The Problem of Malfatti: Two Centuries of Debate

Abstract

G ianfrancesco Malfatti (Figure 1) was a brilliant Italian mathematician born in 1731 in a small village in the Italian Alps, Ala, near Trento. He first studied at a Jesuit school in Verona, then at the University of Bologna. Malfatti was one of the founders of the Department of Mathematics of the University of Ferrara. He died in Ferrara in 1807. As a very active intellectual in the Age of Enlightenment, he devoted himself to the promotion of many new ideas and wrote many papers in different fields of mathematics including algebra, calculus, geometry, and probability theory. He played an important role in the creation of the Nuova Enciclopedia Italiana (1779), in the spirit of the French Encyclopedie edited by Diderot and d’Alembert. His mathematical papers were collected by the Italian Mathematical Society in the volume [7]. His historical figure has been discussed in a series of papers in [1]. This paper was inspired by a conference in 2007 commemorating the 200th anniversary of Malfatti’s death, organized by the municipality of Ala and the mathematics departments of Ferrara and Trento. Malfatti appears in the mathematical literature of the last two centuries mostly in connection with a problem he raised and discussed in a paper in 1803 (Figure 2): how to pack three non-overlapping circles of maximum total area in a given triangle? Malfatti assumed that the solution consisted of three mutually tangent circles, each also tangent to two edges of the triangle (now called Malfatti’s arrangement) and in his paper he constructed such arrangements (for a historical overview see [3]). In 1994 Zalgaller and Los [11] disproved Malfatti’s original assumption and showed that the greedy arrangement is always the best one. The detailed story of this 200-year-old problem is worth telling because it has many paradigms typical of research in mathematics, including the way one formulates a problem, how one interprets it or solves it, and what one should consider trivial and what one should not. In the following section we give the history of the problem. The section after that contains a new non-analytic solution for the problem of maximizing the total area of two disjoint circles contained in a given triangle. In the last section we generalize the two-circle problem for certain regions other than triangles. Our non-analytic approach shows that in various situations the greedy arrangements are the best ones.