# A Platonic Dalí in Washington

Salvador Dalí (1904-1989) is one of the most important surrealistic painters. This eccentric Catalan painter finished in 1955 the Sacrament of the Last Supper after nine months working in it and this painting is mathematically remarkable because of the dodecahedric form over Christ and the twelve apostles. The thirteen sacred figures are having supper when it’s not at night and we can see a beautiful Catalan landscape behind the main scene. If Dalí wanted to capture this Biblical moment, why the dodecahedron was painted over it? The dodecahedron and its twelve sides may be related to the twelve apostles. Furthermore, a philosophic point of view can show to us a platonic idea for this famous painting. Plato (c.428 BC-c.347 BC) wrote in the IVth century BC the Timaeus where he associated the four basic elements to fours regular polyhedra: tetrahedron was the fire, hexahedron was the earth, octahedron was the air and icosahedron was the water. Thus, the universe was in perfect harmony with Mathematics!

Firstly, Plato described two kinds of triangles as the base of all the nature:

In the first place, then, as is evident to all, fire and earth and water and air are bodies. And every sort of body possesses solidity, and every solid must necessarily be contained in planes ; and every plane rectilinear figure is composed of triangles ; and all triangles are originally of two kinds, both of which are made up of one right and two acute angles ; one of them has at either end of the base the half of a divided right angle, having equal sides, while in the other the right angle is divided into unequal parts, having unequal sides. These, then, proceeding by a combination of probability with demonstration, we assume to be the original elements of fire and the other bodies

With these two triangles, Plato built the squares and triangles which had to be the sides of the regular polyhedra:

Now is the time to explain what was before obscurely said : there was an error in imagining that all the four elements might be generated by and into one another ; this, I say, was an erroneous supposition, for there are generated from the triangles which we have selected four kinds: three from the one which has the sides unequal ; the fourth alone is framed out of the isosceles triangle. Hence they cannot all be resolved into one another, a great number of small bodies being combined into a few large ones, or the converse. […]. I have now to speak of their several kinds, and show out of what combinations of numbers each of them was formed. The first will be the simplest and smallest construction, and its element is that triangle which has its hypotenuse twice the lesser side. When two such triangles are joined at the diagonal, and this is repeated three times, and the triangles rest their diagonals and shorter sides on the same point as a centre, a single equilateral triangle is formed out of six triangles ; and four equilateral triangles, if put together, make out of every three plane angles one solid angle, being that which is nearest to the most obtuse of plane angles ; and out of the combination of these four angles arises the first solid form which distributes into equal and similar parts the whole circle in which it is inscribed. The second species of solid is formed out of the same triangles, which unite as eight equilateral triangles and form one solid angle out of four plane angles, and out of six such angles the second body is completed. And the third body is made up of 120 triangular elements, forming twelve solid angles, each of them included in five plane equilateral triangles, having altogether twenty bases, each of which is an equilateral triangle. The one element [that is, the triangle which has its hypotenuse twice the lesser side] having generated these figures, generated no more ; but the isosceles triangle produced the fourth elementary figure, which is compounded of four such triangles, joining their right angles in a centre, and forming one equilateral quadrangle. Six of these united form eight solid angles, each of which is made by the combination of three plane right angles ; the figure of the body thus composed is a cube, having six plane quadrangular equilateral bases.

And then…

There was yet a fifth combination which God used in the delineation of the universe.

Great Plato! Which is this fifth combination? The answer is now so easy: the dodecahedron! The platonic solid which was used by God to draw our universe.  It’s easy to understand the reason why this harmonious figure is presiding the scene.

Location: National Gallery of Art in Washington (map)