The art of abstraction

Mathematics was at the centre of John Ruskin's art. Fascinated by form, pattern, proportion and symmetry in the world around us, Ruskin believed that mathematical knowledge underpinned both the technical proficiency and the 'analytical power' needed to compose a work of art.

Euclid's 300 BCE treatise, The Elements, was at the core of nineteenth-century education and Ruskin owned several editions of Euclid in his own library. In his teaching at school, university and Working Men's colleges, and in his textbooks, he proposed practical exercises based on Euclidean geometry to his students.

Placing geometry as the basic principle of his understanding of structures, Ruskin recognised the fractal qualities of the natural world. Although in the above sketch from Modern Painters, Ruskin was not concerned with the scaling quality of the arrangement of the tree, he composed an exercise that anticipated the definition of fractal geometry: proposing to divide each stem of the tree into two branches at an equal angle, with each branch three quarters the length of the preceding one.

Ruskin used geometry to assist the depiction of complex forms across the natural world, from trees and clouds to shells. He used diagrams to illustrate the use of geometric compositional rulings in drawing curved shapes – in particular, the perspective to the depiction of clouds. He believed that a firm grasp of the rules of proportion and perspective was necessary to capture ‘the expression of buoyancy and space in sky’.

Linear perspective, a device of Renaissance architects Filippo Brunelleschi and Leon Battista Alberti, is the scaffold for the conceptualisation of ideas by architects, engineers and designers.

Ruskin built particularly on German artist Albrecht Dürer, who published the first scientific treatment of perspective in a manual of geometric theory. Ruskin commented that Dürer's line is 'always decisive and always right'.

Although Ruskin's architectural works were steeped in gothic aesthetics, they were also studies in perspective. Building directly on Dürer and Palladio, he captured structural forms and analysed proportions with mathematical accuracy.

Just as Ruskin sought out precision in the world around him, he delivered it in his own work, climbing up ladders in palaces and cathedrals to note accurate measurements in his notebooks and diaries.

Beyond man-made structures, Ruskin used geometry to assist the depiction of a wider range of numerical, spatial and temporal parameters in nature.

Ruskin compared rock features to larger mountain forms in a manner anticipating fractal geometry: 'No mountain was ever raised to the level of perpetual snow, without an infinite multiplicity of form'. His diaries are full of diagrammatic images of particular rock formations in the bid to capture accurate outlines, and record his use of trigonometry to determine 'the angle of the right-hand precipice'.

Ruskin tried to draw the 'governing or leading lines' of natural features with respect to individual geometric shapes, 'not because they are the first which strike the eye, but because like those of the grain of wood in a tree-trunk, they rule the swell and fall and change of all the mass'.

This attempt to find geometrical structures in even the most irregular organic forms such as clouds or waves, highlighted Ruskin's dynamic sense of pattern formation. Ruskin was fascinated by the challenge of observing movement in the skies in the same way that contemporary mathematicians were trying to use calculus to understand dynamic systems but acknowledged he did not have 'the mathematical knowledge required for the analysis of wave-action'.

The impact of mathematics on Ruskin’s artistry brings out his core empirical principles of thought and expression.
He explained: ‘All drawing depends, primarily, on your power of representing Roundness … For Nature is all made up of roundnesses; not the roundness of 25 perfect globes, but of variously curved surfaces. Boughs are rounded, leaves are rounded, stones are rounded, clouds are rounded, cheeks are rounded, and curls are rounded: there is no more flatness in the natural world than there is vacancy’.