ARCHITECTURE, MATHEMATICS & NUMBERS (Part I)

Architecture is related to mathematics as it is with geometry, being mathematics part of Pythagorean theories. Mathematics is then the perfect architecture tool for the creation of harmonious forms and for following the principles of proportion. The value of mathematics and numbers goes beyond and is related with some symbolism and even religious and spiritual meaning as the one conferred by the golden ratio that was used in Egypt, Greece, Rome and the Islamic world.

Geodesic Dome - Buckminster Fuller

In this sense, Mathematics has been an integral part of Architecture since its inception as the base of Geometry. All kind of buildings around the world embodies a complex set of calculated dimensions, follow location principles in the site layout and are based on some geometric pattern in plans based on mathematics formulas. Shapes and orientations of buildings have not only been based on pure geometry but also on trigonometry principles to give a proper approach for the construction of temples and emblematic pieces of architecture such as the Pyramids.

ANCIENT ARCHITECTURE & NUMEROLOGY

The first clear manifestation of Mathematics and numbers applied to architecture are found in the Great Pyramid in Giza. Number Phi and the golden ratio are present in several dimensions and elements of the pyramid plan. Even today it is debated if the Pythagorean Theorem was used or not, although the existence of Euclidean techniques is a real fact.

Great Pyramid at Giza - Source Egyptra.com

Egypt Pyramids Triangles

The proportions of some pyramids may have also been based on the 3:4:5 triangle proportion (face angle 53°8'), which many historians believe Egyptians admired in some ways.

3:4:5 Triangle

The curious thing is that the 3:4:5 triangle is a pure reconstruction of the Pythagorean Theorem. Egyptians were doing their reinterpretation of this theorem before it was formulated. Many historians support the idea that they already knew it and they found their own method to construct a right angle and without using a ruler or a compass.

Islamic Architecture

Some historians believe that some of the most emblematic buildings of Islamic architecture, like Alhambra or the Great Mosque of Cordoba, were designed using the Hispano-Muslim fixed measures system, with a minimal unit named foot or codo ( 0.62m). These proportions are clearly present in spaces such as the Alhambra’s Court of Lions, bracketed by the Hall of Two Sisters and the Hall of the Abencerrajes, which creates a regular hexagon that can be drawn from the centers of these two halls and the four inside corners of the Court of the Lions.

Alhambra Geometry - Source Semantic Scholar

Pythagoras ratios and proportions are also present in the decoration patterns, following the proportions based on √2 that generates squares inside circles and eight-pointed stars and √3 generating six-pointed stars.

Islamic Geometries - Source Catnaps.org

Other buildings like The Selimiye Mosque, built by Mimar Sinan, are based on the octagon geometry. A large central octagonal space is formed by eight enormous pillars and covered by a circular dome of 31.25 meters in diameter and 43 meters high. The octagon creates a square with four half-domes and four tall minarets 83 m tall.

Selimiye Mosque - Architect Sinan (Source Archnet)

The general building's plan is a circle, inside an octagon, inside a square.

ROME, NUMBERS & PROPORTIONS

The clearest example of the use of number and geometry in Roman architecture is The Pantheon in Roma that remains a perfect portrait of scale, proportion, and decoration and the world's largest unreinforced concrete dome.

The building dome has an open oculus of 43,3m at the apex ad is fronted by a colonnade with a triangular pediment. The size of the dome and the diameter of the interior circle are the same, 43.3 m, so the whole interior would fit exactly within a cube, and the interior could house a sphere of the same diameter.

Pantheon Diagram - Source ResearchGate

Although the measures could be considered a bit weird, they match perfectly with the ancient Roman measure system: the dome spans 150 Roman feet and the oculus 30 feet in diameter.

The rest of the measures are perfectly related to a geometrical system of cylinders, spheres and cubes that matches the golden ratio and even Earth proportions.

Pantheon Diagrams - Source SlideShare

THE PERFECT NUMBER & MEDIEVAL ARCHITECTURE

Numbers played a very symbolic role in medieval architecture and society in general. Under the influence of religion, numbers were given powers and special meaning. In these terms, number 1 was the symbol of divinity and the beginning of everything; number 2 was the symbol of duality and polarity (good/evil, earth/heaven, male/female); number 3 was the representation of the Trinity for Christian, Egyptian and Hindu religions; number 4 represented the balance between body and soul, material and immaterial worlds (the elements, the seasons, the cardinal points…); number 5 was the symbol of the power of God to create things (3+2, Trinity + Human Duality); number 6 was a representation of perfect balance (number 4 + number 2); number 7 was one of the most symbolic and powerful ones and the final number of the circle, representing the seven days of the Creation process, the number of days, gospels, sins…; number 8 was the beginning of a new circle, representing the renaissance; number 9 has been very recurrent for the energy division of spaces, as well of being the symbol of light; number 10 is the sum of the first 4 numbers and represents the perfection; the last number of these two circles is number 12, representing the 12 disciples, Israel tribes, months of the year, zodiac cosmology. After these, number 13 is considered a bad-luck number for the mere fact of being the first number of the non-divine circle.

The Perfect Number

Of all the numbers with symbolic values, perfect numbers appear to have been valued more than any other numbers in the Middle Ages. A perfect number is equal to the sum of its divisors. The first three perfect numbers are 6, 28 and 496. In his Etymologiae, Isidore of Seville describes the number six as follows: ‘It has three parts, the sixth, the third and the half. The sixth is 1, the third is 2, the half is 3. These parts added to make a complete 6. The perfect numbers are: 6 within 10; 28 within a hundred; 496 within a thousand’.

St. Michael in Hildesheim - Source Flickr

There are many examples of medieval architecture following these numbers symbolism. In the church of St. Michael in Hildesheim there is a nave of three bays, with four piers, and twelve columns in total, representing the Trinity, four evangelists and twelve apostles.

 Noyon Cathedral South Clerestory - Source ResearchGate

In Noyon Cathedral studies have noted that the southern elevation of the Cathedral’s transept was unusual and differs from every other Gothic elevation the times the building was constructed. The elevation of this south transept has 28 windows (the perfect number of 100).

Canterbury Cathedral

In Canterbury Cathedral has been also designed following the symbolism of the number 28 as the perfect number of 100. This number is made up of 4 times 7. There are a total of seven arches in the lower level, 14 in the middle and 7 on top, for a total of 28 arches. Although the number of arches fits this scheme perfectly, there are only five windows on both the ground floor and clerestory, for a total of 10 windows.