148 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
Goldberg, Fuller, Caspar, Klug and Coxeter and a general approach to local symmetry-preserving operations
Cubic polyhedra with icosahedral symmetry where all faces are pentagons or
hexagons have been studied in chemistry and biology as well as mathematics. In
chemistry one of these is buckminsterfullerene, a pure carbon cage with maximal
symmetry, whereas in biology they describe the structure of spherical viruses.
Parameterized operations to construct all such polyhedra were first described
by Goldberg in 1937 in a mathematical context and later by Caspar and Klug --
not knowing about Goldberg's work -- in 1962 in a biological context. In the
meantime Buckminster Fuller also used subdivided icosahedral structures for the
construction of his geodesic domes. In 1971 Coxeter published a survey article
that refers to these constructions. Subsequently, the literature often refers
to the Goldberg-Coxeter construction. This construction is actually that of
Caspar and Klug. Moreover, there are essential differences between this
(Caspar/Klug/Coxeter) approach and the approaches of Fuller and of Goldberg. We
will sketch the different approaches and generalize Goldberg's approach to a
systematic one encompassing all local symmetry-preserving operations on
polyhedra
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