Taking a ‘Secant’ to Look at Art from a Different Angle

Words by Tia Finch

Trying to link maths and art, you might believe, is the first sine of madness, but understanding how they connect is important for exploring human cognition. Both maths and art are results of the human brain processing its surroundings and expressing solutions to understand its environment. The biggest difference is that maths needs logic, whereas art doesn't. 

If you’ve ever sat in a GCSE art class you were probably taught to simplify still life drawings into simple shapes such as spheres, cylinders, or cubes, which was an idea heavily employed by Cézanne. Braque and Picasso were key in taking inspiration from him to found the Cubist movement of the early twentieth century, merging multiple perspectives into the same artwork. However, the involvement of maths runs deeper in Cubism than at face value.  

Concepts of the 4th dimension were introduced to Picasso by mathematician Maurice Princet, including a work published by another mathematician, Esprit Jouffret.

Figure 1: Diagram of the boundaries of a 4D surface, Esprit Jouffret https://www.lindahall.org/about/news/scientist-of-the-day/esprit-jouffret/

Figure 2: Exploded diagram of a hypercube, Esprit Jouffret  https://www.lindahall.org/about/news/scientist-of-the-day/esprit-jouffret/

Art historian Linda Henderson, noticed a difference in his artworks after Picasso was introduced to the reading. She spotted the similarities between figures 1 and 2 (from ‘Traité élémentaire de géométrie à quatre dimensions’ – meaning ‘Elementary Treatise on the Geometry of Four Dimensions’ – by Jouffret) and ‘Daniel-Henry Kahnweiler’ by Picasso.

Figure 3: ‘Daniel-Henry Kahnweiler’, Picasso, 1910 https://www.artic.edu/artworks/111060/daniel-henry-kahnweiler

Geometry and art go hand in hand (from my angle, they're acute pairing). Being introduced to mathematical ideas influenced Picasso’s artworks - he hasn’t been the only artist to draw inspiration from geometry.

M.C. Escher, a Dutch mathematician and artist, also looked to mathematics to create art. He incorporated impossible objects into his artworks: the impossible cube, the Penrose square, and Penrose triangle, found in ‘Belvedere’, ‘Ascending and Descending’, and ‘Waterfall’ respectively.

Figure 4: ‘Waterfall’, Escher, 1961 https://www.artchive.com/artwork/waterfall-maurits-cornelis-escher-1961/

Additionally, Escher experimented with divisions of planes through tessellations which are evidenced in his many versions of ‘Angels and Devils’. His first version, the ‘Study of Regular Division of the Plane with Angels and Devils’, uses regular tessellations where all tiles must be identical repeating polygons, as opposed to irregular tessellations which would result in gaps and overlaps.

In regular tessellations, there are only three regular polygons that can be used. This is because the total angles around a point must add to exactly 360°. So, for a regular shape to be able to regularly tessellate, 360° must be divisible by the internal angles for no spaces to remain between the shapes. For instance, an equilateral triangle has three internal angles where each is 60°, so 360 ÷ 60 = 6. Therefore, six triangles can be placed around a single point of a regular triangular tessellation. Squares and hexagons are the two other possible polygons. Escher used this maths by viewing the angels and devils as complex polygons, allowing him to join them together to divide the plane. 

He later developed this into 3D space by investigating spherical tessellations, which have five regular tilling possibilities: {3,3}, {4, 3}, {5, 3}, {3, 4} and {3, 5}. These are denoted in the forms {a, b}, where a is the number of regular polygons joined around a vertex, and b is the number of sides the repeated polygon has, so a {5,3} spherical tessellation has 3 pentagons which have 5 sides to a point.

Figure 4: {5,3} and {3,5} tessellations. https://store.steampowered.com/news/posts/?feed=steam_community_announcements&enddate=1715181703

His explorations progressed to hyperbolic space after Escher and mathematician Coxeter began working together after meeting at the Amsterdam 1954 International Congress of Mathematicians. Coxeter produced a diagram that looked like a 3D triangular tiling of a sphere mapped in 2D but, upon closer inspection, was something more complex – a triangular tiling that employed the Poincaré disk. This is a tiling that approaches infinity when mapped onto a plane of negative curvature (for comparison, a sphere’s surface – elliptic space – has positive curvature whilst Euclidean space, the flat plane you are familiar with, has no curvature).

Figure 5: Tiling based on the Poincaré Disk, Coxeter https://www.researchgate.net/figure/A-diagram-of-the-Poincare-disc-model-Credit-Institute-For Figuring_fig1_225365373

Figure 6: ‘Angels and Devils, Circle Limit IV’, Escher, 1960 https://www.d.umn.edu/~ddunham/dunbr09tlk.pdf

Escher adapted Coxeter’s diagram into a piece with ‘Angels and Devils’, resulting in ‘Circle Limit IV’. 

Similarly to the Poincaré disk, fractals can also be used to create tilings. The first fractal images generated by software were created by Benoit Mandelbrot, the most famous being the Mandelbrot set. Images formed based on iterations of z2 + c where ‘z’ is any complex number, z = x + iy, and c is a constant. These iterations to infinity are called Julia sets. Julia sets within the Mandelbrot set are stable and tend towards a fixed point, whereas unstable Julia sets are beyond the Mandelbrot set and tend to infinity. Julia sets can be broken down into isolated points named Fatou dust. Where Fatou dust is thicker, points are densely plotted by Julia sets so the colour appears darker. Therefore, the inside of the Mandelbrot set is black and the less densely plotted margin of the Mandelbrot set is red/orange.

Because Julia sets are iterated infinitely, the boundary of the Mandelbrot set can never be reached. You may be wondering if I’m going off on a tangent about fractal geometry, so here are some magnification images of increasing scale.

Figure 7-11: The Mandelbrot Set https://mathigon.org/step/talks/mandel-zoom

By exploring the connections between art and maths, I hope to show that maths is more than just sums; beneath the layers of problem-solving there is an abstract beauty that can be found. Hopefully, this post will encourage someone to take inspiration from maths and create their own mesmerising artwork.

Glossary

Complex Number – has both imaginary and real numbers.

Fractal - a shape constructed by iteration where self-similarity occurs.

Hyperbolic Space – rejects the fifth hypothesis of Euclidean  geometry which is that two parallel lines will remain equidistant. In  hyperbolic space, two parallel lines will diverge. 

Figure 12: Visual of different spaces https://math.stackexchange.com/questions/4384197/relationship-between-hyperbolas-and-hyperbolic-spaces

Hypercube – known in the fourth dimension as a tesseract, it is a  representation of a square or cube in the nth dimension.  

Imaginary Numbers – √(-1) = i, Therefore, √(-4) = √(4) * √(-1) = 2i

Impossible object – optical illusions that are projected logically onto a 2D plane, but could not exist as a solid, real  3D object. They are paradoxical shapes. 

Poincaré Disk – introduced by Henri Poincaré, one line in the Euclidean plane is an arc on the disk, where two arcs  that meet at right angles are perpendicular lines. The Poincaré disk is a map of Euclidean geometry onto hyperbolic space.

Tessellation - repeated division of a plane.