Surfaces, Meshes, Geometric Structures — Presentations

International Workshop in Admont: July 6-9, 2009
Mikhail Skopenkov: Dissections and orientation

We say that two similar triangles in a plane are oriented oppositely if one of them includes angles α, β, γ clockwise, and another one counterclockwise (angles α, β, γ are supposed to be mutually unequal).

Theorem 1. Let Δ be a triangle in the plane with angles α, β, γ. Let Δ' be a triangle in the plane, equal to Δ but oriented oppositely. Suppose that the triangle Δ can be dissected into 2 polygons such that one can combine them to obtain the triangle Δ' moving the polygons in the plane without flipping. Then k α + l β + m γ = 0 for some integers k l, and m not vanishing simultaneously. This theorem answers a question of Boltyanskiy from 1956.

Example. If α/β=(n+1)/n for some positive integer n then a triangle with angles α, β, π-α-β can be dissected into 2 polygons as required in Theorem 1.

Theorem. Suppose that a triangle with angles α, β, and γ can be dissected into triangles similar to it but oriented oppositely. Then k α + l β + m γ = 0 for some integers k, l, and m not vanishing simultaneously.

The approach is based on a generalization of the Hadwiger-Glur invariant of scissors congruence.

[workshop home page] [program page]