Mathematics and Art: Why can’t we be friends?


The figures of Geometry and Arithmetic by the Coëtivy Master, late 15th century (detail from Philosophy Presenting the Seven Liberal Arts to Boethius)

For most of history, mathematics and the visual arts have been friends. Art was not distinguished from what we now call “craft,” and mathematics – geometry and arithmetic – provided both a source of inspiration and a set of tools. Polykleitos, for example, in the 5th century BC, outlined a set of “ideal” proportions for use in sculpture, based on the square root of two (1.414…). Some later artists used the golden ratio (1.618…) instead.

Symmetry has also been an important part of art, as in the Navajo rug below, as well as a topic of investigation for mathematicians.


Navajo woollen rug, early 20th century (Honolulu Museum of Art)

The Renaissance saw the beginning of the modern idolisation of artists, with Giorgio Vasari’s The Lives of the Most Excellent Painters, Sculptors, and Architects. However, the friendship between mathematics and art became even closer. The theory of perspective was developed during 14th and 15th centuries, so that paintings of the time have one or more “vanishing points,” much like the photograph below.


Perspective in the Galerie des Batailles at Versailles (base image: 1890s Photochrom print, Library of Congress)

Along with the theory of perspective, there was in increasing interest in the mathematics of shape. In particular, the 13 solid shapes known as Archimedean polyhedra were rediscovered. Piero della Francesca rediscovered six, and other artists, such as Luca Pacioli rediscovered others (the last few were rediscovered by Johannes Kepler in the early 17th century). Perspective, polyhedra, and proportion also come together in the work of Albrecht Dürer. Illustrations of the Archimedean polyhedra by Leonardo da Vinci appear in Luca Pacioli’s book De Divina Proportione.


Illustration of a Cuboctahedron by Leonardo da Vinci for Luca Pacioli’s De Divina Proportione (1509)

Some modern artists have continued friendly relations with mathematics. The Dutch artist M. C. Escher (reminiscent of Dürer in some ways) sought inspirations in the diagrams of scientific publications, for example.


Tiling by M. C. Escher on the wall of a museum in Leeuwarden (photo: Bouwe Brouwer)

Today it is possible to follow in Escher’s footsteps by studying a Bachelor of Fine Arts / Bachelor of Science double degree at some institutions. There is also a renewed interest in the beauty of mathematical objects, whether three-dimensional (such as polyhedra) or two-dimensional (such as the Mandelbrot set). The role of the artist then becomes that of bringing out the beauty of the object through rendering, colouring, choice of materials, sculptural techniques, and the like.


View of the Mandelbrot set at −0.7435669 + 0.1314023 i with width 0.0022878 (image: Wolfgang Beyer)

Artistic techniques such as these (“must we call them “craft” or “graphic design”?) are also important in the field of data visualisation, and are recognised by the “Information is Beautiful” Awards. Speaking of which, this year’s awards are now open for submissions.


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Escher and Lego!

Above is a neat model, by Puriri deVry, based on the Escher print Ascending and Descending (click image for details). The illusion is explained by Andrew Lipson here.


Mathematics in action: tiling the plane

In the next post of our mathematics in action series, we look at tessellations of the plane. The most familiar of these are the three regular tilings, using tiles that are regular triangles, squares, or (as below) hexagons.


Photo: Claudine Rodriguez

The great Dutch artist M. C. Escher is famous for his distorted versions of such tilings, such as this tiling on the wall of a museum in Leeuwarden:


Photo: Bouwe Brouwer

Alternatively, the regular tilings can be extended by mixing different kinds of regular polygon. Of particular interest are the eight semiregular tilings, in which the tiles all meet edge-to-edge, and each vertex is equivalent to each other vertex (i.e. each vertex can be mapped to each other vertex through rotations, reflections, translations, or glide reflections). Here is one of the eight:


Photo: “AnnekeBart”

Because of the high level of symmetry, an exhaustive list of the 11 regular and semiregular tilings can be made by considering all possible meetings of polygons at a vertex, such as these two:

    

Penrose tilings, discovered by Roger Penrose in 1974, loosen the regularity and symmetry conditions, while still using a fixed number of kinds of tile, and while still being “almost” symmetrical. In the image below, Penrose is standing on a Penrose tiling. His 1974 discovery goes to show that fairly simple mathematical truths can still be discovered today.


Photo: “Solarflare100”