombus)-faced tilings, as follows: The set of spherical monohedral rhombus-faced tilings consists of (1) the central projection of the rhombic dodecahedron, (2) the central projection of the rhombic triacontahedron, (3) a series of non-isohedral tilings, and (4) a series of tilings which are topologically trapezohedra (here a trapezohedron is the dual of an antiprism.). The set of spherical tilings by congruent kites consists of (1) the central projection T of the tetragonal icosikaitetrahedron, (2) the central projection of the tetragonal hexacontahedron, (3) a non-isohedral tiling obtained from T by gliding a hemisphere of T with π/4 radian, and (4) a continuously deformable series of tilings which are topologically trapezohedra. The set of spherical tilings by congruent darts is a continuously deformable series of tilings which are topologically trapezohedra. In the above explanation, unless otherwise stated, the tilings we have enumerated are isohedral and admit no continuous deformation. We prove that if a spherical (kite/dart/rhombus) admits an edge-to-edge spherical monohedral tiling, then it also does a spherical isohedral tiling. We also prove that the set of anisohedral, spherical triangles (i.e., spherical triangles admitting spherical monohedral triangular tilings but not any spherical isohedral triangular tilings) consists of a certain, infinite series of isosceles triangles I, and an infinite series of right scalene triangles which are the bisections of I.
A polytope is said to be regular, if the symmetry group acts transitively on both the vertices and the faces, and semi-regular if the group does transitively on the vertices but not the faces. The regular polytopes are exactly five Platonic solids. The dual of the semi-regular polytopes consist of an infinite series of triangle-faced polytopes (i.e., n-gonal bipyramids. Namely the dual of n-gonal prisms. (n=3,5,6,7,8,...)), an infinite series of quadrangle-faced polytopes (i.e., n-gonal trapezohedrons. Namely, the dual of n-gonal antiprisms. (n=4,5,6,7,8,...)), 7 triangle-faced polytopes, 4 quadrangle-faced polytopes, and 2 pentagon-faced polytopes. The last thirteen semi-regular polytopes are called Archimedean duals. The vertices of some (semi-)regular polytope are not on a single sphere. The skeleton of a spherical tile-transitive (i.e., isohedral) tiling is exactly that of a regular polytope or the dual of a semi-regular polytope, as we see below:
THEOREM([Grünbaum-Shephard81]). For any normal spherical monohedral tiling,
if the tiling is isohedral, then the tiling is topologically a Platonic solid,
an Archimedean dual, an n-gonal trapezohedron,
or an n-gonal bipyramid (n>2).
Here a skeleton is just a graph, which by no means carries information on the edge-lengths and the inner angles. For spherical tilings by congruent polygons, fact the skeleton is that of the duals of a semi-regular polytope does not necessarily imply the isohedrality. Indeed, although a 6-gonal antiprism P is a semi-regular polytope, there exists a spherical tiling A by 12 congruent concave quadrangles such that the skeleton S(A) of A is that of the dual of P, but the tiling A is not isohedral [Akama13].
On the other hand, the tile's convexity is required for a tiling having such a graph to be isohedral, because there is a spherical monohedral tiling A such that A is topologically a trapezohedron of 12 faces (see the graph below), the tile of A is concave (see the image. the tile has three equilateral silver edges and a distinguished golden edge), but A is not isohedral, according to [Akama13]. To see that the tiling A is not isohedral, note that A has only three perpendicular 2-fold rotational symmetries but no mirror planes. The figures from left to right are view from generic viewpoint, view from the antipodal viewpoint, view from a perpendicular 2-fold rotational symmetry axis, view from the north pole, and view from another perpendicular 2-fold rotational symmetry axis.
Interesting enough, the twelve tiles of the non-isohedral tiling A self-organize A as well as another spherical isohedral tiling (see below) and two tilings have the same graph. This may lead to the self-assembly of supra molecules on the sphere, which is described next.
From viewpoint of classification of spherical monohedral tiling, it is natural to consider a partial inverse of THEOREM: whenever a (semi-)regular polytope other than an atiprism deforms to a spherical tiling T by congruent polygons, T is isohedral. To be more precise,
CONJECTURE. For every spherical tiling T by congruent polygons, for every dual P of a semi-regular polytope, if the skeleton of P is that of T , then T is isohedral or the skeleton of T is that of an n-gonal antiprism for some n. Actually, if the tile of the spherical tiling T is convex, then T is always isohedral.
The conjecture is true for the bipyramids, the seven 3-gon-faced Archimedean duals, and the five Platonic solids, according to [Akama-Yan14] and [Akama-Yan16]. The conjecture is true for the four 4-gon-faced Archimedean duals [Akama16]. In [Akama-Yan16], the combinatorial type of the spherical monohedral polygonal tiling T with the regular dodecahedron skeleton is determined and how T deforms. The degree of freedom of T is two. See the movie and the movie.
Recently in supramolecular chemistry, self-assembly of shell (equicentric spherical) structures by not only biochemical molecules but also metallic materials are found [MacGillivray12]. A complete catalog of spherical tilings by congruent quadrangles is hopefully useful to analyze the spherical structure of materials newly found or yet to be found. The classification of spherical monohedral polygonal tilings shows the possible forms of molecular sphere[Chand et al.02] of synthetic chemistry. The constructions of spherical monohedral polygonal tilings, some of which are given in [Ueno-Agaoka02] and [Sakano-Akama15], suggest finite, spherical coordination networks which self-organize from small components [Tominaga et al.04] of synthetic chemistry. In particular, each spherical monohedral triangular tiling TI16n+8 and the spherical monohedral kite-faced tiling gDIT admit gliding a hemisphere in a fixed angle. On the other hand, some spherical monohedral triangular tilings and the spherical monohedral quadrangular tilings TRPZ2nα admit continuous deformation. We hope that these discontinuous deformations and continuous deformations are related to the stability of such molecular spheres.