Mathematical concepts in Arabic calligraphy: The proportions of the 'Alif

The starting point of every study on the proportions of Arabic calligraphy is the letter 'Alif. It is considered the reference for all other letters. Usually, it is measured in dots. This paper is an attempt to study the mathematical concepts upon which the historical theory of the 'Alif proportions was based, though not mathematically stated. In order to achieve this, the terms and components of the theory were clearly defined, and analyzed in their textual and visual context, historically, and logically according to our present time. In spite of the frequent use of these terms throughout time, their meanings were not always clear and can accept different interpretations. Some terms, even, indicated different meanings and were loosely used to satisfy different functions, for example, the term "Nuqṭah" (dot). Relating the components of the theory was, also, opened for opinions and interpretations, for example, the number of dots for the 'Alif was never agreed upon. This paper starts with historical research and analysis. Then, it presents the mathematical expressions of the 'Alif proportions both numerically and visually. Later, it discusses how they do apply, how other modern interpretations can fit, and how historical misunderstandings can be understood. Finally, it presents the historical account of how to relate other letters to the 'Alif recommending analyzing their mathematical aspects in future studies.