The Why of Mathematics

Mathematics is an essential part of our scientific culture, and also of our vital culture. We relate to the world through mathematics: we count, anticipate or draw conclusions from them. In the architectural monuments that surround us, mathematics is also very present and finds a field of exceptional development. Historically, heritage buildings have been designed according to their criteria, both functional and aesthetic, trying to add beauty through patterns and mathematical patterns, or solidity through studied supports, arches and constructive elements.

This is why our heritage offers an exceptional opportunity to approach applied mathematics, those that were in the minds of the architects, master builders and designers of the great buildings that we admire in our towns and cities. But these mathematics have not always been explicit in the history of art or science. Many of their contributions remain hidden, and others have only partially emerged.

The purpose of this project is to delve deeper into these contributions, to discover guides and mathematical criteria under which these monuments have been built, as a way of approaching their understanding, the usefulness of mathematics and the scientific solutions given at each historical moment, and to inspire us to have other solutions for the future and the challenges of the present.

Some attempts have already been made in this direction, one of them the recent book published by the Editorial Universidad de Granada "Paseos Matemáticos por Granada: un estudio entre arte, ciencia e historia". But this project aims to go further and enter a dimension that has not been addressed until now. It is not a question of the singular study of this or that particularly interesting monument, but of the global study of certain mathematical characteristics in all of them, to see what this aggregate data can provide us with.

This is not the task of a single researcher, or even a team of them, but it is only possible to tackle it with the great tool of Citizen Science, in which the collaboration of a conscious and involved citizens can help to reach where a research team would not reach, neither by time nor by means, providing a wealth of relevant information and extracted with a scientific methodology that is explained in MonuMAI.

In MonuMAI we are going to begin by studying the first aspect to which we have referred, the proportions. We will add cases and build a large database to have a global and meaningful view of the appearance of the same in different artistic styles and architectural objects. You can help us in this task, using MonuMAI and sending your images with proportions searched in them. You will collaborate in the study in a direct way, of the precision and fidelity with which you do it will also depend on the scope of the results that we can obtain in the way of clarifying the interwoven relation between art, mathematics and computation.

In general among all the most relevant mathematical aspects that we can find in monuments are:

  • Art Proportions

    The proportions size the buildings and their most notable elements (doors, windows, architraves, ...) so that they are more aesthetic or with an added symbolism.

    Proportion 8:7

    It is almost a square, but barely elongated on one side. Rarely used in architecture, as the exact square is usually preferred, it appears in some Gothic and Renaissance constructions. It has an exact decimal value of 1.14.

    Proportion of cordoba

    It is defined with the rectangle whose sides have the proportion of the side of a regular octagon and its radius (the radius of the circumference circumscribed to it), with mathematical expression. It was discovered by the Cordovan architect Rafael de la Hoz and made public in 1973, after two decades of studies on Cordovan monuments, where he found it frequently, and hence the name. Used above all in Hispano-Muslim art, its influence sometimes reaches other styles with which it coexists culturally, such as Gothic or Renaissance. It has an approximate value in decimal to 1,306.

    Root ratio of 2

    It is a classic proportion that comes from taking as one side of the rectangle the diagonal of the square built on the other side. Widely used in Hispano-Muslim architecture, it appears in almost all styles. It has an approximate value in decimal to 1,414.

    Golden proportion

    Perhaps the most famous of all proportions, its name, seems to come from the initial of the Greek sculptor Phidias. Mathematically defined as , it is the solution of an algebraic equation associated with the recurrent succession of Fibonacci. But its importance appears when it emerges in numerous forms of nature and in classical architecture from its Greek origins to the renaissance or contemporary art. It is often considered associated with the ideal canon of beauty. It has an approximate decimal value of 1.618.

    Root proportion of 3

    It is a classical proportion that comes from taking as one of the sides of the rectangle the diagonal of a rectangle of sides 1 and . Its use is extended in the Renaissance and Baroque, stylizing elevations and architectural parts. It has an approximate value in decimal to 1.732.

    Proportion 17:9

    It is a proportion associated with audiovisual formats. It was introduced in 1953 by Universal Pictures, and is the most widespread standard in cinemas. It is slightly longer than the 16:9 format, the standard used in high definition television. It has an approximate decimal value of 1.888.

    Proportion 2:1

    It is the proportion that results from stacking two squares. It works as a constructive module when you want to extend proportions in an easy way. It has an exact value in decimal to 2.

    Root ratio of 5

    It is a classic proportion that comes from taking as one of the sides of the rectangle the diagonal of a double square, 1 x 2. Its use extends in the Renaissance and Baroque, being common in cartels and other elements with elongated shape in vertical or landscape. It has an approximate value in decimal to 2,236.

    Proportion of silver

    It is known as one of the metallic numbers and is associated with an algebraic equation of the type x² - 2x - 1 = 0. One of its solutions is δ = √2 + 1, which is known as the silver number. Along with the gold number, it maintains curious mathematical properties, such as its inverse 1/δ = δ - 2, similar to what happens with the golden reason, where 1/δ = δ - 1. It has an approximate value in decimal to 2.414.

  • Polygons

    The polygons and more in general the constructions with ruler and compass, that make patterns in plants or facades, compose the decoration and modulate spaces and measures.

  • The arches

    The arches evolve historically with the architectural styles, some to almost extinguish with a change of cultural epoch. From those of a centre such as the rounded arch, the segmental arch or the horseshoe arch, linked to Hispano-Muslim art, through those of two centres, such as the ogival arches, almost exclusive to the Gothic period, to more complex arches, such as the conopial arches of 4 or 3 centres, also Gothic, or the more complex carpanels of 3, 5 or 7 centres. And there are also mixtilinear arches, which mix classic constructions with period inspirations to make them unique and more refined.

  • Conicals and curves

    The conics and curves, which finish off pediments, frame decorative figures or inspire artistic interpretations. Circumferences, ellipses, hyperbolas, ... are frequent in architecture for their mathematical properties and ease of layout. Other notable curves such as spirals, astroids, ovoids, cycloids, catenaries or lemniscatas, are also used, to name just a few. More present in styles with a more elaborate decoration, such as Baroque and scarcer in others such as Renaissance.

  • The friezes

    The friezes, as mathematical forms to fill the decorative space following a line or main direction. There are 7 different types of them, and they all appear in their use in monumental architecture in different periods and styles, although the most symmetrical are used, they all emphasize the idea of a plinth or decorative line: they frame, festoon, finish off or separate spaces in an elaborate way.

  • Tessellations

    Tessellations, as a way of combining symmetries, turns and displacements to expand a pattern or minimum motif in an entire space or decorative cloth. There are 17 different types of them, discovered mathematically at the end of the XiX century. From their omnipresent use in tiles and plasterwork in Hispano-Muslim architecture, linked to the idea of infinity, to their use in sgraffito and other Renaissance or Baroque motifs, with mere decorative interest.