[1] The idea of footprint- (and mind-print-) literacy was introduced by Tsion Avital; see his papers in the present issue of VM. Different linear symmetry groups (frieze groups) "printed" by animals were briefly discussed by Wolf and Wolff (1956). I should recall here the Japanese director A. Kurosawa's movie Dersu Uzala. This title is the name of an old man living in the Siberian forest. He has an extraordinary ability to "read" footprints. For example, he tells the visitors that the actual footprints are belonging to an old tiger and they should be very careful because the tiger is very hungry, etc. I am sure that this ability of Dersu Uzala was a common knowledge in the hunting and gathering society.

[2] Ratios of length of a vibrating string with the modern names (and notations) of intervals

1/1 - prime or unison (from C to C)

8/9 - second (from C to D)

4/5 - third (from C to E)

3/4 - forth (from C to F)

2/3 - fifth (from C to G)

3/5 - sixth (from C to A)

8/15 - seventh (from C to B)

1/2 - octave (from C to the next C)

We may consider semitones and in that case we should make a clear distinction between the new minor and the original major intervals, e.g.

5/6 - minor third (from C to Eb)

4/5 - major third (from C to E)

and

5/8 - minor sixth (from C to Ab)

3/5 - major sixth (from C to A)

In the same time, we may speak about "perfect" unison, forth, fifth, sevenths, and octave.

Notes:

- The given here system is called in various names, one of them is "just tuning"

- The frequency of a vibrating string is very nearly inversely proportional to its length. Thus, if we would like to give the ratios of frequencies, as many works on acoustics do, we should take the inverse (reciprocal) ratios (e.g., 2/1 is the octave).

[3] The possibility of some link between Polykleitos (or Polyclitus) and the Pythagoreans was suggested already by Diels in 1889. Raven (1951) presented his arguments that probably there was a Pythagorean source where the Canon of Polyclitus was summarized and this work was used by both Vitruvius and Galen. Pollitt (1974, p. 18-21), continuing Raven's work, went even further and suggested that this link was mutual. Specifically, Polykleitos was influenced by, and perhaps contributed to, the Pythagorean doctrine of number and symmetria (commensurability). The latter view has got a further support by Stewart (1978, pp. 127 and 131).

Raven, J. E. (1951) Polyclitus and Pythagoreanism, Classical Quarterly, 45 [= New Series, Vol. 1], 147-152.

Pollitt, J. J. (1974) The Ancient View of Greek Art: Criticism, History, and Terminology, New Haven, Connecticut: Yale University Press. [See the chapter "Polyclitus's Canon and the idea of symmetria", pp. 14-22].

Stewart, A. F. (1978) The canon of Polykleitos: A question of evidence, Journal of Hellenic Studies, 98 122-131.

[4] Carmel Konikoff (1973) gave a deep analysis of the prohibition of image-making in the Second Commandment and its influence on the art of ancient Israel, including the passive tolerance and even some blessing towards figural representation in various cases. Although the book does not have illustrations, it refers to many works of art and presents an extensive bibliography (pp. 101-113):

Konikoff, C. (1973) The Second Commandment and its Interpretation in the Art of Ancient Israel, Genève: Imprimerie du Journal de Genève, 113 pp.

[5] Sholem, G. (1988) Kabbalah, Jerusalem: Keter, 492 pp. [See Part 2, Chap. 14 "Magen David", 362-368].

[6] Nagy, D. (1995) The 2500-year old term symmetry in science and art and its `missing link' between the antiquity and the modern age, Symmetry: Culture and Science, 6, No. 1, 18-28.

Curchin, L. and Fischler [Herz-Fischler], R. (1981) Hero of Alexandria's numerical treatment of division in extreme and mean ratio and its implications, Phoenix [The Journal of the Classical Association of Canada], 35, 129-133.

Kidson, P. (1990) A metrological investigation, Journal of the Warburg and Courtauld Institutes, 53, 71-97. [See the Theon of Smyrna's method on p. 77 and its possible extension to the golden number on p. 92].

