The Egyptian God Thoth, usually represented as a sitting baboon, was a God of wisdom, knowledge, and by extension, mathematics.
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THE ELEMENTS How indeed did this art of mathematics begin? There is an early story that is fascinating as it explains how some very practical ideas gave birth to the mathematical art. Although this is not the first occurrence or event in the history of mathematics, it is noteworthy as it traces the emergence of a particular search for beauty and the ways in which this art further developed. WHATEVER’S HANDY – BEGINNINGS OF GEOMETRY Centuries before the Tomb of Menna was excavated at the beginning of the 20th century, children the world over were familiar with the origin of the word geometry. It comes from ‘geo’, meaning earth, and ‘metria’ meaning measurement. According to legend, geometry began in ancient Egypt. The river Nile flooded its banks and surrounding areas so often, that a need arose to establish boundaries on maps and re-establish them after each flood. But to do so, one had to measure the land, mark it, and then transfer the boundaries onto maps and plans. Hence the emergence of the first known surveyors, known as rope-stretchers or harpedonaptai.
Rope-stretchers, likely from the Tomb of Menna. Metropolitan Museum of Art, New York.
The rope-stretchers had one particularly neat little trick, which involved marking their rope with knots. You can just about make out these knots on the image above. The knots marked equal distances and would be helpful if you wanted to measure land, but also if you needed to make a right angle out of a rope. In that case, you could have a triangle with sides of 3, 4 and 5. The Pythagorean triangle is always a right-angled triangle, where the squares on the two shorter sides equal the square on the longest side, known as the hypotenuse. 21
< Lighthouse of Alexandria (1572), by Philip Galle, Metropolitan Museum of Art, New York. A legend tells that the Elements were written by Euclid for King Ptolemy I Soter, who tried learning mathematics but couldn’t find all he needed to learn in one place. When the King then asked, after having seen the thirteen books, whether there is a faster way to learn it, Euclid famously (and supposedly) replied “There is no royal road to geometry”.
THE MAGICAL FOREST OF LINEAR PERSPECTIVE Whilst perspective as a method of drawing and painting to create a realistic illusion of depth and space wasn’t completely unknown, its mathematical foundations weren’t laid until the 15th century. Between 1415 and 1420, Italian architect Filippo Brunelleschi (1377–1446) realised that linear perspective could be used to create pleasing effects in the built environment by drawing buildings before constructing them. This allowed architects to see how their creations would appear to those walking around and approaching them from various angles, enabling this to be factored into the planning stages. Apart from piquing architects’ interest in using linear perspective, it also inspired painters to create images of great beauty. A young friend of Brunelleschi, Leon Battista Alberti (1404–1472), crafted a mathematical framework in both its theoretical and practical aspects, transforming linear perspective into a usable method. He began with the actual frame of the image: this was defined by the horizontal line – which is where our scene begins – while the floor was composed of tiles conveniently placed in such a way that they proceeded from that horizontal line. The unit he used was braccia meaning an arm in Italian, a foundational unit of measurement that would scale well with the rest of the human body. The Renaissance artists adopted this mathematical method en masse. One of the most famous, and earliest, of these works was The Hunt in the Forest, painted around 1470 by Paolo Uccello (1397–1475), but there were others later, and in other parts of Europe. 44
The Ideal City, c. 1470, tempera on panel. In Galleria Nazionale delle Marche, attributed to Luciano Laurana, formerly to Piero della Francesca, and earlier to Francesco di Giorgio.
> Figure from the 1804 edition of De Pictura by Leon Battista Alberti, composed in Latin in 1435 and published in 1450, showing how to convey the sense of space in a painting using a series of lines on the floor that converge into a central point (the vanishing point).
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Allegory of Geometry (1649) by Laurent De La Hyre, Fine Arts Museum, San Francisco. The painting on the easel depicts the approach to perspective used by La Hyre and his colleagues at the French Royal Academy which he helped establish.
A PATH FROM ANGELS TO DEVILS Repeating patterns such as these are usually called wallpaper patterns, even if they are found on pavements, as they cover a flat plane. What’s even stranger (although perhaps not entirely surprising given our story of Kepler and the beautiful snowflakes – which are, after all, crystals) is that these so-called wallpaper patterns exhibit specific types of symmetries and were discovered by people interested in crystallography, the science of crystal formations and their properties. It turns out that there are exactly seventeen wallpaper symmetries. The classification of these symmetry groups is done by looking at what type of rotations, reflections, translations and glide-reflections a pattern has. Many patterns seem unlike one another at first sight, yet beneath their elaborate and beautiful faces they share the same wallpaper symmetry group. An example is shown in the next two images. Though different artistically, their underlying mathematics is exactly the same. They both belong to what is called the Wallpaper group p4g.
Regular division drawing, sometimes called Angels and Demons. No. 45, M. C. Escher, Christmas 1941.
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An example of p4g wallpaper pattern, from an undated historical pattern from China.
To understand this group of wallpaper patterns, let’s start with an object like, for example, a square sandwich! You cut your sandwich in half along a diagonal, and the cut will be your line of reflection. Then slide one half along the cut edge a bit, and you will have combined the reflection with a glide; this is known as a glide-reflection. The ‘g’ in the name of our group above, applied in the two images, stands for this glide-reflection. The ‘p’ stands for a primitive in a strictly mathematical sense, meaning this is a unit cell of this pattern, repeated then across the plane. And finally, the 4 in the name of this group stands for 4-fold rotation, meaning that you can rotate the motif around the centre four times, with each rotation being through a right angle, or 90 0. It is worth spending time to decipher this comparison between the two images and discover how they are similar and mathematically the same. Imagine that whilst doing this you are using paints, playing at recreating these, or similar, images, rather than using mathematical terms such as ‘reflection’ or ‘rotation’. If you then look at our two previous images they should, in your mind, blend together with ease. That is what the art of mathematics does to you when you engage with it. The devil may, indeed, be in the mathematical detail, and we will use this image of Escher again (as he did) when we look at some other mathematics created by playing with mathematical ideas, in Chapter 7. 71
Lycurgus Consulting Pythia, painting by Eugène Delacroix portraying the ancient Greek king Lycurgus seeking guidance from the Oracle of Delphi, 1835–1845, University of Michigan Museum of Art.
