ARCHITECTURE & GEOMETRY (Part I)

A simple glimpse at modern architecture will make each of us soon realize that modern architecture has become a constant exercise of exotic shapes reproduction. No matter how complex these shapes are, all these buildings have been influenced by earlier time’s architecture that was based on pure geometrical shapes and basic principles such as symmetry.

Both historical and modern developments show that geometry plays an essential role in architecture history.

The London Gherkin - Foster & Partners (Source Millenium Mathematics Project)

THE SACRED GEOMETRY

The origin of Geometry is primarily found in nature patterns that occur in the universe and that are inherent to its existence and being. This deeper meaning beyond what we can understand gives Geometry shapes a symbolic and sacred meaning that has been used in religious iconography and all art manifestations, especially in architecture.

These Sacred Geometries, deeply related with mathematics, are manifested in Golden Ratio, Fibonacci Sequence and Divine Proportion and found in many natural elements such as hexagonal honeycombs, shell spirals or the human body proportions and configuration

It is believed that taking inspiration from nature or using sacred geometry in buildings leads to the development of a structure that is perfect in almost every sense.

Sacred Geometries - Source Rethinking the Future

GREEK & ROMAN PROPORTIONS

The origin of Geometry is directly linked with Mathematics and although the rules are present in nature and the Universe, the basic rules that we know today were put down by the Greek mathematician Euclid of Alexandria in the year 300 BCE. Later, in the first century BCE, the ancient Roman architect Marcus Vitruvius set more geometry rules in his book collection De Architectura, or Ten Books on Architecture. Vitruvius is considered responsible for the proportion rules used in Architecture.  

The Golden Ratio

Vitruvius believed that any construction should always use precise ratios based on symmetry and proportions, to give temples a regular plan. These symmetries and proportions would be modeled after the human body, shaped according to a universal precise and uniform ratio. This ratio could link the human face with the total body height in a proportion of 1/10 and foot in a proportion of 1/6.  

After Vitruvius, Scientists found that this ratio was present in every element of nature, based on the proportion 1 to phi (Φ) or 1.618. Later called the Golden Ratio, these numbers have been the foundation and construction code of architecture since ancient times.

Golden Ratio & Fibonacci Sequence - Source Wikimedia Commons, Online Math Learning

Although it was Greek mathematicians the first ones officially documenting studies about the Golden Ratio, claims have been made about golden ratio proportions in Egyptian, Sumerian, Chinese, Olmec, Cretan and Mycenaean civilizations.

The Great Pyramid of Giza, built in the year 2570 BC by Hemiunu also exhibits the golden ratio without the use of mathematical techniques but with other simpler geometrical concepts. Some historians and mathematicians have set alternative theories based on the triangle proportions that were known in Egyptian times: the 3:4:5 triangle.

Pyramid of Khafe and Golden Ratio - Source Pinterest

Parthenon Proportions

The Parthenon follows the ancient Greek ideals of harmony and perfect proportions. Following these proportions, the Parthenon’s facade is claimed to be made from a progression of golden rectangles, also present in the vertical-horizontal proportions as well as in the space between columns.

The Greek sculptor Phidias also used the divine proportion in some of his sculptures, being the main reason why the American mathematician Mark Barr proposed the Greek letter phi (φ), the first letter of Phidias's name, to name the golden ratio.

Parthenon Golden Ratio - Source SlideShare

DECORATION PATTERNS IN ISLAMIC ARCHITECTURE

As with some other new techniques they introduced,  Muslims incorporated logic into their Architecture, combining geometry with traditional art and creating a new form of art based on replicated patterns and corresponding shapes.

The Islamic philosophers had learned about geometry by looking at the works of the Greek philosophers and mathematicians Euclid and Pythagoras.

From these first geometric and mathematic concepts, they discovered a new way to construct geometric patterns as decorated tessellations, as repeated shapes that cover a whole surface.

Islamic Patterns - Source Pinterest

Starting from Euclidean Geometries, some Islamic artists developed a basic system of complex shapes that could be infinitely replicated.

Islamic Art Geometric Patterns - Source Istanbul Hides

From these tessellations and patterns, Islamic art created complex compositions, avoiding and figurative images. This Aniconism in Islamic culture caused artists to explore non-figural art, and creating a new aesthetic tendency based on mathematically-based decoration. It was a matter of how complex the final decoration mural would be and the preferred geometric pattern.

Islamic Geometric Tiles pattern - Source Reddit

The Topkap? Scroll, made in Timurid dynasty Iran in the late-15th century, contains a summary of 114 patterns including variance of designs that give an idea of the range of combinations from the basic geometries and circles.

Topkapi Scroll - Source Wikipedia

Combining squares and circles, overlapped, reflected and interlaced, the composition constituted the entire decoration, mainly from floral inspiration and with some or calligraphic touch, without any references to human or animal bodies. Some recurring motifs were the 8-pointed star, made of two squares, or the polygons. All these geometries were repeated metaphorically bringing the idea of the infinite.  

Shah Imam Mosque, Isfahan - Source Inhabitat

Another great example of these patterns is found in the stucco decorations on the walls of the Alhambra Palace in Granada with such a level of complexity that some studies have found most of the pattern groups on those walls. Reaching a number of 17 patterns,  

Stucco Patterns Alhambra - Source Dreamstime