Geometric Ornaments in Istanbul By: Lisa Berry, Sarah DeBruin and Stephanie Kimber

History Tessellation comes from the Latin word tessella, which means a small cube (such as a die or a piece of mosaic) or tile. Although the appearance of tessellations can be traced back to ancient cultures, the mathematical and scientific examination of this subject is fairly recent. In 1891, E.S. Fedorov, a Russian crystallographer proved that every form of tiling is constructed using one of seventeen distinct groups of isometries (the method of repeating tiles over a plane.) Fedorov’s research marked the beginning of the study of tessellations. Even though the mathematical investigation of tessellations is only about 120 years old, much progress has been made. During our research, we only focused on two-dimensional tessellations, but advance mathematical analysis of tiling in more than two dimensions and to non-Euclidean geometric systems exist. Examples of tessellations can be found throughout northern Africa, the Middle East and Asia. Historically, many of these geometric tessellations show up in the form of art on temples, mosques, windows, doors, railings and on the sides of buildings. In this way, ancient geometers were able to showcase mathematical techniques to everyday citizens by utilizing geometric shapes, bright colors and basic transformations. Also, in the past and even currently, some cultures utilize tessellations in clothing, basket weaving and in the creation of elaborate carpet and blanket designs. The majority of our research on tessellations is from the Middle East, which is largely influenced by Islamic religion. According to Islamic law, artists are prohibited from portraying human and animal figures in their art. Thus, geometric tessellations along with calligraphy, arabesque and floral patterns emerged. Unlike religions such as Christianity and Buddhism, Islam offers no image of God. The only material image of God that the Koran offers is Nur, which means light. This belief inspired many tessellation designs, which fused basic geometric shapes and stars to create magnificent works of art.

Some keywords to keep in mind Tessellation A piece of artwork that is created using a single unit that is repeated over and over again Fundamental Repeat Unit A tile that is repeated over and over again in a Geometric artwork Grid The shapes you actually see in the final design Subgrid The shapes that are drawn in order to create the design, but are usually not visible in the final design.

Breaking Down the Artwork Ornament from the Beyezid Mosque in Istanbul In this case, the fundamental Repeat Unit was a hexagon. From here, we must figure out what other shapes are included in each of the fundamental repeat units

Breaking down the artwork Step 1: Boundary of repeat unit Create regular hexagon Divide each side into three equal parts Step 2: First subgrid Connect hexagon vertex to three opposite vertices

Breaking down the Artwork Step 3: Second subgrid Draw a circle with center at hexagon vertex and radius equal to two-thirds of one hexagon side Divide arc inside hexagon into four equal parts using points of intersection Step 4: Repeat above steps for each hexagon vertex

Breaking down the Artwork Step 5: Create inner hexagon: Create six segments, connecting intersections of circles with hexagon Connect midpoints of above segments to form inner hexagon Step 6: Complete repeat unit Hide circles and segments connecting hexagon vertices

Variations on the Same Piece

The Minbar from the Sultan Ahmed Mosque

Courtyard in the Sehzade Mosque

Tomb of Mahmut Paşa

Entrance aT the Sultan Ahmed Mosque This particular piece uses proportions moreso than diverse Geometric shapes. Each repeat Unit consists of  a subgrid of 27 congruent rectangles. The proportion of this particular piece is  1 to square root of 3, which is a common proportion in many ancient Islamic designs.

References Most of the material for this project is based on an article entitled, “Geometric Ornaments in Istanbul” by Miroslaw Majewski from the New York Institute of Technology.