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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Nature journals

Having said something about phenology wheels, I thought that I should mention nature journals too. Some years ago, I blogged about the professional aspects of this, but nature journals are a powerful educational tool, because of the way that they focus observational attention. John Muir Laws has good advice on getting started, including “Do not focus on trying to make pretty pictures. That just leads to journal block. Open your journal with the intention of discovering something new. Use the process to help you slow down and look more carefully.



Mother and child nature journaling examples from Nature Study Australia Instagram and website

The very useful Nature Study Australia website also has good advice and several examples, as well as other nature study resources for Australians. Artist Paula Peeters, aiming more at adults, runs nature journaling workshops around Australia, and offers an introductory book for sale or free download.


Nature journaling example from Paula Peeters, who runs workshops around Australia

Nature journals need not only contain pictures and text: a spiral-bound sketchbook will easily accomodate flat objects such as leaves, pressed flowers, feathers, and sun prints. Drawings are an essential aspect, however.


The CNPS curriculum

The California Native Plant Society offers a superb nature journaling curriculum for free download. It includes the observational prompts “I notice… I wonder… It reminds me of…” It advises parents and teachers not to say things like “that is really pretty” or “what a good drawing,” but instead to say things like “Oh, you found a spider on top of the flower! Great observation.” It also provides excellent practical advice on drawing, poetry, and other activities.

With so many excellent guides to nature journalling, why not get started on your own?


A drawing of mine (from quite some time ago)


Happy Christmas to all my readers!

The Christmas fresco above, by Giotto, shows the Star of Bethlehem as a comet (top centre). It is likely that this fresco depicts Halley’s Comet, which Giotto saw in 1301, about two years before he began the series of frescos of which this is part. This work by Giotto was celebrated in the name of the Giotto spacecraft, which observed the comet in 1986.

Let me take this opportunity to wish all my readers a very happy Christmas (and apologies for duplicating a previous post).


Art Forms in Nature: a book review


Art Forms in Nature by Ernst Haeckel

I finally got my own copy of the classic Art Forms in Nature (Kunstformen der Natur) by Ernst Haeckel. Yes, the prints are all online, but that isn’t quite the same thing. This collection of 100 prints, first published as a set in 1904, is a true classic.

   

Is it an art book or a science book? It doesn’t really matter, does it? It’s beautiful, and it’s informative. Everyone should cast their eyes over these pictures at some point. These images influenced the Art Nouveau movement, and have also found their way into many books on the relationship between mathematics and science. And they are just fun to peruse.

* * * * *
Art Forms in Nature by Ernst Haeckel: 5 stars


Citrus by Pierre Laszlo: a book review


Citrus: A History by Pierre Laszlo

This fun little book from 2007 contains everything you ever wanted to know about the history, economics, and properties of oranges, lemons, limes, grapefruit, and other citrus fruit.

Somewhat unusually for a book of this kind, it even includes a number of recipes – for fried valencia orange slices, sea bass with tangerine juice, marmalade, tarte au citron, citrus sabayon, orange mousse, and a few other things. For example:

LIME CHUTNEY

12 limes, halved
1 medium onion, peeled and quartered
4 hot green chilli peppers
1 inch ginger root
4 oz seedless raisins
7 green cardamom pods
1 tablespoon black peppercorns
1 tablespoon coriander seeds
1 tablespoon mustard seeds
4 dried red chilli peppers
1 1/2 cups cider vinegar
3 tablespoon coarse salt
1 lb light brown sugar

  • Juice the limes. Discard 6 lime halves.
  • In a food processor, combine remaining 18 lime halves, green chilli peppers, onion, ginger and raisins. Chop finely. Place mixture in a non-metal bowl.
  • Open cardamom pods. In a heavy skillet, toast peppercorns, cardamom seeds, mustard seeds, coriander seeds, and the dried red chillies for about 3 minutes, stirring constantly. Let the spices cool on a dry plate, then grind finely.
  • Add spices, lime juice, sugar, and vinegar to the chopped fruit mixture. Stir thoroughly, cover, and let steep at room temperature for 2 days.
  • On the third day, pour mixture into an enamelled pot (no stainless steel), add salt, and bring to a boil slowly. Simmer, uncovered, for 30 minutes.
  • Place in prepared clean jars. Close jars with a tight-fitting lid. Store in a cool place.


Still-Life, Jan Davidsz. de Heem, 1642

I found the chapter on art a little frustrating, though. There are many oranges in art and lemons in art, but authors who discuss such works should provide a few more illustrations. And equating oranges with “golden apples” in Greek mythology was rather dubious, I thought. But overall, I enjoyed reading this book.


Citrus: A History by Pierre Laszlo: 3½ stars


Blogroll: John Muir Laws / Drawing Frogs

Author and nature artist John Muir Laws blogs many interesting things at www.johnmuirlaws.com. One interesting recent post explains how to draw a frog step-by-step (the image below shows steps 4 and 17). He has also written about drawing insects, plants, and birds. Anyone interested in nature should take a look!

