1,090 research outputs found
Pentagons and rhombuses that can form rotationally symmetric tilings
In this study, various rotationally symmetric tilings that can be formed
using pentagons that are related to rhombus are discussed. The pentagons can be
convex or concave and can be degenerated into a trapezoid. If the pentagons are
convex, they belong to the Type 2 family. Since the properties of pentagons
correspond to those of rhombuses, the study also explains the correspondence
between pentagons and various rhombic tilings.Comment: 50 pages, 42 figure
Construction of the discrete hull for the combinatorics of a regular pentagonal tiling of the plane
The article 'A "regular" pentagonal tiling of the plane' by P. L. Bowers and
K. Stephenson defines a conformal pentagonal tiling. This is a tiling of the
plane with remarkable combinatorial and geometric properties. However, it
doesn't have finite local complexity in any usual sense, and therefore we
cannot study it with the usual tiling theory. The appeal of the tiling is that
all the tiles are conformally regular pentagons. But conformal maps are not
allowable under finite local complexity. On the other hand, the tiling can be
described completely by its combinatorial data, which rather automatically has
finite local complexity. In this paper we give a construction of the discrete
hull just from the combinatorial data. The main result of this paper is that
the discrete hull is a Cantor space
Spherical Tiling by 12 Congruent Pentagons
The tilings of the 2-dimensional sphere by congruent triangles have been
extensively studied, and the edge-to-edge tilings have been completely
classified. However, not much is known about the tilings by other congruent
polygons. In this paper, we classify the simplest case, which is the
edge-to-edge tilings of the 2-dimensional sphere by 12 congruent pentagons. We
find one major class allowing two independent continuous parameters and four
classes of isolated examples. The classification is done by first separately
classifying the combinatorial, edge length, and angle aspects, and then
combining the respective classifications together.Comment: 53 pages, 40 figures, spherical geometr
Rotationally symmetric tilings with convex pentagons belonging to both the Type 1 and Type 7
Rotationally symmetric tilings by a convex pentagonal tile belonging to both
the Type 1 and Type 7 families are introduced. Among them are spiral tilings
with two- and four-fold rotational symmetry. Those rotationally symmetric
tilings are connected edge-to-edge and have no axis of reflection symmetry.Comment: 13 pages, 16 figures. arXiv admin note: text overlap with
arXiv:2005.08470, arXiv:2005.1270
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