TY - JOUR

T1 - Classification of spherical tilings by congruent quadrangles over pseudo-double wheels (II)—the isohedral case

AU - Akama, Yohji

N1 - Funding Information:
The author was supported by JSPS KAKENHI Grant Number 16K05247. 2010 Mathematics Subject Classification. Primary 52C20; Secondary 05B45, 51M20. Key words and phrases. graph, skeleton, spherical monohedral tiling, spherical quadrangle, spherical trigonometry, symmetry, tile-transitive.

PY - 2019/3

Y1 - 2019/3

N2 - We classify all edge-to-edge spherical isohedral 4-gonal tilings such that the skeletons are pseudo-double wheels. For this, we characterize these spherical tilings by a quadratic equation for the cosine of an edge-length. By the classification, we see: There are indeed two non-congruent, edge-to-edge spherical isohedral 4-gonal tilings such that the skeletons are the same pseudo-double wheel and the cyclic list of the four inner angles of the tiles are the same. This contrasts with that every edge-to-edge spherical tiling by congruent 3-gons is determined by the skeleton and the inner angles of the skeleton. We show that for a particular spherical isohedral tiling over the pseudodouble wheel of twelve faces, the quadratic equation has a double solution and the copies of the tile also organize a spherical non-isohedral tiling over the same skeleton.

AB - We classify all edge-to-edge spherical isohedral 4-gonal tilings such that the skeletons are pseudo-double wheels. For this, we characterize these spherical tilings by a quadratic equation for the cosine of an edge-length. By the classification, we see: There are indeed two non-congruent, edge-to-edge spherical isohedral 4-gonal tilings such that the skeletons are the same pseudo-double wheel and the cyclic list of the four inner angles of the tiles are the same. This contrasts with that every edge-to-edge spherical tiling by congruent 3-gons is determined by the skeleton and the inner angles of the skeleton. We show that for a particular spherical isohedral tiling over the pseudodouble wheel of twelve faces, the quadratic equation has a double solution and the copies of the tile also organize a spherical non-isohedral tiling over the same skeleton.

KW - Graph

KW - Skeleton

KW - Spherical monohedral tiling

KW - Spherical quadrangle

KW - Spherical trigonometry

KW - Symmetry

KW - Tile-transitive

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U2 - 10.32917/hmj/1554516036

DO - 10.32917/hmj/1554516036

M3 - Article

AN - SCOPUS:85067230090

VL - 49

SP - 1

EP - 34

JO - Hiroshima Mathematical Journal

JF - Hiroshima Mathematical Journal

SN - 0018-2079

IS - 1

ER -