The mathematics of cubism
The twentieth century was the scenario of great and
big transformations. Changes occurred in almost
every field of human knowledge. Everywhere, great
minds pushed firmly toward a deconstruction and
reconstruction of the conventional wisdom.
Of a 3D cube we can at most only see 3 of
the 6 sides it has. However, a 4-dimension
being can see all 6 sides at the same time.
What cubists did was to deconstruct
the 3D cube in all of 6 facets. Then
paint the faces on a 2D plane: the
canvas. Now we can “see the cube”
as if we were from the fourth
dimension.
The theory of relativity was of enormous
impact in many adjacent sciences to physics
like astronomy and mathematics. The dilation
of time and space introduced by the equations
by H. Lorentz, the expansion of the universe
observed by E. Hubble, the introduction of
new kinds of infinite numbers, consequences
of the theory of set by G. Cantor, all
contributed to new horizons and frontiers in
Deconstruction and reconstruction of a cube.
the human thought.
Picasso: Portrait of Dora Maar (1937)
Juan Gris: The Guitar (1918)
Thinking of time as a new dimension to be added to our
physical world, thus making the universe a fourdimensional structure, was immediately a great impact
in the arts as in the sciences. For some reasons, the
plastic artists immediately grasped the importance of
time as a dimension and as a new opportunity to
incorporate the time element into their paintings.
Braque: Viaduct at L’Estanque (1908)
The idea that in the fourth dimension a body can be seen from many points of view
at the same time was the lightning spark that artists were unconsciously waiting to
rebel against the traditional flat visualization of the environment. In Picasso’s
Portrait of Dora Maar, he painted the lady as looking to the front and to the left
simultaneously. She also seems to be sitting and standing at the same time.
Painting by itself is a difficult activity, not merely because of applying
new concepts of color harmony, composition, and perspective, but
because it is placing a 3D world into a 2D canvas. So, bringing a fourth
dimension into a planar 2D canvas seemed an impossible dream.
However, in the hands of the cubists, they faceted the objects into 2Dsubstructures and placed them again one near the other, as if they
were seen from the fourth dimension. With some antecedents, this was
the way that cubism was born. Cubism is also an abandonment of the
traditional perspective. Cubism was a subject of intense study by Pablo
Picasso, Juan Gris, and Georges Braque.
Albert Einstein (1879-1955).
Father of the theory of
relativity. He introduced time
as the fourth dimension.
Pablo Picasso (1881-1973).
Deeply explored the
deconstruction and
reconstruction of reality
Ref.: Aczel, A. D. (2006). The Artist and the Mathematician: The story of Nicolas Bourbaki, the Genius Who Never Existed.
© E. Perez http://4DLab.info
Thunder’s Mouth Press.