### Dürer, rhinos, and snowflakes

10th April, 2009

I’ve been a fan of the 16th century German artist Albrecht Dürer ever since I saw his print of a rhinoceros. It was put together from descriptions and someone else’s sketch: he never saw a real one. I love the scaly reptilian legs poking out beneath the sheet-metal armour, the shoulder horn, and the serrated hind-quarters.

Dürer was also a master of religious art to rival any of his Italian contemporaries. But it’s only recently that I’ve learned of his interest in mathematics^{[1]}.

In fact it was over at the Walking Randomly blog, in a post about pentaflakes: snowflake-like fractal constructions built from pentagons (check out the link for pictures). It was Dürer who first discovered them, in the second volume of his work *Underweysung der Messung* (‘Instruction in measurement’) in 1525 (almost 400 years before the discovery of the Koch snowflake).

(On the subject of beautiful snowflakes, please have a look at Kenneth Libbrecht’s stunning photographs of some real ones [via The Filter].)

In the spirit of my recent post, Dürer’s 1538 revision of the *Underweysung* is also significant for his thoughts on polyhedra. This is the first known use of nets to analyse these shapes. Here, he can also claim discovery of two of the Archimedean solids: the truncated cuboctahedron and the snub cube.

Another solid associated to Dürer is the so-called Melancholy Octahedron from his allegorical engraving *Melancholia I*:

Schreiber (1999) identifies it as a cube, first distorted to give rhombus faces with angles of 108° and then truncated so that its vertices lie on a sphere.

Also depicted in that picture is Europe’s first magic square:

As well as the rows, columns, diagonals, quadrants, corners and other significant 4-tuples all summing to 34, the bottom row also serves as a signature: the date 1514 is positioned inside the numbers 4 and 1: alphanumeric code for *D* and *A*.

It’s a delightful trick. But Dürer’s influence on mathematics goes deeper. He contributed to the theory of ruler and compass constructions, and studied a variety of algebraic curves in some depth, including an account of logarithmic spirals a hundred years before Descartes or Bernoulli.

The ultimate fusion of his artistic and mathematical interests came in his work on perspective, or more generally the problems of accurately representing 3-dimensional objects on a 2-dimensional space: so-called descriptive geometry. This is a fundamental question for artists, architects, and astronomers, as well as mathematicians, and its first systematic study is generally attributed to Gaspard Monge, almost 300 years later.

In short, Dürer was not an artist toying with mathematics, but a genuine polymath, whose broad interests and talents led to inspiring achievements in both art, and science.

[1] He even has his own MacTutor biography, which is where I got most of the information for this post.

Reference:

Schreiber, P. *A New Hypothesis on Dürer’s Enigmatic Polyhedron in His Copper Engraving ‘Melancholia I.’*, Historia Math. 26, 369-377, 1999.

April 10th, 2009 at 2:20 pm

[...] Elwes has a delightful blog post this morning: Dürer, rhinos, and snowflakes. The post is primarily about the art and mathematics of Albrecht Dürer (1471-1728) but also [...]

April 11th, 2009 at 9:27 am

Great post – I’ve been meaning to research Dürer in a bit more detail for some time and this post will start me off nicely. Thanks for the mention of WR too

You may be interested in this as well:

http://www.walkingrandomly.com/?p=697

April 14th, 2009 at 8:34 am

Thanks Michael – and sorry your comment got held up in moderation for a bit…

June 2nd, 2009 at 1:13 am

Hi, good post. I have been wondering about this issue,so thanks for posting. I’ll definitely be coming back to your site.

June 4th, 2009 at 10:53 pm

Great post! Just wanted to let you know you have a new subscriber- me!

June 22nd, 2009 at 3:03 pm

You are such a hero! And love the website.

October 13th, 2009 at 4:25 pm

Thanx.i have art homework on durer and i have to draw some of his paintings

October 18th, 2010 at 10:40 am

[...] Richardson, Pierre Fatou and Gaston Julia have a greater claim (and much earlier thinkers such as Albrecht Dürer might have something to say about it [...]