The geometric reason for the non-existence of a MOL(6)

The problem of Euler concerning the 36 officers, (Euler, in Leonardi Euleri Opera Ser I 7:291–392, 1782), was first solved by Tarry (Comptes rendus Ass Franc Sci Nat 1 (1900) 2:170–203, 1901). Short proofs for the non-existence were given in Betten (Unterricht 36:449–453, 1983), Beth et al. (Design Theory, Bibl. Inst. Mannheim, Wien, Zürich, 1985), Stinson (J Comb Theory A 36:373–376, 1984). This problem is equivalent to the existence of a MOL(6), i. e., a pair of mutually orthogonal latin squares of order 6. Therefore in Betten (Mitt Math Ges Hamburg 39:59–76, 2019; Res Math 76:9, 2021; Algebra Geom 62:815–821, 2021) the structure of a (hypothetical) MOL(6) was studied. Now we combine the old proofs and the new studies and filter out a simple way for the proof of non-existence. The aim is not only to give still other short proofs, but to analyse the problem and reveal the geometric reason for the non-existence of a MOL(6)- and the non-solvability of Euler’s problem.

Emilio Camps Cazorla and the Search of a Geometric Ratio for Islamic Architecture

Islamic architecture has always been surrounded by an exotic aura which usually shows an impression of fanciful and exuberant forms as those romantic images suggested in stories like “The Arabian Nights” or “Tales of the Alhambra”. In architecture, this vision lead to studies more concerned on ornamentation or historical chronology of the buildings than its own proper architecture. Facing this view, there is a number of studies attempting to find a geometric reason for that architecture, which could not only analyse the ornament individually but also look for a formula able to explain how spatial, ornamental and constructive compositions remain constant along time. Emilio Camps Cazorla was one of the first theorists in searching that geometrical ratio which he called “Caliphal module”. Previously some authors had investigated the same subject and many others did after him, with mixed results. This paper documents all initiatives taken in order to determine this geometric ratio, their common characteristics as well as aspects to consider for future studies.

Emilio Camps Cazorla and the Search of a Geometric Ratio for Islamic Architecture

Islamic architecture has always been surrounded by an exotic aura which usually shows an impression of fanciful and exuberant forms as those romantic images suggested in stories like “The Arabian Nights” or “Tales of the Alhambra”. In architecture, this vision lead to studies more concerned on ornamentation or historical chronology of the buildings than its own proper architecture. Facing this view, there is a number of studies attempting to find a geometric reason for that architecture, which could not only analyse the ornament individually but also look for a formula able to explain how spatial, ornamental and constructive compositions remain constant along time. Emilio Camps Cazorla was one of the first theorists in searching that geometrical ratio which he called “Caliphal module”. Previously some authors had investigated the same subject and many others did after him, with mixed results. This paper documents all initiatives taken in order to determine this geometric ratio, their common characteristics as well as aspects to consider for future studies.