Program
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

Abstract:

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.

 

摘要:
2002年,上野裕佳子和阿賀岡芳夫将球面的用全等三边形以边对边方式的拼图完全分类。我们试图做类似的五边形拼图的分类。

作为预备工作,我们首先建立一些关于“统计类”的结果,我们也引进一些关于球面几何的结果。然后我们以边长的组合将球面的五边形拼图分为三类:

(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.

第一类几乎完成,结果应该是正规多面体的五边细分或加倍细分。第二类必须采用和第一类完全不同的做法,结果是一共有八个拼图。第三类最难,我们发现了好几类拼图,但离完全分类还有相当距离。

此工作是与日本东北大学的赤间阳二,香港科技大学的陆海平、王二小的合作结果。