AI 中文总结
本文研究有理四边形与有理四面体上的有限群作用，利用Heegner与Regge对合从已知解构造丢番图方程新解，并揭示其与Kummer曲面理论的联系。
AI 中文摘要
本文致力于研究有理四边形和有理四面体的相关问题。在这两种情形下，均存在一个有限群作用于相应的集合。对于有理四边形的情形，该有限群由Kurt Heegner发现；在第二种情形下，存在两种变体，其一同样归功于Heegner，另一种则是Tullio Regge所引入的对合的直接推论。这些对合使得人们能够从已知解出发，构造相应丢番图方程的若干新解。最初的问题由Kummer提出。本文与四面体曲面（tetrahedroid）理论及Kummer曲面理论有着密切的联系。
英文摘要
This paper is devoted to questions concerning rational quadrilaterals and rational tetrahedra. In both cases there is a finite group which acts on the set. In the case of rational quadrilaterals it was discovered by Kurt Heegner; in the second case there are two variants, one again due to Heegner the other a consequence of an involution introduced by Tullio Regge. These allow one to construct a number of solutions to the appropriate diophantine equation from given ones. The original question was posed by Kummer. There is an intimate connection with the theory of the tetrahedroid and so Kummer surfaces.