[8] Zeising, A. (1854) Neue Lehre von den Proportionen des menschlichen Körpers, aus einem bisher unerkannt gebliebenen, die ganze Natur und Kunst durchdringenden morphologischen Grundgesetze entwickelt und mit einer vollständigen historischen Uebersicht der bisherigen Systeme begleitet, [A New Theory of the Proportions of the Human Body, Developed from a Hitherto Unrecognized Basic Law of Morphology Penetrating the Whole Nature and Art and Accompanied by a Complete Historical Survey of the Earlier Systems, in German], Leipzig: Weigel, 1854, xxii + 457 pp.

[9] There are various names for "rhythmic pattern": iqa'at in many Arabic countries, durub in Egypt, mazim in the Maghrib (North-Western Africa), usul in Turkey, and darb in Iran. This concept is discussed in many works on Islamic music, see, for example, the following brief survey:

Malm, W. P. (1967) Music Cultures of the Pacific, the Near East, and Asia, Englewood Cliffs, New Jersey: Prentice-Hall, 169 pp. [See rhythmic patterns on pp. 49-51, "finger modes" on p. 48.].

[10] Hogendijk, J. P. (1986) Discovery of an 11th-century geometrical compilation: The Istikmal of Yusuf al-Mu'taman ibn Hud, King of Saragossa, Historia Mathematica, 13, 43-52.

[11] Mathematical works on art-related geometrical questions:

- Abu Kamil's Kitab [...] al-mukhammas wa'l-mu`ashshar (Book on the Pentagon and the Decagon) is available in modern Italian (G. Sacerdote, 1896), German (H. Suter, 1909-10), and more recently in English translation:

Yadegari, M. and Levey, M. (1971) Abu Kamil's "On the Pentagon and Decagon", Japanese Studies in the History of Science, Supplement 2, Tokyo: History of Science Society of Japan.

- Abu'l-Wafa' al-Buzjani's Kitab fima yahtaju ilayhi al-sani` min a`mal al-handasa (Book on What is Necessary from Geometric Constructions for the Artisans) is available in various manuscript versions, some of them are partly or fully translated into modern languages:

Paris manuscripts (Persian): Bibliothèque Nationale, Ancien fonds persan 169:

Woepcke, F. (1855) Analyse et extraits d'un recueil de constructions géométrique par Aboûl Wefa, [Analysis and extracts of a book of geometrical constructions by Abu'l-Wafa', in French], Journal asiatique, 5th series, 5, 218-256 and 309-359.

Milan manuscript (Arabic): Biblioteca Ambrosiana, Arab. 68:

Suter, H. (1922) Das Buch der geometrischen Konstruktionen des Abûl Wefa, [The book of geometrical constructions by Abu`l-Wafa', in German], Abhandlungen zur Geschichte der Naturwissenschaften und Medizin, 94-109. [Expository paper on the manuscript].

Istanbul manuscript: Ayasofya 2753 (eleven of the thirteen chapters are extant):

Krasnova, S. A. (1966) Abu-l-Vafa al-Buzdzhani, Kniga o tom chto neobkhodimo remeslennika iz geometricheskikh postroenii, [Abu'l-Wafa' al-Buzjani, Book on What is Necessary for the Artisan from Geometrical Constructions, in Russian], Fiziko-matematicheskie nauki v stranakh vostoka [Physical-Mathematical Sciences in the Countries of the East, in Russian], 1, No. 4, 42-140. [Russian translation of the manuscript with comments].

Also see the following survey on the manuscripts:

Özdural, A. (1995) Omar Khayyam, mathematicians, and conversazioni with artisans, Journal of the Society of Architectural Historians, 54, 54-71. [Appendix, pp. 67-68].

- Al-Kashi's book Miftah al-hisab (The Key of Arithmetic) was translated into Russian (B. A. Rozenfeld, 1954) and its architectural chapter was discussed in an additional paper (L. S. Bretanitskii and B. A. Rozenfeld, 1956). The section "On measuring the area of the muqarnas" was published more recently in a bilingual Arabic-English form with additional commentaries:

Dold-Samplonius, Yvonne (1992) Practical Arabic Mathematics: Measuring the Muqarnas by al-Kashi, Centaurus: International Magazine of the History of Mathematics, Science, and Technology, 35, 193-242.

[12] Works on geometrical methods in art, which were written by artisans:

[13] Documents on discussions between mathematicians and artisans: