103 research outputs found
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
Tilings of the Sphere by Edge Congruent Pentagons
We study edge-to-edge tilings of the sphere by edge congruent pentagons,
under the assumption that there are tiles with all vertices having degree 3. We
develop the technique of neighborhood tilings and apply the technique to
completely classify edge congruent earth map tilings.Comment: 36 pages, 34 figure
Angle Combinations in Spherical Tilings by Congruent Pentagons
We develop a systematic method for computing the angle combinations in
spherical tilings by angle congruent pentagons, and study whether such
combinations can be realized by actual angle or geometrically congruent
tilings. We get major families of angle or geometrically congruent tilings
related to the platonic solids.Comment: 58 pages, 5 figure
Spherical tilings by congruent quadrangles : forbidden cases and substructures
In this article we show the non-existence of a class of spherical tilings by congruent quadrangles. We also prove several forbidden substructures for spherical tilings by congruent quadrangles. These are results that will help to complete of the classification of spherical tilings by congruent quadrangles
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