Religious design

Summary: Sacred spaces have often been designed with special attention paid to their geometry.

Houses of worship are places dedicated to spiritual and religious practices. Because spatial metaphors are universally important to humans, many faiths tie their religious ceremonies or practices with special architecture, decoration, and visual symbols that promote spiritual contemplation or changed states of consciousness or, in some cases, serve to teach aspects of the religion. Mathematically rich areas of study and practice related to houses of worship include sacred architecture and sacred geometry. The mathematical patterns from these studies are frequently investigated within both mainstream science and religions, for example, under the umbrella of ethnomathematics.

Houses of Worship as Motivators for Mathematics and Science

Because houses of worship typically had special meanings, they were often constructed differently than other buildings. At times, this elevated the architectural and aesthetic demands on both designers and builders, who found themselves challenged with erecting unusual and often very large structures, which promoted applied mathematics. Their methods remain a source of debate in modern scientific circles, since it is not clear exactly which engineering methods were used to lift the enormous monoliths that make up Stonehenge in England, to fit together large blocks forming the ancient Egyptian Luxor temples or pyramids, or to orient Sumerian temples with compass directions. However, it is clear that the desire for special sacred architecture could provide justification to spend time, materials, and other resources in mathematically rich ways.

A Sampler of Mathematical Features in Houses of Worship Through History

Sophisticated mathematical ideas and principles can be found in Sumerian ziggurats, which were built in ancient Mesopotamia. These structures have a characteristic “gigantic step” shape made of two to seven receding tiers. The top tier, where historians assume rituals were performed, could only be reached through narrow ramps. This feature would have isolated the priests and made the defense of the top tier easy. This was important, as ziggurats also served as city administrative centers, and the shrine complexes contained within them housed kings who performed rituals or who may have been considered deities themselves by the people they ruled. Such ties between ruler and divinity, or the concept of the divine right of kings, were found in many of the ancient cultures, including Egypt, as well as in many European monarchies. This principle may also have played some role in design. The ziggurat tier shape is still frequently used in modern architecture.

In Japan, Shinto shrines and Buddhist temples encouraged congregants to bring small wooden tablets called sangaku as offerings to gods. They have been traced back to the beginning of the Edo period of the seventeenth century. Temple visitors painted colorful sangaku to share Euclidian geometry puzzles, named for Euclid of Alexandria, or solutions and variations on puzzles others shared earlier.

Hindu temples were based on vedic mathematics. Their shape was usually square, with sides divided into eight or nine parts. This defined 64 or 81 smaller squares within the temple, dedicated to different gods. The whole temple, in a fractal manner that appears in many houses of worship, represented both the universe and the inner space of a person and the idea that the two are similar. Town plans often followed these temple plans, to add another level of recursion. Mathematical formulas in India were frequently used as descriptions or metaphors for sacred ideas. For example, one of the words for “temple” is vimana, which literally means “well-proportioned.” Ratios within the temple symbolized the harmony within the universe and were strictly followed during construction. Even images within temples displayed the sacred ratios of “iconometry.”

Ancient Egyptian temples, built strictly along the east-west line, were believed to be symbolic models of the universe. For example, the floor was slant upward from outer to inner courts, symbolizing the rise of the world out of primordial waters. Precise measurements and astronomical observations were required for construction of the temples, which also served as libraries and education centers.

Islamic architecture typically includes repeating geometric patterns, symbolizing Allah’s infinite power. The mathematics underlying some of the patterns is so intricate that research papers connecting it to recent discoveries are still published in the twenty-first century. For example, girih design consists of tessellating polygons overlaid with networks of lines, described by the areas of mathematics now called “quasicrystalline structures” and “Penrose tilings,” named for Roger Penrose.

These designs can be infinitely extended without repetition. Muquarnas (or stalactite vaults) are nested, self-similar structures consisting of niches hanging from the ceiling. Their two-dimensional projections consist of tessellating shapes of decreasing size, with ratios set in such a way that shapes still fit together. As with compass orientations and direction in some other sacred spaces, the alignment of mosques toward Mecca, the direction of which varies by geographical location, is another topic that is of interest to mathematicians and others.

The sweat lodges used by Native Americans for purification ceremonies employ the effects of raised temperature, humidity, and plant smoke rather than specific architectural elements to achieve spiritual effects by affecting the body. Sophistication of the lodge experience comes from matching the rhythm of changes in temperature and humidity to the pace of the ceremony.

Traditional southern African settlements display a fractal structure based on circles. The village is built in a large circle made of smaller extended-family circles, and the large circle in the middle for the chief’s family that includes circular huts for honoring the spirits of ancestors. Within each house, the shape is repeated on a miniature scale, with a circular sacred altar in the middle.

