[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.

Avital, T. Footprint Literacy: the Origins to Art and Prelude to Science and Mindprints: the Structural Shadows of Mind Reality?

Wolf, K. L. and Wolff, R. (1956) Symmetrie: Versuch einer Anweisung zu gestalthaftem Sehen und sinnvollem Gestalten, systematisch dargestellt und an zahlreichen Beispielen erläutert, 1. Textband, 2. Tafelband, [Symmetry: An Attempt towards an Instruction in Seeing Gestalt and Meaningfully Creating Gestalt, Systematically Described and with Numerous Examples Explained, 1. Text-Volume, 2. Plate-Volume, in German], Münster: Böhlau, viii + 139 and vi + 192 pp.

[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)


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.


- 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.

[7] Hero (or Heron) of Alexandria's use of 5/8 as the approximation of the golden number is not clearly stated, but it is "hidden" in his calculations. This fact was pointed out by Curchin and Fischler (1981). Theon of Smyrna's successive approximation of Ö2 is discussed, beyond some mathematical works, by Kidson (1990) in the context of artistic questions:

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:

- The anonymous Persian manuscript Fi tadakhul al-ashkal al-mutashabiha aw mutawafiqa (On Interlocking Similar and Congruent Figures), Bibliothèque Nationale, Ancien fonds persan 169, is available in the same collection that includes, among others, a Persian version of Abu'l-Wafa' al-Buzjani's Book on What is Necessary from Geometric Constructions for the Artisans [11]. The manuscript was translated into Russian by A. B. Vil'danova with additional commentaries and analysis of the figures by M. S. Bulatov in the following book:

Bulatov, M. S. (1978) Geometricheskaya garmonizatsiya v arkhitekture Srednei Azii IX-XV vv., Moskva: Nauka; 2nd ed., ibid., 1988, 360 pp. [See "Prilozhenie 2", Appendix 2, pp. 315-340].

This manuscript was rediscovered by Chorbachi and she suggested a better translation of the title, which is used here, and some further corrections to Bulatov's interpretations:

Chorbachi, W. K. (1989) In the tower of Babel: Beyond symmetry in Islamic design, Computers and Mathematics with Applications, 17, 751-789. [See especially pp. 755, 764-765, 776-778].

A more recent work shows the importance of the Persian manuscript in a new context although emphasizes that is not so original mathematically as the earlier works suggest:

Özdural, A. (1995) Omar Khayyam, mathematicians, and conversazioni with artisans, Journal of the Society of Architectural Historians, 54, 54-71. [See pp. 64-67].

- The Tashkent Scrolls were first analyzed by G. I Gaganov in 1940. The author tragically died during WW2 and his work was not published until 1958:

Gaganov, G. I. (1958) Geometricheskii ornament srednei Azii, [Geometrical ornament of Central Asia, in Russian], Arkhitekturnoe nasledstvo, 11, 181-208.

The first detailed publication on the subject is:

Baklanov, N. B. (1947) Gerikh: Geometricheskii ornament Srednei Azii i metody ego postroeniya, [Girih: Geometrical ornament of Central Asia and the methods of its construction, in Russian with a French summary], Sovetskaya arkheologiya, 9, 101-120.

Here girih, originally in Persian, refer to geometric grid systems. Note an interesting fact: this paper is one of the first papers in the humanities that suggests applying the theory of symmetry worked out by the crystallographer Shubnikov.

Also see the more recent work by

Notkin, I. I. (1995) Decoding sixteenth-century muqarnas drawings, Muqarnas: An Annual on Islamic Art and Architecture, 12, 148-171.

and Necipoglu (next item).

- The Topkapi Scroll was fully published, together with a brilliant survey on geometry in Islamic art, by

Necipoglu, G. (1995) The Topkapi Scroll--Geometry and Ornament in Islamic Architecture, Santa Monica, California: The Getty Center for the History of Art and Humanities, 412 pp.

We recommend her survey to all interested mathematicians and artists.

[13] Documents on discussions between mathematicians and artisans:

- Abu'l-Wafa' al-Buzjani's book was discussed earlier [11]. The concrete statement on meetings "held among a group of artisans and geometers" is translated into English in the paper by

Özdural, A. (1995) Omar Khayyam, mathematicians, and conversazioni with artisans, Journal of the Society of Architectural Historians, 54, 54-71. [See pp. 54-55].

- Omar Khayyam's anonymous paper where he solves a geometrical problem related to "simple ideas" and refers to a meeting where his highness, unfortunately the name is not specified, was present is available in English translation by

Amir-Moéz, A. (1963) A paper of Omar Khayyam, Scripta Mathematica, 26, 323-337. [See the reference to the meeting on p. 336].

A brilliant study on the possible links of Omar Khayyam's treatise to artistic problems, which were discussed in the anonymous Persian manuscript "On Interlocking Similar and Congruent Figures" [12], is presented by

Özdural, A. (1995) Omar Khayyam, mathematicians, and conversazioni with artisans, Journal of the Society of Architectural Historians, 54, 54-71. [See pp. 64-67].

- Al-Kashi's letter to his father was translated into English and published by

Kennedy, E. S. (1960) A letter of Jamshid al-Kashi to his father: Scientific research and personalities at a fifteenth century court, Orientalia, Nova Series, 29, 191-213. [See the meeting with the master mason on pp. 198-199, the cooperation with the master coppersmith on pp. 199-200].

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