This article is based on a presentation given at the Canada International Conference on Education [CICE-2019] at The University of Toronto. Mississauga, Canada.

A form of art was created in the Islamic world between the 9th and 16th centuries that integrated rhythmic geometric patterns, calligraphy and illumination. These illuminated patterns were created by artists who worked in collaboration with mathematicians. Islamic monuments from Central Asia to Spain were decorated with these patterns.

This article describes the evolution of specific mathematical concepts in Islam and their artistic expression. The primary function was to transpose abstract mathematical thinking into visual beauty. We present the historical development of this art, its philosophy and the mathematical dimension. Modern expressions of geometry in African American art are also presented. Ideas for an interdisciplinary curriculum at the high school level that combine the learning of mathematics with the learning of art and the artistic culture of Islamic and African American civilizations are presented. The major themes for discussion are: the use of symmetry transformations, illumination, the interplay of light and geometry and the mathematical dimension.

Humanistic Mathematics
The model upon which this approach is developed is based on humanistic mathematics. Humanistic mathematics is an interdisciplinary approach that links the humanities with mathematics. In the 1990s, a group of mathematicians expressed their ideas in Essays in Humanistic Mathematics (edited by Alvin Murray White, 1993). The concept has developed from the understanding that mathematical thought lies both in the humanities and the sciences and universal mathematical themes appear in art and music across civilizations.

The Use of Geometry
The use of geometry is prominent in the art of many cultures. The Greeks, the Chinese, Native American civilizations, and many African civilizations created this form of art. In Islamic art, the use of geometry reached a very high level of development. This is attributed to the close collaboration between the mathematicians and the artists. The evolution of this art is described by Oleg Grabar in The Alhambra (1993). "After the 10th century a second type of ornament appears alongside the earlier one emphasizing polygons and stars. It makes geometric pattern almost the only pattern of decoration…. From evidence, which is now being discovered, it seems that this art was made possible by a conscious attempt on the part of the professional mathematicians to explain and to guide the work of the artisans. A recent study has pointed out that the mathematics behind this art is linked to geometric transformations observed in crystals."

The Interplay of Light and Geometry
A design technique frequently used was the interplay of light and geometry. The primary function was to transpose abstract mathematical forms into visual beauty. The aesthetics of proportion and light were discussed extensively by the 11th century physicist Ibn al Haytham in his work, Book of Optics. Geometry was a gateway to higher forms of knowledge. This idea was first expressed by Greeks in the writings of Plato and Pythagoras and later developed by Islamic scholars such as the Ikhwan al Safa (Brethren of Purity, 12th century).

Muslim Influence on African American art
A significant portion of African Americans in the United States are descended from the Mandinka or Malinke, people in West Africa, the majority of whom are Muslim. The Mande style creates narrow strips of geometric forms that are staggered to accentuate the colors. These textiles are similar to the “off-beat phrasing of melodic accents” typical of African American music. The use of fractal geometry is seen in the intricate geometric quilts created by African American artists. 

The following images are examples from Islamic and African American civilizations.
The Star-Cross Pattern 
This is a frequently used pattern in Islamic Art. It is found in many areas of the Middle East and Spain. The pattern has 4-fold symmetry with rotations and reflections. The stars are often decorated with divine names and verses from the Quran. This pattern has significance beyond aesthetics and geometry. It represents the convergence of three religious traditions that coexisted during the middle ages, Islam, Judaism, and Christianity. The 8-pointed star is the Seal of Solomon, the cross is from the Christian tradition and the calligraphy decorating the cross consists of verses from the Qur’an.

The Interplay of Light and Geometry
This image shows the interior of a mausoleum, the tomb of I’timad-ud-Daula, in India built during the Moghul period (17th century). In some architecture, no medium was used except geometry to transmit the incoming light. The stars and polygons disperse the incoming light into geometric shapes and project the illuminated forms into the space enclosed by the building. The rays of the sun create an interplay of light and geometry. This subtle design creates a transformation of space within the enclosures of sacred buildings. 

Rosie Lee Tompkins (1936–2006)
Rosie Lee Tompkins was a widely acclaimed African-American quiltmaker. She was deeply religious and developed art as a meditative experience. The staggered design is used in two of her paintings. Half-squares show a staggered display of geometric forms in alternating colors and the use of fractal geometry.
Bales, Judy, "Fractal Geometry in African American Quilt Traditions" (2009). 

The Belle Rive Jamatkhana, Edmonton, Canada 
This architecture blends modern technology with tradition. The design of the Jamatkhana que is based on geometric motifs developed from two interlaced squares. In the spiritual realm, the square is the most perfect geometric shape owing to its symmetry along the horizontal and vertical, and diagonal axis. The interlacing of squares is

symbolic of man's search for perfection. All patterns within this building are variations of the interlaced squares reflecting man's continuous search for perfection in a house of worship. (Images from The Ismaili Canada, July 1998)

The Mathematical Dimension
The use of 2-dimensional transformations was prominent in Islamic and African American art. This makes it a valuable tool for teaching concepts of transformation and symmetry. Four types of symmetry transformations were used: Translations, Rotations, Reflections, and Glide Reflections. Traditionally it was believed that geometric patterns were created by lines drafted with a straightedge and compass. However, a mathematically sophisticated approach appeared by 1500 CE using symmetry transformations and tessellations. This was reported in Science by two physicists (“Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture”. Peter J. Lu and Paul J. Steinhardt. Science, 2007). 

Applying Symmetry Transformations
The application of symmetry transformations is demonstrated in a variation of the Andalusian pattern found in Spain. The pattern has 6-fold rotational symmetry. It uses a triangular or “bird” motif. 60-degree rotations through point O are implemented on the generating motif. The tessellation displays kinetic geometry with rotational movement as the underlying principle.

In exploring the world of Islamic and African American art we can bring together the mathematical, artistic, and spiritual dimensions of two cultures. This article highlights elegant and sophisticated mathematical thought expressed in the art of these civilizations. Creating art is a significant human experience, not restricted to professional artists but an integral part of the life of an ordinary man in Islam and African American civilizations.