A congruence problem for polyhedra

TL;DR

This paper investigates the minimal measurements needed to determine convex polyhedra and polygons up to congruence, revealing that generally the number of edges dictates the count, but specific shapes require fewer measurements.

Contribution

It establishes that the number of measurements needed equals the number of edges for most polyhedra, and identifies cases where fewer measurements suffice, including special polygons.

Findings

Number of measurements equals the number of edges for most convex polyhedra.

Nine measurements suffice to determine a unit cube.

Certain quadrilaterals can be determined with only four measurements.

Abstract

It is well known that to determine a triangle up to congruence requires three measurements: three sides, two sides and the included angle, or one side and two angles. We consider various generalizations of this fact to two and three dimensions. In particular we consider the following question: given a convex polyhedron $P$, how many measurements are required to determine $P$ up to congruence? We show that in general the answer is that the number of measurements required is equal to the number of edges of the polyhedron. However, for many polyhedra fewer measurements suffice; in the case of the unit cube we show that nine carefully chosen measurements are enough. We also prove a number of analogous results for planar polygons. In particular we describe a variety of quadrilaterals, including all rhombi and all rectangles, that can be determined up to congruence with only four measurements, and we prove the existence of $n$-gons requiring only $n$ measurements. Finally, we show that one cannot do better: for any sequence of $n$ distinct points in the plane one needs at least $n$ measurements to determine it up to congruence.