Painting By Numbers

Fractals have a property of self-similarity. They are similar to themselves in different regions, and similar to themselves on different length scales, so an enlargement of part of a fractal looks similar to the whole fractal. Benoit Mandelbrot pointed out that many objects, including coastlines, have a fractal structure. The mathematics of fractals dates back to Gaston Julia, who was seriously injured during the First World War and completed much of his work while recovering in hospital. Computer-generated imagery (CGI) is widely used in the video game and film industries. One technique for generating images of virtual scenes is known as ray-tracing. Filmed footage can be combined with computer-generated imagery using a technique known as chroma key. One of the most creative examples of computer art is the beautiful panoramic video in Pursuit of Venus by Lisa Reihana.

Burning Down the House

Algorithms are important in the modern world, but what is an algorithm? At school we learn algorithms called arithmetic, which is much easier in Hindu–Arabic rather than Roman numerals. Chaucer calls them augrim numbers. Augrim is a variant of algorithm and derives from the name of the ninth-century Persian mathematician al-Khwārizmī. In ‘The Miller’s Tale’ Chaucer tells of the scholar Nicholas with his augrim stones and astrolabe. This could be a caricature of Nicholas of Lynn. There was a long rivalry between those who performed calculations using arithmetic and those who used a counting table. The Exchequer gets its name from the similarity between a chess board and the counting table used to calculate taxes. For many centuries, debts were recorded on tally sticks. In 1834 the Houses of Parliament were destroyed when the vast collection of accumulated tally sticks were burnt, and the fire was immortalized by J. M. W. Turner.

Interstellar Overdrive

Kepler sought patterns and symmetry in the laws of nature. In 1611 he wrote a booklet, De Niva Sexangular (The Six-Cornered Snowflake), in which he attempted to explain the structure of familiar symmetrical objects. Almost 300 years before the existence of atoms was definitively established, he concluded that the symmetrical shape of crystals is due to the regular arrangement of the atoms of which they are formed. He also investigated the structure of geometrical objects such as the Platonic solids and the regular stellated polyhedra, known today as the Kepler–Poinsot polyhedra. Like Kepler, today’s theoretical physicists are seeking patterns and symmetries that explain the universe. According to string theorists, the universe includes six extra hidden spatial dimensions, forming a shape known as a Calabi–Yau manifold. No-one knows whether string theory will revolutionize physics like Kepler’s brilliant insights, or whether it will turn out to be a red herring.

The End of the World is Nigh!

Astronomy was the first science. Even in the fourteenth century, astronomers could accurately predict the date and time of an eclipse that lay one hundred years in the future. But early astronomers also developed some strange ideas which still resound today. Astrologers of the past identified conjunctions of the planets, especially the outer planets Jupiter and Saturn, with disastrous events such as floods, schisms, and pestilence. These ideas were related to the notion that world history can be understood as a series of 1,000-year cycles. This idea dates back to ancient Persian and Babylonian astrologers, but it has been perpetuated within Zoroastrianism, Judaism, and Christianity and is known today as ‘millennialism’. It is quite remarkable that the sequence of conjunctions of Jupiter and Saturn was also the key that led Johannes Kepler to dedicate himself to astronomy and ultimately to transform astronomy into a modern science.

Mapping the Cosmos

There is no way to transcribe the features of the Earth’s spherical surface onto a flat map without some distortion. All maps distort the geography of the sphere. The familiar Mercator maps inflate regions close to the poles compared to regions in the tropics. In 1973, Arno Peters promoted the Gall–Peters projection that compensates for the expansion of polar regions compared to the tropics. Buckminster Fuller invented a map called the Dymaxion in which the globe is projected onto an icosahedron, which is then unfolded into an icosahedral net. Another interesting projection is the Pierce Quincuncial projection invented by Charles Sanders Pierce. The Milky Way galaxy was recently mapped using data from NASA’s Wide-field Infra-red Survey Explorer (WISE) and shown to be a barred spiral galaxy. Pablo Carlos Budassi has created a map of the entire visible universe using NASA images by representing radial distances on a logarithmic scale.

The Bride Stripped Bare

Hinton’s writing on higher dimensions influenced artists as well as writers. Chapter 19 looks at how higher-dimensional geometry influenced the development of the visual arts in the twentieth century. Hinton’s influence was both direct through his own books and through the spiritual movement known as the Theosophists who latched onto his more mystical ideas. The cubists were the first modern artists to abandon the use of traditional perspective, and they were rapidly followed by other art movements. A number of the pioneers of abstract art were influenced by the Theosophists, including Kandinsky, Mondrian, and Malevich. Marcel Duchamp played a key role in determining the future direction of the visual arts, and some of his major works were developed around ideas of higher dimensions. These include Nude Descending a Staircase and Bride Stripped Bare By Her Bachelors, Even. Duchamp also led the way toward today’s conceptual art.

Optical Skullduggery

Chapter 14 takes a close look at The Ambassadors by Hans Holbein the Younger. The painting includes many interesting features, including a representation of the Cosmati Pavement on the floor, and the famous anamorphic image of a skull. The text discusses various explanations of the painting, in particular some of those proposed by the astronomer John North, who shows that several astronomical instruments in the painting record the date of the painting as 11 April 1533, which was Good Friday. The text considers the possibility that the painting was commissioned by Anne Boleyn to commemorate her marriage to Henry VIII and her coronation.

The Golden Number

The golden number or divine proportion was defined by Euclid. It is sometimes claimed that it was used in classical architecture, but it is not mentioned by Vitruvius, so this seems unlikely. The illustrations for Luca Pacioli’s book The Divine Proportion were drawn by Leonardo. The golden number is related to the structure of polyhedra with five-fold symmetry. Chapter 13 considers some of the properties of the regular and semi-regular or Archimedean polyhedra, and also considers the suggestion that the pupil in the famous painting of Luca Pacioli is a young Albrecht Dürer.

The Vanishing Point

Filippo Brunelleschi was a pivotal figure in the Renaissance. He designed and constructed the dome of the cathedral in Florence whose size has only been surpassed using modern building materials. Even more influential was his invention of single-point perspective as a technique for producing geometrically realistic paintings. His methods were disseminated by Alberti and developed by painters such as Paolo Uccello and Piero della Francesca. Chapter 12 provides a simple explanation of how a perspective grid is constructed and discusses examples of its use, including Raphael’s School of Athens.

There’s Magic in the Air

Carl Friedrich Gauss showed remarkable mathematical ability from a young age. As a child he found a formula for calculating the sum of sequences of consecutive numbers, a type of arithmetic progression. The formula is explained and used in the text. The mythological origins of the 3x3 magic square known to the Chinese as the Luo Shu are discussed. Larger magic squares were introduced to Europe through the book Shams al-Ma’arif, written by sufi mystic al-Buni. In this work, the magic squares were associated with astrology. This association was perpetuated by European writers such as Cornelius Agrippa, and Luca Pacioli included magic squares in his book De Viribus Quantitatis (On the Power of Numbers). The most famous use of a magic square in art is in Albrecht Dürer’s engraving Melancholia I. Here the meaning of this enigmatic work is discussed.