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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August Calendar: Australian National Science Week

The next calendar (click for hi-res image). Australian National Science Week is coming up on the 10th. Why not get involved?

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The legacy of Novum


Image credits: 1, 2, 3, 4, 5, 6, 7, 8

In 2017, the University of Michigan produced a stunning monohull solar car, Novum, which went on to take second place in the World Solar Challenge (it is shown mirror-reflected above). Their 2019 car, Electrum, has a pointier nose, and a more streamlined tail.

The new Top Dutch team has very sensibly taken Novum as a starting point for their car, no doubt feeling that the Netherlands already has too many catamarans (every new team should, if possible, strive to emulate one of the leaders of the last World Solar Challenge). Top Dutch appear to have independently made tail modifications very similar to those of Electrum. The Covestro Sonnenwagen from Aachen also shows signs of being influenced by Novum, but with a quite different nose.

Not shown are the Japanese monohulls, which look a little different, and the unique asymmetric monohull from Stanford. It will be very interesting to see all these monohulls take on the compact catamarans from Delft, Twente, and Belgium in the race this October!


And returning him safely to the earth

In 1961, John F. Kennedy told Congress: “I believe that this nation should commit itself to achieving the goal, before this decade is out, of landing a man on the moon and returning him safely to the earth.

The Moon landing on 20 July 1969 achieved the first part of that goal. The second part was yet to come (in 1970, that would prove to be the hard part).

But on 21 July 1969, at 17:54 UTC, the spacecraft Eagle lifted its metaphorical wings and took off from the Moon (well, the upper ascent stage took off, as shown in the photograph below). There followed a rendezvous with Columbia, a flight back to Earth, and an eventual splashdown on 24 July. Mission accomplished.


The Eagle has landed!

Fifty years ago, on 20 July 1969, at 20:17:40 UTC, the spaceship Eagle landed on the Moon. Here is the landing site – below, as it was, and above, as seen by the Lunar Reconnaissance Orbiter Camera (LROC). Clearly visible in the LROC image, and illustrated with inset photographs from 1969, are:


A tale of two arrivals

Fifty years ago, on 19 July 1969, the spaceship duo Columbia / Eagle entered orbit around the Moon, roughly 3 days and 4 hours after its launch, as part of the Apollo 11 mission. Eagle (with Neil Armstrong and Buzz Aldrin) went on the land on the moon on 20 July while Columbia (with Michael Collins) continued to orbit the moon. When he announced the space programme, Kennedy had said:

We choose to go to the moon. We choose to go to the moon in this decade and do the other things, not because they are easy, but because they are hard, because that goal will serve to organize and measure the best of our energies and skills, because that challenge is one that we are willing to accept, one we are unwilling to postpone, and one which we intend to win, and the others, too.

Much can be learned from doing hard things, and an enormous amount was learned from the space programme. Solar car teams also learn a great deal from doing hard things. Fifty years after Columbia and Eagle entered orbit, also after hard effort, the University of Michigan Solar Team’s solar car Electrum arrived in public view (at 17:30 Michigan time). They intend to win too!


A tale of two launches

Fifty years ago, on 16 July 1969, a Saturn V rocket carrying three men (Neil Armstrong, Buzz Aldrin, and Michael Collins), plus the command module Columbia, and the lunar module Eagle, took off from Florida, en route to the first human landing on the moon (above).

Fifty years later, the Dutch Vattenfall Solar Team launched their 10th solar car, Nuna X (below). Catamaran-style solar cars seem to have reached an optimum: the front of Nuna X looks a lot like Nuna 9 and the rear looks a little like Twente’s Red E and Agoria’s Punch 2 – but only on the left side! The new car weighs just 135 kg (298 lbs), which is probably the lightest ever solar car. I think that Twente, Agoria, and everybody else will have their work cut out trying to outrace this latest car from Delft!