Stairs: The Only Way to Go

J Shilleto ; James T Smith

Antiquitates Mathematicae, Tome 17 (2023), pp. 5-27.

Résumé

In plane Euclidean geometry, any polygonal regions Q, Qʹ with equal area can be dissected into the same number n of pairwise congruent subregions. The smallest possible n is called their degree of equivalence. Around 1930, the renowned logician Alfred Tarski, teaching elementary geometry in Warsaw full-time and leading research on logic at its University part-time, noticed a dissection for a square Q and a nonsquare rectangle Qʹ with n = 3. He reported it in the pioneering journal Parametr for Polish secondary-school mathematics teachers and students, then asked, are there such dissections with n = 2? Henryk Moese, a schoolteacher, responded that m-step staircase dissections are possible when Qʹ has dimensions (m+1)/m by m/(m+1). Moese conjectured that the only such dissections of Q,Qʹ with n = 2 are those. Tarski reported that this had been confirmed, but the proof was too complex to publish there. Decades later, he presented this material in the same way to general audiences that included the present authors. We regarded Moese’s conjecture as a challenge, devised a way to verify it, and present that argument here. Using only elementary geometry, it suggests what methods Tarski might have expected for solving some other problems that he posed in the same journal. It is unlike any arguments that we have found in related literature, but seems to anticipate some current techniques in computational geometry. Information is included about remarkable Polish efforts in the 1930s to improve secondary-school mathematics instruction, and about the role of Tarski’s Warsaw research seminar.

Détail

DOI : 10.14708/am.v17i1.7160

Classification : 52B45, 97-03, 97G30
Mots-clés : Tarski, Polygonal Dissection, Geometry Instruction