@article{oai:miyazaki-u.repo.nii.ac.jp:00000199,
 author = {Uda, Hirohumi and 宇田, 廣文 and Uda, Hirofumi},
 journal = {数学教育学研究紀要},
 month = {Mar},
 note = {In this paper we study the following
i)the structure of polygons by making use of congruent transformations,
ii)the concrete constructions of polygons with a given structure of congruent transformations,
iii)an investigation of the problem presented by T.W.Shilgalis.
Let P be a polygon and G(P) be the set of all congruent transformations of P. Then G(P) is a subgroup of a suitable dihedral group D2 n, where D2n is the dihedral group of a regular polygon of n sides. Therefore, we began with a decision on subgroups of the dihedral group D2n.
The outline of conclutions is as follows.
(1)We determined the subgroups of the dihedral group D2n and showed the existence of polygons with a given subgroup of the dihedral group D2n as a group of congruent transformations. Moreover, we gave the concrete constructions of polygons with a given group-Structure of congruent transformations.
(2)We classified polygons by using groups of congruent transformations and determined the fundamental polygons in a viewpoint of congruent transformations. Moreover, we considered the number of symmetry lines and rotations in polygons, and made these properties clear.
(3)T.W.Shilgalis denonted by f(n) the maximum number of symmetry lines in an irregular polygon of n sides and suggested the following:
f(n)=the largest divisor of n (except n itself).
This fact can be given by considerations in (2). This implies that a concept of groups is useful for investigations of polyogns.
(4)Concerning a polygon-in particular a dodecagon-having two given symmetry lines, we examined its properties concretely and determined its group-theoretical structure.},
 pages = {177--192},
 title = {合同変換による多角形の考察},
 volume = {19},
 year = {1993},
 yomi = {ウダ, ヒロフミ}
}