MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

66 Mathematical Properties
Figure 2.55: Malfatti’s Problem [231]
In 1826, J. Steiner published, without proof, a purely geometrical construction
while, in 1853, K. H. Schellbach published an elementary analytical
solution [84]. Then, in 1929, H. Lob and H. W. Richmond showed that, for
the equilateral triangle, the Malfatti circles did not solve the marble problem.
≈ 0.739 of the
The correct solution, shown on the left of Figure 2.55, fills 11π
27 √ 3
triangle area while the Malfatti circles occupy only π √ 3
(1+ √ 3) 2 ≈ 0.729 of that area
[2]. Whereas there is only this tiny 1% discrepancy for the equilateral triangle,
in 1965, H. Eves pointed out that if the triangle is long and thin then the
discrepancy can approach 2:1 [231]. In 1967, M. Goldberg demonstrated that
the Malfatti circles never provide the solution to the marble problem [322].
Finally, in 1992, V. A. Zalgaller and G. A. Los gave a complete solution to the
marble problem.
(a)
(b)
Figure 2.56: Group of Symmetries [108]
Property 71 (Group of Symmetries). The equilateral triangle and the regular
tiling of the plane which it generates, {3}, possess three lines of reflectional
symmetry and three degrees of rotational symmetry as can be seen in Figure
2.56 [327].
(c)