The two central figures  also define a basis for the foundation of knowledge, human thought, philosophy, for all of mankind. Their extended arms, Plato’s towards the skies, and Aristotle’s horizontal extension, palm facing down, also form an orthogonal basis, in such a way that they define both a pictorial and mathematical frame of reference:

The School of Athens, detail of Plato & Aristotle

Plato’s face is actually Leonardo da Vinci’s, who illustrated Luca Pacioli’s mathematical work entitled ‘De divina proportione’, or ‘Divine Proportions’.  The subject was mathematical and artistic proportion, especially the mathematics of the golden ratio $latex \varphi = 2 \cos(\pi/5) = (1+\sqrt{5})/2 \approx 1.618$, and its application to art and architecture. Leonardo da Vinci drew the illustrations of the regular solids in ‘De divina proportione’, while he took mathematics lessons from Pacioli. This remarkable irrational and algebraic number is the only positive number that has the property that multiplied with itself amounts to adding (the number) one to it: $latex \varphi^2 = \varphi + 1$.

There is an intimate relation between the Golden Ratio and Fibonacci numbers. For a historical overiew and miconceptions of the usage of the golden ratio, please read the article “Misconceptions about the Golden Ratio” by George Markowsky. Another worthwhile, more accessible publication on the topic: “The Golden Ratio: The Story of Phi, the World’s Most Astonishing Number“, by Mario Livio.

Additionally, because of the intimate relation between certain Platonic Solids, their Symmetry Groups, and the Golden Ratio, some of the most intricate 4-Dimensional models of the Universe based on such symmetric transformations are almost entirely described by $latex \varphi$. For more details, please read the corresponding section entitled “Cosmology and the Golden Ratio“.

Islamic Art, since the 7th century AD, is also one of the original forms of geometric abstraction:

Bowl, 12th–13th century – Ghaznavid – Afghanistan

For more details on Islamic Art and Geometric abstraction, please see the following links:

• Geometric Patterns in Islamic Art
• Pattern in Islamic Art

An interesting fact underlying Islam (and pre-islamic cults) lies in the geometry associated with the concept of the “Tawaf“, or circumambulation of the Kaaba cube at Mecca, during the holy pilgrimage of the “Hajj” and the “Umra“. The geometric figure described by the combination of the human motion and the architecture of this sacred stone is that of the spherical cube:

Circumambulation (or Tawaf) of the Kaaba at Mecca

Da Vinci’s Vitruvian Man and the Spherical Cube:

Then came the revolution of the 20^th century abstraction, and the revival for mathematics and geometry as a subject for fine arts. Mondrian believed that mathematics and art were closely connected. He used the simplest geometrical shapes and primary colors (blue, red, yellow), used in the logo of a well known search engine. It is believed that he often used the Golden Ratio in the elaboration of some of his works.

Composition in Red, Yellow, and Blue – Piet Mondrian – (1926)

Mondrian’s point of view lies in the fact that it shape is possible to create any object with basic geometric shapes as well as any color can be created with different combinations of red, blue, and yellow. For more on Mondrian, Music and Mathematics, see the french article “POETIQUE DE MONDRIAN : LA MUSIQUE D’AVANT TOUTE CHOSE“.

50 years later, this is mathematically formalized in numerical methods such as finite element methods, where a finite number of similar geometric elements (i.e. squares, or triangles), of different sizes, approximate a certain region (2D, 3D, …):

Unfitted Finite Element Method – 3d “Deco-Cube”

Wassily Kandinsky, and more notably László Moholy-Nagy, both early participants and teachers at the Architecture / Social / Design school of the German Bauhaus set in Weimar and then Dessau, were also adepts of using basic geometric figures as the fundamental blocks of their art work:

Composition VIII (1923) – Wassily Kandinsky

László Moholy-Nagy, Composition A.XX, 1924, oil on canvas, 135.5 x 115cm (Musée national d’art moderne, Centre Georges Pompidou, Paris)

Clearly there are many uses of simple geometric figures in Cubism, Surrealism and Metaphysical Art (see Giorgio de Chirico), and I will note of one particularly less known surrealist painter, the American artist Kay Sage, companion of French leading surrealist painter Yves Tanguy.

I saw three cities – Kay Sage, 1944 (Princeton University Museum)

In this late composition of Kay Sage, we see an interesting mix of free-flowing curves, and still geometric shapes, creating a deep sense of perspective into a line of no-escape, where surrealism and metaphysics converge in the realm of pure ideas, beyond the physical world.

The organized or structured surrealist art of Kay Sage is one step removed from total abstraction, where abstract geometric objects give rise to matter. For those somewhat versed in astrophysics, this not too far removed from Einstein’s general theory of relativity, where mass and energy are related to the geometry of the underlying space, also in the spirit of the Platonic theory of form, whereby ideal geometric shapes (and ideas) take precedence over the physical world.

Another closely related 20^th Century Hungarian artist, founder of the Op Art movement, also makes use of simple geometrical shapes to create optical illusions, and mesmerizing works. He was especially interested in the concept of the spherical cube.

Victor Vasarely, Vega Nor Improvisation

The spherical cube, perhaps the mathematical and geometrical reference that has been most used by artists throughout times, starting with Leonardo da Vinci with the drawing of the Vitruvian man. It may also be found in another very popular art style from the orient, Mandalas, and more specifically those of Tibetan origin. Mandalas are Buddhist icons using sacred geometry and symbols. The philosophy of sacred geometry is symmetry, balance, and relations among basic geometric forms that picture the cosmology of the universe.

Painted 19th century Tibetan mandala of the Naropa tradition, Vajrayogini stands in the center of two crossed red triangles, Rubin Museum of Art

A contemporary art piece centered on the Buddhist theory of the transmigratrion of the soul, also appealing to sacred geometry and spherical cubes, by Namgoong, Whan:

The Portrait of Transmigrating Multi-Morphologies

There is an additional relation between sacred geometry, spherical cubes, and today’s higher dimensional models of cosmology. See the article on cosmology and platonic solids. Many artists have since used the spherical cube, see for example the mathematician turned artist Piotr Kowalsky, shown at the Musée Pompidou:

Piotr Kowalsky – Identité n°2

Piotr Kowalsky – Identité n°2

Piotr Kowalsky – Identité n°2

Piotr Kowalsky – Identité n°2 ’73, Musée National d’Art Moderne

more on Piotr Kowalsky…

Jean-Pierre Raynaud

Container Zero, Jean-Pierre Raynaud, 1988

Container Zero, Jean-Pierre Raynaud, 1988, Musée National d’Art Moderne