Heronian单形的格嵌入
Lattice Embedding of Heronian Simplices
摘要：
一个有理三角形有有理边长和面积;一个有理四面体有有理面和体积;当它的边长是整数时，它是Heronian的，当它的内容是非零时，它是真的。利用复数GCD给出了已知结果的一个变形证明，即任何Heronian三角形都可以嵌入到笛卡尔格Z ^2中，然后证明了对于一个真三角形，这种嵌入是唯一的模格等距;最后，通过四元数GCD将该方法扩展到Z^3中的四面体，其中不再获得唯一性，并且还存在通过这种构造无法获得的嵌入。必要的复杂和四元数数论背景是事先总结。随后的章节涉及附属的实现问题：初始的合理嵌入，规范化约简，对通过GCD产生的嵌入的穷举搜索;并提供了说明性的数值例子。一个反例表明，这种方法在高维空间中一定会失败。最后，其他作者的替代方法进行了总结。
英文摘要：
A rational triangle has rational edge-lengths and area; a rational tetrahedron has rational faces and volume; either is Heronian when its edge-lengths are integer, and proper when its content is nonzero. A variant proof is given, via complex number GCD, of the previously known result that any Heronian triangle may be embedded in the Cartesian lattice Z^2; it is then shown that, for a proper triangle, such an embedding is unique modulo lattice isometry; finally the method is extended via quaternion GCD to tetrahedra in Z^3, where uniqueness no longer obtains, and embeddings also exist which are unobtainable by this construction. The requisite complex and quaternionic number theoretic background is summarised beforehand. Subsequent sections engage with subsidiary implementation issues: initial rational embedding, canonical reduction, exhaustive search for embeddings additional to those yielded via GCD; and illustrative numerical examples are provided. A counter-example shows that this approach must fail in higher dimensional space. Finally alternative approaches by other authors are summarised.
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