Laws also helped develop the CNPS Nature Journaling curriculum, which may be of interest to parents of young biologists.

Culture




Folk culture (left), popular culture (centre), and “high” culture (right). The photo of the blue handmade pottery cup at top left is by “Wildfeuer,” and the photo at bottom right is by Jessica Spengler.

I recently read a book discussing the now-traditional distinction between folk culture, popular culture, and “high” culture (see pictures above). Folk culture includes traditional arts and crafts, hand-made objects, and fairy tales or recipes passed down from generation to generation. Popular culture is characterised by mass-produced objects made from cheap materials or ingredients, aimed at immediate gratification and at the lowest common denominator of taste. “High” culture consists of things that only the well-educated cognoscenti can appreciate. “High” culture should be distinguished from what is simply the more expensive end of folk culture – craftspeople have always been able to produce more sophisticated items, made with more expensive materials and more elaborate decoration, if they were paid for the extra time and cost (“for you, my lord, I can fletch the arrows with eagle’s feathers instead of hen’s feathers”).


Bristol Cathedral

Now, I think this classification is missing a few things. For a start, there’s the important category of religious culture, which includes things like the great cathedrals (or, elsewhere in the world, temples and mosques) and the religious music of, for example, J. S. Bach. Such cultural artefacts were aimed at ordinary people (not the cognoscenti), but they were dedicated to God. They were meant to inspire devotion, and quite literally to point to heaven.

The Agnus Dei from Bach’s Mass in B minor, sung by Kathleen Ferrier

In Bach’s case, this intent was genuine – he was a deeply religious man, who ended his musical manuscripts with the phrase Soli Deo gloria (Glory to God alone), or with the initials S.D.G. In religious culture, it is neither the artist nor the listener/viewer to whom honour is intended to accrue. In a weaker form, this attitude can be found in, for example, movie soundtracks, whose primary purpose is not to be appreciated on their own, but to help the audience enter into the story.

In many cases, what we call “high” culture is actually folk culture or religious culture that has lost its original context. Folk-culture artefacts from the past (like the Greek vase at top) first enter the antique store and then, as they become even older, move to the museum, where they become objects of “high” culture. Folk-culture artefacts from other countries appear to become objects of “high” culture as soon as they are transported from their place of origin. Religious music becomes “high” culture when it shifts from the cathedral to the concert hall. When a degree of context is restored, some objects of “high” culture can actually become extremely popular, as in movie adaptations of Shakespeare or of classic novels. Other objects of modern “high” culture bemuse even well-educated individuals outside the cognoscenti.


Traditional slit drums from Vanuatu in the Australian Museum, Sydney (my photo)

Movies are a key part of popular culture. The chart below relates the “percentage of professional critic reviews that are positive for a given film” from rottentomatoes.com to total revenues (as at a few years ago – data is from here). It is not clear, however, exactly what the “professional critic reviews” are measuring. The relationship between revenues and quality is hazy as well, although some really terrible movies do seem to make a great deal of money.

For several decades now, a growing rebellion against popular culture has been emerging. There was the Arts and Crafts movement around 1900, and since then a steadily increasing interest in traditional forms of music, hand-crafted objects, and home cooking. There have also been a number of other interesting movements. The slow food movement attempts to resurrect a comprehensive folk culture of food, and offers a superior alternative to “fast food.” The maker movement (as defined by e.g. Make magazine) merges modern technology with traditional crafts. It seeks to ally modern technology with folk culture, rather than with mass-produced popular culture. At the simple end, maker culture includes minor customisations of high-tech devices, like this Macbook sticker:


Macbook sticker (photo by Denis Dervisevic, slightly modified)

At the more sophisticated end, there is this classic steampunk computer monitor and keyboard by Jake von Slatt:


Steampunk computer monitor and keyboard (photographer & maker: Jake von Slatt)

Steampunk culture also rebels against the cheap plastics used in so many popular-culture artefacts (although Robert M. Pirsig notes that “Mass-produced plastics and synthetics aren’t in themselves bad. They’ve just acquired bad associations. A person who’s lived inside stone walls of a prison most of his life is likely to see stone as an inherently ugly material, even though it’s also the prime material of sculpture, and a person who’s lived in a prison of ugly plastic technology that started with his childhood toys and continues through a lifetime of junky consumer products is likely to see this material as inherently ugly”). Steampunk culture prefers older materials like brass and copper, as in this 1994 Jules-Verne-inspired steampunk makeover of the Arts et Métiers Métro station in Paris:


The Arts et Métiers Métro station (photo: Stephen Butterworth)

Aspects of the emerging maker movement can also be seen in the solar cars developed for the World Solar Challenge, where high-tech electronics and solar panels are combined with carefully engineered and hand-crafted car bodies made of quality materials like carbon-fibre composites, resulting in vehicles of aesthetic beauty as well as practicality and speed.


Solar Team Twente’s solar car Red One (photo: Jérôme Wassenaar)