![]() Moreover, further examples of cuboctahedra found in four other buildings were dated from the first half of the sixteenth century to late eighteenth century. 4 and 5 below) are cuboctahedra, but dodecahedra and octahedra were found in two buildings. All examples found in fifty-nine buildings in twenty towns of Turkey (see Figs. Since 2012, we have visited most of the towns to record examples of polyhedra, which can be dated to between the early twelfth to early fifteenth centuries. Following our earlier discovery of cuboctahedra at the capitals of aiwan in the hospital part of Gevher Nesibe Complex (Hisarlıgil and Bolak-Hisarlıgil 2009), which was built between 12 in Kayseri, we began to study Anatolian Seljuk structures from the literature to find whether there are more examples. In this context, numerous examples of cuboctahedra indicate the theoretical interest of medieval scholars in this form as well as the practical skill and knowledge in geometry of medieval builders. Furthermore, Thabit ibn Qurra, one of the first translators of Euclid’s Elements into Arabic, was recorded as being the first to specificially study and illustrate cuboctahedron in the ninth century. Among these scholars, Abu’l-Wafā’ and al-Kashi studied polyhedral geometry in general, and the cuboctahedron in particular. According to Özdural ( 2000: 171–172), the manuscripts written by Abu al-Wafā’ al-Būzjānī in the tenth century in Baghdad, Omar Khayyam in the eleventh century in Isfahan, al-Kashi in the fifteenth century in Samarkand, and Cafer Efendi in the late sixteenth and early seventeenth centuries in Istanbul demonstrate the effects of this collaboration between mathematicians and artisans on the art and architecture of the era. ![]() One of his best-known studies (Özdural 2000) reveals the systematic meetings between mathematicians and artisans in the tenth-seventeenth centuries. Similarly, in Turkey, Alpay Özdural has presented extensive works on the scientific content of medieval art in Anatolia (Özdural 1995, 1996, 1998, 2000, 2015). Recently, however, some scholars have used several examples to show how the artisans of the era constructed an ancient mathematical problem free of symbolic, linguistic, or visual associations imposed by historians (Chorbachi 1989 Makovicky 1992 Bonner 2000 Lu and Steinhardt 2007 Cromwell 2009). These complex geometrical patterns, which were apparently formulated by skilled designers, have generally been regarded as serving only decorative purposes with symbolic content, like other types of art. Among these, the abstract geometrical patterns are the most prevalent and have been of great interest in Anatolia. 10-sided trapezohedron (2 sets of 10 sides = 20).The visual elements of medieval-era Islamic Art can generally be grouped into calligraphy, vegetal motifs, and abstract geometrical patterns.10-sided bipyramid (2 sets of 10 sides = 20).9-sided antiprism (2 sets of 9 sides + 2 ends = 20).Even though all the faces are congruent, the rhombic icosahedron is not face-transitive.Ĭommon icosahedra with pyramid and prism symmetries include: It can be derived from the rhombic triacontahedron by removing 10 middle faces. The rhombic icosahedron is a zonohedron made up of 20 congruent rhombs. These symmetries offer Coxeter diagrams: and respectively, each representing the lower symmetry to the regular icosahedron, (*532), icosahedral symmetry of order 120. These lower symmetries allow geometric distortions from 20 equilateral triangular faces, instead having 8 equilateral triangles and 12 congruent isosceles triangles. Tetrahedral symmetry has the symbol (332), +, with order 12. Pyritohedral symmetry has the symbol (3*2),, with order 24. ![]() If all the triangles are equilateral, the symmetry can also be distinguished by colouring the 8 and 12 triangle sets differently. ![]() This can be seen as an alternated truncated octahedron. There exists a kinematic transformation between cuboctahedron and icosahedron.Ī regular icosahedron can be distorted or marked up as a lower pyritohedral symmetry, and is called a snub octahedron, snub tetratetrahedron, snub tetrahedron, and pseudo-icosahedron. Its Schläfli symbol is Ī regular icosahedron is topologically identical to a cuboctahedron with its 6 square faces bisected on diagonals with pyritohedral symmetry. The great icosahedron is one of the four regular star Kepler-Poinsot polyhedra. A detail of Spinoza monument in Amsterdam ![]()
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