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PREDICTION
Wouldn’t it be great if we could predict the future? If we knew when danger was coming, we could take steps to avoid it! People have tried using various techniques to forecast the future, and with mathematics they have managed to do so to a certain degree. Many people since then have prayed to their gods, craving certainty. Part of the need to understand or predict the future is also about seeking clarity and guidance about what to do in certain situations, and the fear of what might happen if we don’t know the right action to take. What steps should we take to avert disaster? Many prophecies were made in an attempt to navigate uncertainty. YOU CAN ALWAYS ASK AN ORACLE Perhaps the oldest legend of this kind, and one that is directly related to mathematics, is the story of the Delian oracle. A great plague was ravaging a number of Greek islands and, after all their individual prayers to stop the plague went unanswered, the inhabitants of the Island of Delos, decided to ask an oracle for help. Therefore, they visited Pythia, the high priestess of the Temple of Apollo. Pythia gave them an answer: they could stop the plague if they prayed in double measure to their god Apollo. And their altar, which was a cube, should be double its current size. A cube which is double that of an existing cube thus became a mathematical problem and, in the history of mathematics, is known as the Delian problem.
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> Boy Building a House of Cards, by Jean-Siméon Chardin, 1735, oil on canvas. Waddesdon Manor, Buckinghamshire, England.
THE ASCENT BEYOND THE VISIBLE A couple of decades after Abbott wrote about this flat world, a son of his friend, Oxford mathematician Charles Hinton (1853–1907), wrote a book about the fourth dimension. In contrast to Abbott, he tried to teach his readers how to see objects in a dimension higher than ours – the fourth dimension. Apparently, by quickly circling your eyes and scanning the differentcoloured cubes, you might be able to visualise a four-dimensional cube, or hypercube. This method didn’t work with most people (you can try it for yourself), but it did with one of his wife’s younger sisters, Alicia Boole Stott (1860–1940). This talented girl later became a mathematician, without spending a day at a university. She did have three personal mathematics teachers, though – Hinton, and her own parents, Mary and George Boole. Mary wrote mathematics textbooks for teaching children, and George was a famous mathematician (also without the usual training) who invented a method that now underlies all computer systems. We mentioned him in the last chapter with Leibniz and his work on binary system. Boole (1815–1864) wrote a book An Investigation of the Laws of Thought, bringing humanity closer to mechanising thoughts by using only two values, 1 and 0. Each statement can be broken down to the smallest units which can, through logic, then be assigned values of either 1 or 0, where 1 would be true, and 0 untrue. But let’s return to George’s daughter, Alicia Boole Stott, who was the only person at the time to visualise the four-dimensional bodies. In her study of them, she used the same method that Abbott introduced whereby a three-dimensional body such as a sphere, when passing through a two-dimensional world, leaves a two-dimensional trace (a circle). Only now that she was dealing with four-dimensional bodies, she sought to find the three-dimensional traces they would leave behind when they passed through our three-dimensional world. She worked out these three-dimensional traces, made their models, and through this insight influenced further developments in higher-dimensional geometry.
> Frontispiece of The Fourth Dimension, Charles Hinton, 1904, London.
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Circle Limit IV (Heaven and Hell), M.C. Escher, woodcut in ochre and black, printed from two blocks, July 1960.
Hyperbolic geometry can be imagined through a model of space known as the Poincaré disk, after its inventor Henri Poincaré (1854–1912). This is a two-dimensional representation of hyperbolic geometry where the infinite curved plane is mapped onto a disk. As you approach the boundary of this disk, you become infinitely small, never to reach the actual edge. Maurits Cornelis Escher (1898–1972) who played with symmetries of angels and demons (Chapter 3), applied this symmetry pattern to the Poincaré disk and called this image Circle Limit IV. In this case all the angles are preserved, but the distances are compressed as you go further from the centre. The angels and demons meet again, but now in a curved space. Whilst the invention of non-Euclidean geometries didn’t immediately affect how we teach geometry in schools (and still doesn’t), it led to further research which saw the birth of relativity theory and its aftermath. Such work persists and mathematicians (and others) still work on trying to understand and determine the geometry of space and, ultimately, its physical structure. This was a further example that even a system considered for millennia to be impervious to objection, could be seen as just one of several possible mathematical systems. Whilst new adventures in modern geometry may be exciting, many may also miss the old comforts of non-curved spaces, and feel that something should be done about that. But how to free ourselves from such existential insecurities?
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< Portrait of Luca Pacioli, attributed to Jacopo de’Barbari (1495), Museo di Capodimonte, Italy.
Mathematics is art – created not with colours or forms, but through abstract thought. This is the central idea behind Art of Mathematics by acclaimed author Snezana Lawrence. In this work, she explores the history of mathematics through themes such as the elements, ratio, and dimensionality, highlighting the inherent beauty of the discipline. Along the way, she presents a rich selection of artworks inspired by mathematical ideas. Through this approach, Lawrence reveals what mathematical beauty truly is: the ability to trace a thread through history that weaves together the stories of people and cultures around a single, original abstract concept.