Geoffrey C. Shephard’s research while affiliated with University of East Anglia and other places

... A tiling or tessellation is a set of regular polygons which are isometric and cover the entire manifold without overlappings or voids. In the Euclidean case there exist exactly three isometric tilings, which are the well-known square, triangular and honeycomb lattices [93]. The corresponding characteristic lattice spacing h can be scaled freely. ...

Reference:

HYPERTILING — a high performance Python library for the generation and visualization of hyperbolic lattices

... us is the scheme developed by them [56] that allows us to generate an isonemal weave pattern with two simple operations: (1) create a random binary matrix (M ) called the fundamental block and (2) repeat/tile the fundamental block in the Euclidean plane. In this construction, M can be visualized as a square grid with a cell being either black (0) or white (1). ...

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ABC‐Auxetics: An Implicit Design Approach for Negative Poisson's Ratio Materials

... A configuration may have more than one self-duality maps (in our case we have altogether 6, as we have seen above), thus the following definition makes sense. The rank r(C) of a self-dual configuration C is the minimum value of r(δ) over all self-duality maps δ of C. (We note that this notion was introduced by Grünbaum and Shephard [18] in case of geometric objects for which self-duality can be defined, in particular, for polyhedra and configurations; for further details related to configurations, see [16] and the references therein). ...

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On a New $(21_4)$ Polycyclic Configuration

... There exists extensive literature on fabric construction and mathematics. The classic work of Grunbaum and Shephard [4][5][6][7] pioneered research into which periodic weaves are isonemal. Fabrics that are not isonemal are not really fabrics at all, as a set of warp and weft threads (that are themselves mutually interwoven) can lift off from the rest. ...

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A 6-Letter ‘DNA’ for Baskets with Handles

... An interesting description of homogeneous polyhedra can be found in [14,15] (more general results for homogeneous polytopes in multidimensional Euclidean spaces can be found also in [16]). Another description of homogeneous polyhedra follows from the classification of all tilings on the 2-sphere with natural transitivity properties [8]. Here we collect some useful results on the structure of homogeneous polyhedra in order to study more narrow classes. ...

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On m-point homogeneous polyhedra in 3-dimensional Euclidean space

... (see Danzer, Grünbaum & Shephard [10], and [37])). Here the relaxation of the congruence requirement for the tiles creates many new possibilities for their metrical shapes. ...

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Combinatorial Space Tiling

... In the two adjoining chapters he also describes several with more than one kind of regular polygon as faces. In the 1979 paper by Grünbaum and Shephard [19], the beginnings of the application of incidence symbols to infinite collections of polygons in periodic surfaces in 3-dimensional the electronic journal of combinatorics 16 (2009), #R22 space were described. This was one of the ingredients that led to the enumeration approaches of the present paper. ...

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The {4, 5} Isogonal Sponges on the Cubic Lattice.

... Figure 6 shows two patterns with overlapping motifs that cannot be distinguished by the base types in [4]. An extension to the catalogue of 1-sided periodic pattern types that allows patterns with intersecting or disconnected motifs can be found in [3]; the base types in Table 3 that contain a hyphen refer to this list. ...

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The henomeric types of 2-sided patterns in the plane

... Thus, E must belong to either 2-uniform tilings of the plane or the 1-uniform tilings of the plane. We know from [10,11,16] that there are twenty 2-uniform tilings and 7 Archimedean tilings which are not always vertex transitive on torus. Here are these tilings. ...

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2-uniform toroidal maps, classification and asymptotic behavior

... The mentioned result was developed in various directions. In the case of toroidal graphs the weight is at most 15, see Grünbaum and Shephard [6]. Zaks [16] and Ivančo [9] analysed the weight of graphs embedded in orientable surfaces of a positive genus, while Jendrol' and Tuhársky [12] were interested by weights of graphs embedded in non-orientable surfaces. ...

Reference:

On the Maximum Weight of a Sparse Connected Graph of Given Order and Size