Rational Elliptic Surfaces and the Trigonometry of Tetrahedra

Joseph Wolstenholme与四面体三角学

Joseph Wolstenholme and the Trigonometry of Tetrahedra

本文整理并介绍Joseph Wolstenholme晚年在四面体三角学方面留下的研究结果，并援引Richardson在1897年的记述说明其方程手稿的来源。

The ancient Greeks present: Rational Trigonometry

Pythagoras' theorem, the area of a triangle as one half the base times the height, and Heron's formula are amongst the most important and useful results of ancient Greek geometry. Here we look at all three in a new and improved light, using quadrance not distance. This leads to a simpler and more elegant trigonometry, in which angle is replaced by spread, and which extends to arbitrary fields and more general quadratic forms.

Space vectors forming rational angles

We classify all sets of nonzero vectors in $\mathbb{R}^3$ such that the angle formed by each pair is a rational multiple of $π$. The special case of four-element subsets lets us classify all tetrahedra whose dihedral angles are multiples of $π$, solving a 1976 problem of Conway and Jones: there are $2$ one-parameter families and $59$ sporadic tetrahedra, all but three of which are related to either the icosidodecahedron or the $B_3$ root lattice. The proof requires the solution in roots of unity of a $W(D_6)$-symmetric polynomial equation with $105$ monomials (the previous record was $12$ mono…

由Desmic四面体产生的六次K3曲面

K3 surfaces of degree six arising from desmic tetrahedra

本文研究由Desmic四面体产生的六次K3曲面，探讨其双有理模型、Picard格、射影自同构群及低次有理曲线。