Hugo Hadwiger’s influence on geometric dissections with special properties

On the occasion of the Swiss Mathematical Society’s centennial, it is both an honor and a pleasure to contribute an article to this special edition of the Elemente der Mathematik on a mathematical subject from the work of Hugo Hadwiger. Professor Hadwiger was a prolific and influential Swiss mathematician known for his work in geometry, combinatorics, and cryptography [2]. Here we shall note how his ideas in dissection geometry have been extended into the present. The focus will proceed from his article, “Zerlegungsgleichheit ebener Polygone” [13], co-authored with Paul Glur, which appears to be the most-referenced of his articles on dissection. In the almost sixty years since this article appeared, several popular books on geometric dissections [4, 5, 6, 15] have been published, a superb website [17] has been created, and an early manuscript [12] has been recovered. A dissection possesses translation with no rotation ,o r istranslational, if the pieces can be moved from one figure to the other without rotating them. It is not always possible to find dissections between two figures that possess this property, no matter how the figures are oriented with respect to each other. Hadwiger and Glur gave a simple characterization of which figures can have a dissection that has this property. For each figure, walk around its boundary in a clockwise direction, and identify the slopes of each line segment in the boundary. Each line segment has a slope and a direction, either to the right or to the left, over which it is walked. (For vertical line segments, replace right and left by up and down.) For each figure, and for each slope of line segments in that figure, add the lengths of the line segments walked to the right and subtract from this sum the lengths of line segments walked to the left. There will be a translational dissection if and only if for every slope, the sum for one figure equals the sum for the other.