Desargues定理的推广、逆、对称与Enumeration
The Extension of the Desargues Theorem, the Converse, Symmetry and Enumeration
摘要：
这篇文章部分地概括了A.A. Bruen，T.C Bruen和J.M.McQuillan在arXiv中讨论Desargues定理：2007.09175[mathCO] 2020年7月17日。我们用4个不同的论证证明了Desargues定理在所有维数上的推广，并给出了逆定理。证明了射影n-空间中的Desargues构形对应于包含n-空间的n+1维射影空间中的弧，其中n ≥ 2.因此，原则上，当基础场是有限的时，可以枚举n维的笛沙格组态的数目。在n维的Desargues配置进行了详细研究，并表现出新的自我复制或分形性质。在第11节中，我们推广了关于半单形的一个经典定理，其中n ≥ 3。
英文摘要：
This work is, in part, a generalization of the article by A.A. Bruen ,T.C Bruen and J.M.McQuillan on Desargues Theorem in arXiv:2007.09175[mathCO]July 17,2020. We prove the extension of Desargues theorem in all dimensions, using 4 different arguments.We also show the converse theorem. It is shown that the Desargues configuration in projective n-space, for all n at least 2, corresponds to an arc in a projective space of dimension n+1 containing the n-space. Thus, in principle, one can enumerate the number of Desargues configurations in n dimensions when the underlying field is finite. The Desargues configuration in n-dimensions is studied in detail and is shown to exhibit new self-replication or fractal-like properties. In section 11 we extend a classical theorem on semi-simplexes for all n at least 3.
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