ThatsMaths

Sometimes the “obvious” answer to a mathematical problem is not the correct one. The case of Malfatti’s circles is an example of this. In an equilateral triangle of unit side length, we must draw three non-overlapping circles such that the total area of the circles is maximal.

The solution seems obvious: draw three identical circles, each one tangent to two sides and to the other two circles (above figure, left). This is certainly the most symmetric arrangement possible. However, it turns out not to be the optimal solution. There is another arrangement (above figure, right) for which the three circles have greater total area.

Giovanni Francesco Malfatti (1731-1807)

Giovanni Francesco Malfatti, an Italian mathematician, was born in Ala, Trentino, Italy and died in Ferrara. After studying in Bologna, Malfatti moved to Ferrara in 1754 and became a professor at the University of Ferrara in 1771.

Malfatti studied the problem of carving three circular columns out of a triangular prism of marble. The aim was to use as much of the marble as possible. He conjectured that three columns whose cross-sections were mutually tangent circles inscribed within the triangle would provide the optimal solution. The proposed arangement is usually known as Malfatti’s circles although the problem had been studied earlier, in particular by the Japanese mathematician Ajima Naonobu.

The symmetric solution proposed by Malfatti in 1803 was, for more than a century, believed to be optimal. In fact, Malfatti’s solution is never the optimal solution, irrespective of the shape and size of the triangle!

Equilateral Triangle

For an equilateral triangle, it is not too hard to compute the total area for Malfatti’s solution. The area of the triangle is A = √3 / 4. The radius of the circles is r = 1 / 2 ( √3 + 1 ). Thus, the ratio of the area of the three circles to that of the triangle is

However, this is not the optimal solution. We can cover a greater area by drawing the largest possible circle (it is tangent to all three sides) and then adding smaller circles in two of the corners. This is called the “greedy algorithm”: at each stage, the largest possible circle is drawn. The area ratio can be calculated with simple if tedious geometry. The radius of the large circle is r = 1 / 2√3 and its area is π / 12. The radii of the two smaller circles is one third that of the large circle, so the total area is (11/108 ) π and the ratio of the areas of the three circles to the area of the triangle is

which is significantly more than for the three identical circles. This is quite surprising.

The Greedy Algorithm

The optimal solution can always be found by the greedy algorithm: find the largest circle in the triangle. Then find the largest circle within the three regions outside the first circle. Finally find the largest circle within the five regions that remain. This procedure was formulated in 1930, but it was not until 1994 that Zalgallar and Los’ proved that is always provides the optimal solution. Zalgallar and Los’ described the problem thus:

“Algebraically this is a question of choosing nine variables (six coordinates of centers and three radii) that will maximize the sum of the squares of the radii subject to twelve constraints in the form of inequalities. Nine of the inequalities are linear (conditions for placing the centers on a particular side from each of the sides of the triangle at a distance not less than the radius) and three are nonlinear (conditions insuring that the circles do not overlap).”

Elongated Triangle

For an equilateral triangle, Malfatti’s solution is close to optimal ( 729 / 739 * 100 = 98.6% ). However, for an elongated triangle, with a very small angle, the solution provided using the greedy algorithm is nearly twice the area of the Malfatti circles. In the limiting case where a vertex moves to infinity, two sides become parallel and the solution is clearly three identical circles with co-linear centres.

In recent years, the problem of constructing Malfatti’s circles and finding the optimal solution has been used as a test problem to validate computer algebra packages. The illustration above, from the website of Eryn M Stehr, shows the optimal solution in two cases. The solutions were obtained using the GeoGebra package. In neither case dos the optimal solution correspond to Malfatti’s circles.

Sources:

Website of Eryn M. Stehr at WordPress here.

Zalgaller, V.A. and Los’, G.A. (1994): The solution of Malfatti’s problem, J. Math. Sci., 72, 3163–3177, doi:10.1007/BF01249514