References


[1a] Jay Kappraf, Connections: the geometric bridge between art and science, McGraw-Hill, 1991, pp. 172-173.

[1b] "Mandalas," Grolier Encyclopedia, 1995.

[2a] James Barr, Journal of the American Academy of Religion, "The Question of Religious Influence Ė The Case of Zoroastrianism, Judaism, and Christianity," LIII / 2, June, 1985, 201.

[2b] Farhang Mehr, The Zoroastrian Tradition, An Introduction to the Ancient Wisdom of Zarathustra, Element Inc., Rockport, MA, 1991.

[3] Sacred Architecture, A. T. Mann, p. 47, Barnes & Noble Books, 1996.

[4] Other Native Americans included the circular form in their constructions. 

"Mound Builders refers to the peoples who constructed earthen mounds in North America during prehistoric times (especially to about AD 400). . . . The largest and most elaborate mounds, were often associated with complex geometric earthworks in the form of circles, rectangles, octagons, parallel walls, hilltop fort-like constructions, and other forms." 

      Grolier Encyclopedia, 1995.

[5] Bible, King James Version, Isaiah 40:18,21-22.

[6] The Oxford Dictionary of Quotations, Angela Partington, ed., Oxford University Press, 4th ed., 1992, p. 16, no. 17.

[7] Mathematics, David Bergamini, Time, Inc., 1963, p.44.

[8] Contemporary Abstract Algebra, Joseph A. Gallian, 4th ed., Houghton Mifflin Co., NY, 1998, p. 458.

[9a] In Poincaréís model, the empty space outside the boundary disk can be considered as a region for centers of circles orthogonal to the bounding circle. Let P be a point outside this boundary circle, and PT be a line passing through P and tangent to the bounding circle at T. Then the arc of a circle with center P and radius PT is itself a line in the Poincaré model.

[9b] Jay Kappraf, ibid., p.176.

[10] Visions of Symmetry: Notebooks, Periodic Drawings, and Related Work of M. C. Escher, Doris Schattschneider, W. H. Freeman and Co., New York, 1990.

[10a] Douglas Dunham, Transformation of Hyperbolic Escher Patterns, http://members.tripod.com/vismath/dunham/index.html

[11] Metaphysics, Aristotle, (Book 12, section 8), 350 BC.

[12] The Six Aenneads, Tractate 3 - Are the Stars Causes? Plotinus, section 17., 250 AD.

[13a] Mann, ibid., p. 43. [13b] Bruce Martin, 1994.

[14] The Tragedy of King Lear, W. Shakespeare, 1606.

[15a] Mann, ibid., fig. 69-70.

[15b] Kappraf, ibid., pp. 5-6.

[16a] Mann, ibid., p. 101.

[16b] For information on Hildegard von Bingen, see, for example, the web site of Bonnie Duncan,
        http://www.millersv.edu/~english/homepage/duncan/medfem/mandala.jpg
        from the English Department, Millersville University.


[17a] Mann, ibid., p. 144-145.

[17b] Chartres, Émile Mâle, Harper & Row, 1983.

[17c] Rose Windows, Painton Cowen, Thames and Hudson Ltd., London, 1979, p. 126.

[18] Aldrich Catalog Handbook of Fine Chemicals, 1996, item 37, 964-6, Milwaukee, WI.

[19] The Problems of Mathematics, Ian Stewart, 2nd ed., Oxford University Press, 1992, "Sphereful Symmetry," chapter 6.

[20] Advanced Inorganic Chemistry, F. A. Cotton and G. Wilkinson, Fourth edition, John Wiley & Sons, New York, 1980, chapter 1.

[21] The Golden Section, Garth E. Runion, Dale Seymour Publ., Palo Alto, CA, 1990, p. 15.

[22] The most recent translation of Rumiís poetry (The Essential Rumi, by Coleman Barks) has sold more than a quarter of a million copies. The Christian Science Monitor, Alexandra Marks, November 25, 1997, page 1.

[23a] Reza Sarhangi, in Bridges: Mathematical Connections in Art, Music, and Science; Conference Proceedings, 1998. Reza Sarhangi, ed., Winfield, Kansas, 1998, p. 106.

[23b] Mystical Poems of Rumi, 1, Translated from the Persian, A. J. Arberry, University of Chicago Press, Chicago, 1968, poem 23.

[24] Arberry, ibid., poem 32. The Kaaba is the black rock cube in Mecca, toward which the Islamic pray as a symbol of heaven. This is also the object that a religious person must circle around repeatedly in the symbolic ritual of the hajj pilgrimage.

[25] The Essential Rumi, Translated by Coleman Barks, with John Moyne, A. J. Arberry, Reynold Nicholson, Harper San Francisco, 1995.

[26] Faust, Johann Wolfgang von Goethe, G. M. Priest, trans., 1808.

[27] "The Girl Without Hands," Grimmís Household Tales, Jakob and Wilhelm Grimm, Edgar Taylor, trans., 1812.

[28] An Essay Concerning Human Understanding, John Locke, 1690.

[29] "Facing West from Californiaís Shores," Leaves of Grass, Walt Whitman, 1855.

[30] Black Elk Speaks, John G. Neihardt, MJF Books, New York, 1991, pp. 198-200.

[31] Sobre la muerte de la ciudad del valle germinará mi canto, Jorge Carrera Bolaños, cited in Bridges: Mathematical Connections in Art, Music, and Science; Conference Proceedings, 1998. Reza Sarhangi, ed., Winfield, Kansas, 1998, p. 292.

[32] Familiar Quotations, John Bartlett, Emily Morison Beck, ed., Little Brown and Co., Boston, 1980, "Outwitted," Edwin Markham, p. 671.



*)  Portions of this paper were published earlier in Bridges: Mathematical Connections in Art, Music, and Science; Conference Proceedings, 1998. Reza Sarhangi, ed., Winfield, Kansas, 1998, pp. 93-111.



 

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