PUMaC 2020 · 加试 · 第 4 题
PUMaC 2020 — Power Round — Problem 4
题目详情
Problem 4.1. Let ABCD be a regular tetrahedron, assume the face ABC is horizontal, and denote this plane by α . Prove that it is possible to roll the given tetrahedron on this plane such that the faces of the tetrahedron form a triangular tiling of the plane. Moreover, prove that it is possible to assign letters a, b, c, d to the vertices of this tiling such that the vertex A of the tetrahedron always lands on the vertex of the tiling marked by a , and similarly for the vertices B, C, D . Having constructed this useful rolling of the tetrahedron on the plane, we can start dealing with ant-paths. Assume the S ( ABCD ) contains an ant-path C . Roll the tetrahedron ABCD along this ant-path, until we come back where we started on the ant-path. In the plane, this ant-path will have the following form: its endpoints will be on the edges of the tiling marked by the same letters, and oriented the same way (e.g. both endpoints will be on the edges ab of the tiling, and a is left of b on both edges). Moreover, as long as such a plane segment does not contain any vertices of the tiling, it will be possible to uniquely bring it back on the surface of ABCD .
解析
Problem 4.5. Let ∆ be a tetrahedron with vertices ABCD . Prove that the following statements are equivalent: • ∆ is equihedral in the sense of Definition 4.A. • The perimeters of all faces ABC , ABD , ACD , and BCD are equal. • The pointiness of all vertices are equal, i.e. p ( A ) = p ( B ) = p ( C ) = p ( D ). • The dihedral angles at the opposite edges are equal. In other words, the angle between planes ABC and BCD is equal to the angle between the planes ACD and ABD (and similarly for other pairs of planes). • The solid angles at each vertex have the same measure. Proof 4.5. (Sketch) From Problem 4.4, there is a rigid motion that brings the equihedral tetrahedron to coordinates A = (0 , y, z ) , B = ( x, 0 , z ) , C = ( x, y, 0) , D = (0 , 0 , 0). From these coordinates, it is an easy to calculation to prove that ∆ equihedral implies the other bullets. • (2nd bullet implies first) Let x = AD, y = BD, z = CD, a = BC, b = AC, c = AB . Then x + z + b = x + y + c = y + z + a = a + b + c . This gives for example z + b = y + c , x + b = y + a, x + z = a + c . This gives z = c . This allows us to get y = b, a = x , proving that the tetrahedra is equihedral. • (third bullet implies first) From the previous formula on the sum of pointiness of tetrahdra, the pointiness of each vertex is π . Thus, if we unravel the tetrahedra into a net, we get a triangle where ABC is the medial triangle. This immediately implies that the triangles are congruent. • (fourth bullet implies first) UNFINISHED • In a tetrahedron, the solid angles are the sum of the adjacent dihedral angles minus π . Thus, this bullet implies the first bullet follows from the previous part. Equihedral tetrahedra turn out to be exceedingly interesting when discussing ant-paths. The following problems show us several examples of this correlation.