Demistifying Malfatti's marble problem.
This article provides more insight in my recent work:
Giancarlo Lombardi, Proving the solution of Malfatti’s marble problem,
Rendiconti del Circolo Matematico di Palermo Series 2
DOI: 10.1007/s12215–022–00759–2
This work deals with the Problem of Malfatti, posed in 1803 by Gianfrancesco Malfatti:
https://en.m.wikipedia.org/wiki/Gian_Francesco_Malfatti
and corresponding to find within a triangle three not overlapping circles of total maximal area:
https://en.m.wikipedia.org/wiki/Malfatti_circles
Indeed, I had an opportunity of exchange with some scholar, believing that Zalgaller and Los 1994 (https://link.springer.com/article/10.1007/BF01249514) provide a proof of the solution. But this is not the case, as I will show here in more detail than in my published work, wherein the careful reader should already understand that. Therefore, the merit of the proof after 220 years is in the paper above.
The gaps in the solution
Since the start of their work, Zalgaller and Los declare the existence of gaps in the original “proof” of Los in 1988 and declare that “the present paper enabled them to flesh out and fill in those gaps”. However, this is not the case, as it will be shown.
The first gap is in the definition of “rigid systems” and in the Lemma. The definition is qualitative and the lemma is unproven, as it is specified merely that “it is easy to indicate the direction in which when the circle is displaced its radius admits increase”.
The major and foremost gap is however in the “proof” related to Arrangement 9, the most complex one. Indeed, as the works of Andreatta and Ninjbat show, all arrangements, except Arrangements 1,2 (the greedy solution) and Arrangements 3,6,9, may be excluded with a geometric reasoning, made rigorous in my paper. The exclusion of Arrangement 6 (Malfatti circles) follows by the statement of an inequality for acute triangles by Zalgaller 1994 (https://link.springer.com/article/10.1007/BF01249513), proven with some minor omissions, related essentially to the properties of the curves delimiting the domain of partial derivation, in a separate paper. Moreover, other omissions regard the properties of the polynomials involved in the inequalities, which are positive as concave or monotone functions: Zalgaller does not indicate it.
The essential and complex “proof” for the exclusion of Arrangement 9 is not however a proof for the following reasons.
1) It is stated that R3 < R1, as otherwise “we could enlarge K3 by displacing it along AC”. This is not correct and a proof, which is necessarily more complex than this statement, is instead in my paper.
2) The proof of the expression for R3 is wrong, as it is unclear why one has to choose the “–” sign and not the “+” sign among the two solutions. The proof in my paper is on the contrary correct, as showing that one solution is negative.
3) On page 3171 the entire reasoning is very qualitative.
4) On page 3172 it is stated that “f(t) = α1’’(t) – α0(t) […] strictly decreases on 1 ≤ t ≤ √2 vanishing when t1 ≈ 1.299896”. This important statement is not proven and the presence of the tables makes to appear that it is based merely on plotting f(t). Indeed the two functions, which are subtracted from each other, are both decreasing, so that it is not obvious that their difference remains decreasing.
5) In point 11, page 3173, it is stated “Table 1 shows the nature of α0, γ0, β0’, α0 + β0’ + γ0, in particular, their monotonicity”. This fundamental statement is based on plotting and is unproven.
6) Last, but foremost, the entire reasoning in points 12–18, on pages 3173–3175 is based on a tabular verification of an inequality by sample t points. Indeed, we have statements as “We next verify V2 > 0 for various t” in section 15 and “it is enough to verify by direct calculation that V5 >0, V6 > 0” in section 17. Sections 17 and 18 are explicit about the use of numerical plotting, by taking “values from Table 2” and by statements, as “the results of the computations are given in Table 3”, “the results of the computations are given in Table 4”.
The manner used in providing the proof mentioned in point 2) makes clear that the conclusions in points 4) to 6) are really reached by plotting and by numerical simulation (as in Goldberg 1967): indeed such conclusions may not be trivially reached and require a complexity not smaller than the one involved in point 2). Therefore, they are unproven according to mathematical standards. Indeed, the use of computational aid is admitted for proving in the following cases:
a) Performing logical calculus, as in the Four Color Problem (https://en.m.wikipedia.org/wiki/Four_color_theorem), in order to speed up a rigorous, but repetitive reasoning by algorithmic verification;
b) To evaluate bounds, which inherently require a finite precision, as in the first Hales’ proof of the Kepler Conjecture (https://en.m.wikipedia.org/wiki/Kepler_conjecture), which is also a packing problem.
In the case of the computations in Zalgaller and Los none of cases a) and b) is satisfied. It follows that the exclusion of Arrangement 9 is verified by pure numerical computation (in the sense of Goldberg 1967), and is thus unproven.
Finally, the exclusion of Arrangement 3 appears also to be not correctly shown, as the statement “Diminishing α, radius R3 decreases while radius R4 increases. Therefore, for α ≥ 0.462, the inequality R4 > R3 is preserved.” is also unproven. The entire proof appears quite qualitative, on top of that.
What my paper does: a real proof
My paper merges the exclusion of Arrangements 3,9 together. Moreover, it is proven that for t ≤ t1, α1’’(t) ≤ α0(t) and the monotone “nature of α0, γ0,
β0’ ”. It shows also that α0 + β0’ + γ0 has a singular derivative, but still has a minimum zero, corresponding to t2, what is sufficient. Above all, it removes all the numerical computations in point 6) and replaces them with a true proof based on the theory of convex functions, wherein only one numerical lower bound on γ is imposed (thus as in case b) above).
Moreover, it provides a sound analytic definition of rigid systems, proves the lemma and shows the conditions for the existence of said systems. Finally, it makes more rigorous the proofs of Zalgaller/Los, Andreatta and Ninjbat about the list of arrangements, using topological considerations.
Therefore, my paper finally provides a proof and concludes after 220 years the solution of Malfatti’s marble problem.