Light and Sound

Stained glass windows have long been used in temples and churches. Geometric elements of stained glass windows include reflections, rotations, symmetry, and tessellations. The chemical knowledge necessary for coloring glass and for connecting it with metal strips helped promote scientific development. Dividing large stained glass windows into panels for structural stability was an engineering problem. The colored light, filtering through stained glass, which is variable depending on time of day and weather, is also an important element of the internal atmosphere of the religious space. Another application of light is in the Abu Simbel temples in Egypt, the design of which was reportedly astronomically aligned so that the light rays would reach the innermost sanctum on the birthday of Ramses II.

The dynamic play of light and shadows may also lead to mathematical investigations. Mathematician Thomas Banchoff has commented on the abundance of geometry in houses of worship, which provided him with both spiritual and mathematical inspiration. For example, as a student, he noted, “there was plenty of time to contemplate the shadows advancing across the tiles at the base of the altar rail. When we first arrived, the narrow of the altar rail covered only a small portion of the triangular tiles, and by the end of Mass, almost the entire triangle was in shadow. When, I asked myself, did the shadow cover half the area? I hadn’t studied any formal geometry yet, but I figured that if you cut an isosceles right triangle in half by a line perpendicular to the hypotenuse, then one of those halves could be rotated to give the triangle that remains when the shadow was covering half of the original triangle. It surprised me that the line did not pass through the centroid of the triangle! To this day, I still use that example when I teach calculus students about centroids.”

Sound as well as light is an important feature of houses of worship. The acoustics of houses of worship are determined by the architecture, including the shape and size of spaces as well as construction materials. Resonance is the property of a system in which it oscillates with larger amplitude at some frequencies than at others. Very high ceilings used in many houses of worship amplify sound and make group singing more resonant and subjectively powerful.

Domed ceilings and other shapes may reflect sound back toward the speakers or singers at many different angles at the same time. This reflection causes interference and reverberation in the sound waves that may make individual words less intelligible, which may promote a feeling of the merging of individuals into the congregation.

Size

Places of worship tended to be some of the largest buildings within each settlement, since they were often the only gathering place for the community for both spiritual and secular uses. Historically, group trading and entertainment also attracted large numbers of people. To symbolize the higher importance of the house of worship compared to secular buildings, some places also required it to be the highest building. This led to sacred architectures that had tall narrow towers, such as steeples on churches and minarets on mosques, soaring high over the town. Large constructions could also provide security, such as the Sumerian ziggurats. Another reason for height was the need for sounds related to religious practices to carry through the settlement, such as calls to prayer from minarets or tolling bells marking the start of services.

Conversely, some spiritual practices call for small, temporary houses of worship. These ephemeral structures evoke meanings opposite to large, permanent houses of worship, such as unity in hard times or intimacy within families; for example, some Native American sweat lodges and the Jewish sukkah—a temporary structure decorated with branches and gourds that is used for a week-long harvest festival. It symbolizes joyful but temporary shelter from the wilderness.

Unusual Modern Designs

Several houses of worship built in the last decades of the twentieth century and the first decade of the twentieth-first century feature interesting mathematical concepts. The Baha’i House of Worship in Delhi, India, also known as the Lotus Temple, is based on a nine-sided polygon. It looks like a half-open lotus flower, with all walls consisting of curved “petals.” Modern software allows for the design of such complex surfaces. Wotruba Church, in Austria, consists of 152 asymmetrical concrete blocks and resembles an abstract sculpture. Balancing such blocks and calculating safe loads within the structure presented interesting spatial geometry problems. The Cathedral of Christ the Light in California employs many traditional features in unique ways. From the outside it appears as a truncated cone composed of many semiopaque windows in a steel grid. In the tradition of stained glass windows, the cathedral uses light to create atmosphere and convey images. The church’s Omega Window is a representation of a traditional Christian symbol known as Christ in Majesty, which often includes a mandorla frame. The image was taken from an eleventh-century stone sculpture and digitally converted into a pixel-like pattern of 94,000 holes that were drilled into aluminum panels. The holes’ varying diameters transmit different degrees of sunlight to create the image.

This is one of many light effects created by the curved internal and external geometry and features like curved beams, folded gothic-style arches, and slats that tilt to manipulate light.

Bibliography

Albers, Donald. “The Third Dimension and Beyond.” Math Horizons 3 (February 1996).

Koetsier, T., and L. Bergmans. Mathematics and the Divine: A Historical Study. Amsterdam: Elsevier, 2005.

Strachan, Gordon. Chartres: Sacred Geometry, Sacred Space. Edinburgh: Floris Books, 2003.