Hiroshima Mathematical Journal

Classification of spherical tilings by congruent quadrangles over pseudo-double wheels (I) -- a special tiling by congruent concave quadrangles

Yohji Akama

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Abstract

Every simple quadrangulation of the sphere is generated by a graph called a pseudo-double wheel with two local expansions (Brinkmann et al. "Generation of simple quadrangulations of the sphere.'' Discrete Math., Vol. 305, No. 1-3, pp. 33--54, 2005). So, toward a classification of the spherical tilings by congruent quadrangles, we propose to classify those with the tiles being convex and the graphs being pseudo-double wheels. In this paper, we verify that a certain series of assignments of edge-lengths to pseudo-double wheels does not admit a tiling by congruent convex quadrangles. Actually, we prove the series admits only one tiling by twelve congruent concave quadrangles such that the symmetry of the tiling has only three perpendicular 2-fold rotation axes, and the tiling seems to be new.

Article information

Source
Hiroshima Math. J., Volume 43, Number 3 (2013), 285-304.

Dates
First available in Project Euclid: 7 January 2014

Permanent link to this document
https://projecteuclid.org/euclid.hmj/1389102577

Digital Object Identifier
doi:10.32917/hmj/1389102577

Mathematical Reviews number (MathSciNet)
MR3161319

Zentralblatt MATH identifier
1295.52024

Subjects
Primary: 52C20: Tilings in $2$ dimensions [See also 05B45, 51M20]
Secondary: 05B45: Tessellation and tiling problems [See also 52C20, 52C22] 51M20: Polyhedra and polytopes; regular figures, division of spaces [See also 51F15] 05C10: Planar graphs; geometric and topological aspects of graph theory [See also 57M15, 57M25]

Keywords
Monohedral tiling spherical quadrangle pseudo-double wheel

Citation

Akama, Yohji. Classification of spherical tilings by congruent quadrangles over pseudo-double wheels (I) -- a special tiling by congruent concave quadrangles. Hiroshima Math. J. 43 (2013), no. 3, 285--304. doi:10.32917/hmj/1389102577. https://projecteuclid.org/euclid.hmj/1389102577


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References

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