IPICTURE OF A CUBOCTAHEDRON FLATTENED FREE

Furthermore, recent findings in Scotland of stones carved with lines that correspond to the edges of regular polyhedra have introduced a new dimension to this topic (Hann 2013: 123). The origin of polyhedra has been dated back to the Babylonians they appeared long before the times of the Ancient Egyptians and Ancient Greeks (Friberg 2007: 351). The Cuboctahedron as a “Geometric Solid” in Medieval Art in Anatolia

Although this study suggests that the examples of cuboctahedra progressed linearly from solid to surface and from surface to lines, the results show that they were evidently designed as vectors.

The results demonstrate the accuracy of the craftsmanship behind the complexity of form and reveal the mathematical content of 800-year-old artworks. Next, using the data in the third part, we explore the relations between the edges and the complex surface profiles by removing the surfaces of examples as vectors that are associated with Fuller’s “vector equilibrium” (VE) and “isotropic vector matrix” (IVM), and we use Thabit ibn Qurra’s studies on the cuboctahedron as intersecting hexagonal faces. Then, the examples of cuboctahedra that envelope complex carved-out surfaces as “Polyhedral Clusters” are extensively analyzed. In what follows, first the cuboctahedron that appeared as an individual figure is evaluated as a semi-regular Archimedean solid, which was applied as an engaged column capital in architecture. Among the extensive number of medieval era examples, the examples that provide complex information for further geometrical content of cuboctahedra in Ağzıkara Caravanserai, Sarı Caravanserai, Tomb of İzzeddin Keykavus I, and the tombstone in Bursa are particularly analyzed in this present study. 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).

IPICTURE OF A CUBOCTAHEDRON FLATTENED FREE

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. The visual elements of medieval-era Islamic Art can generally be grouped into calligraphy, vegetal motifs, and abstract geometrical patterns.

Ipicture of a cuboctahedron flattened