Tiling of Sphere by Congruent Pentagons
Student No.:50
Time:10:30-11:30, 2017-6-26
Instructor:Min Yan  [Hong Kong University of Science and Technology]
Place:Conference Room 1, floor 1, Jinchunyuan West Building
Starting Date:2017-6-26
Ending Date:2017-6-26


In 2002, Ueno and Agaoka completed the classification of edge-to-edge tilings of sphere by congruent triangles. We try to classify tilings by congruent pentagons.
First we have some “statistical results” regarding the distribution of degrees of vertices, special tiles, and angles. We also have constraints from spherical geometry. Then we divide into three cases according to the edge length combination of the pentagon:
(1) Variable edge length: a^2b^2c, a^3bc, a^3b^2.
(2) Equilateral: a^5.
(3) Almost equilateral: a^4b.
The first case is almost complete, where we expect the only tilings are what we call pentagonal subdivisions and double pentagonal subdivisions. The method for the second case is completely different from the first, because no edge length information can be used. We find there are exactly 8 tilings for this case. The third case is the most complicated. We find some interesting families of tilings, but the classification is far from being complete.
This is a joint work with Yohji Akama of Tohoku University, and Hoiping Luk and Erxiao Wang of the Hong Kong University of Science and Technology.




(1) 边长变化: a^2b^2c, a^3bc, a^3b^2.
(2) 等边: a^5.
(3) 几乎等边: a^4b.