Writers, as well as composers and visual artists, sometimes let mathematical ideas inspire their art. Join us as we compose with Fibonacci, take golden photographs and listen to the number π.

Fiction with Fibonacci

When I search my memory for mathematically inspired fiction, I immediately think of Dan Brown's bestseller The Da Vinci Code. The main character, Robert Langdon, gets involved in a murder investigation and has to solve mathematical puzzles to find the killer. But when I look for a more precise link between literature and mathematics, I instinctively turn to poetry. Just like mathematics, poetry is filled with patterns. Verse measures such as hexameter, haiku and sonnet follow an almost mathematically determined structure with a certain number of lines, stressed syllables and rhythmic rhymes. Admittedly, modern poetry has in many ways freed itself from such formal requirements, but in the collection Alfabet from 1981, the Danish poet Inger Christensen lets the collection of poems be curbed by a very special mathematical structure. The number of lines in the poems increases according to the famous sequence of Fibonacci numbers, where each number (from the third onwards) is the sum of the previous two.

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610…

The book begins with poems with 1, 2, 3 and 5 lines respectively, and ends with poem number fourteen with 610 lines. Does Christensen want to say something with her choice of the Fibonacci's sequence? Maybe that the poems – or we – are the sum of what preceded them.

A golden photograph

When comparing the numbers of the Fibonacci's sequence in pairs, something remarkable emerges. The ratio of two consecutive elements approaches a specific number the further into the sequence of numbers you get.

The number that the ratio is approaching is

Given two consecutive numbers in the Fibonacci sequence, the latter number will thus be approximately 61.8% larger than the previous one. Although the numbers are constantly changing, this ratio between them will persist. This curiosity becomes remarkable when one discovers that the ratio of 1: 1.618 is not a ratio like any other. It has been known by mathematicians since ancient times and has a reputation for being particularly beautiful. When you divide a distance into two parts, so that the longer part is 61.8% longer than the shorter one, you say that you have divided the distance into the golden section. This relationship has come to be perceived as particularly pleasing to the eye and has throughout history influenced both visual art and architecture as well as photography.

But it is not only established artists who take advantage of the golden ratio. Image editing software like Photoshop, and photo apps like Wise Camera, have a golden grid (phi grid) that you can use to compose a harmonious photo. The golden grid divides each side of the photograph into three parts, so that the two outer parts are 1.618 times as long as the middle part. By placing the parts of the motif in line with the grid, and prominent objects where two lines meet, the idea is that you can create a well-balanced photo.

Mozart’s golden section

Creating a harmonious composition is not a problem reserved for photographers and artists. Even composers need to balance the parts of their musical works. Mozart, who was known to be interested in mathematics, often chose to use the golden section to create the right balance in his compositions. Almost 50% of his piano sonatas are composed with some regard to the golden section. This is most clearly seen in his very first piano sonata in C major. Of the first 100 measures, the opening exposition is exactly 38 measures, a division as close to the golden ratio as he could get (cf. 61.8 / 38.2 = 1.618…). The British pianist Nicholas Ross is convinced that the division was intentional: “…that you’re not even one measure away, is quite striking. This was planned ", he commented on the radio program Art of now on BBC4.

Listen to π

The golden ratio is not the only mathematical constant that has inspired musicians. Even π – the most significant constant in all of mathematics – has made music creators curious. Some of them have even asked the quirky but creative question Can we listen to π? – and came to the conclusion that the answer is yes. By translating each digit in the decimal development of π into a tone, the number π has become a piece of music of its own.

A patchwork quilt of mathematics

For those who need a reminder, the number π is the ratio between the circumference of a circle and its diameter. That π ≈ 3.14 thus means that the circumference is a little more than three times as long as the diameter.

The decimal expansion of the number π starts 3.141526… and never ends. When you think about it, it is an amazing idea that infinity is inherent in every circle. But we are so used to the simple geometry of circles, that they do not offer any visual enjoyment. Artists who want to visualize π must therefore take other approaches. One of them is the mathematician and artist John Sims. He has created multicolored patchwork quilts, where each square corresponds to a decimal in the number π, and each number between 0 and 9 has its own color. The decimals emerge in a spiral pattern from the center of the blanket and illustrate the infinitely alternating stream of numbers in the decimal expansion of π. They roll forward completely without discernible patterns.

Tessellation

The squares in John Sims patchwork quilts completely cover a flat surface. To cover the plane with geometric figures in this way, without overlap or spaces, is called tessellation. We tesselate every time we put tiles in the bathroom or lay tiles in the garden path.

How many possible tessellations are there? The answer to that question depends on the type of figures allowed, but one condition must always be met: the angles in the corners where the figures meet, must add up to a full revolution, 360°. If you want to do the tessellation with a single type of polygon, where all sides are the same length, there are therefore a limited number of alternatives. Since four right angles and six 60° angles both form a whole revolution (4 · 90° = 360° and 6 · 60° = 360°) you can – as you can see above – tessellate the plane with both squares and equilateral triangles. Are there more opportunities?

The table shows that only the angle of one other polygon evenly divides the 360° – the hexagon’s 120°. We can thus cover the plane with regular hexagons if we let three hexagons meet in each corner (3 · 120 ° = 360 °). In fact, no other regular polygon does the trick. In the table we see that the angle becomes larger as the number of corners in the polygon increases. The only numbers greater than 120 that evenly divide 360°, are 180° and the number 360° itself, but 180° and 360° cannot be angles in a polygon.

If we only allow regular polygons, there are only three ways to tessellate the plane: with squares, triangles or hexagons. If we allow other types of figures, however, there are plenty of alternatives. One who explored these possibilities was the Dutchman and artist M.C. Escher (1898–1972). He has become world famous for his iconic tessellations with figures such as birds, angels or demons.

To complete many of his works, Escher needed to solve mathematical problems, but Escher himself was not a trained mathematician. However, he corresponded with several mathematicians, including Roger Penrose and Coxeter, whose impossible figures and hyperbolic tessellations formed models for some of Escher's most striking artwork.

Summary

M.C. Escher has said:

With all expressions of art, whether it is music, literature or visual arts, it is first and foremost a matter of communicating to the outside world and make sensory perceptible a personal thought, a striking idea or an inner emotion.

As the examples above show, the striking idea can be mathematical. Sometimes, as in the case of Christensen's poems, Mozart's piano sonata and the golden grid of photographs, the mathematical idea is a framework for the creative process – it constitutes a form to populate with content. In other cases, the mathematical idea is the very object of creation. This is the case, for example, for the artists and musicians who have tried to make us see and listen to the speech π. The idea behind the artwork in these cases becomes as breathtaking as – or perhaps more breathtaking than – the visual or auditory experience. It's a bit like a song where you appreciate the music, but where the truly emotional experience comes only when you listen to the lyrics.

For a mathematician, however, an idea does not need to be dressed in colors, shapes or tones for it to offer an aesthetic experience. For them, mathematics is its own art form.

References and further reading

BBC4, Art of Now: A mathematicians guide to beauty

Christensen, Inger (1981) Alfabet, Modernista

Christersson, Malin, Tesselering

Offical webpage M.C. Escher: www.mcescher.com

Ornes, Stephen (2019) Math Art – Truth, Beauty and Equations, Sterling Publishing

Posamentier, A.S. & Lehmann, I. (2007) The (Fabulous) Fibonacci Numbers, Prometheus books

Wikipedia, Escher https://en.wikipedia.org/wiki/M._C._Escher

Wikipedia, Tessellation https://en.wikipedia.org/wiki/Tessellation