Escher’s mathematical understandings was mostly visual and intuitive. His architectural constructions and geometric compositions are remarkable for their perspectival distortions: at first sight, they may appear perfectly plausible, but after careful inspection, they are revealed to be impossible.
The stars that hang in Escher’s black outer space are geometric forms with symmetry of faceted jewels.
Surrounded by geometric models in his studio, Escher readily admitted his awe of these forms and included them in many of his prints.
Admired and exalted from earliest times, the Platonic solids are the most perfectly symmetric of all convex polyhedra. Each form is faceted with copies of a single regular polygon – all sides and all anglesof all faces are equal.
In addition, every corner (vertex) of the solid is the same – the same number of faces meet there, and the angle at which adjoining faces are inclined to each other is always the same. Mathematicians call these regular polyhedral.
In each of them, every possible requirement that aspects of the form be alike (congruent) is fulfilled. These requirements are so severe that only five forms can meet them all.
Their names come from the original Greek and tell us how many faces they possess. Three are faceted with equilateral triangles: tetrahedron, octahedron, and icosahedron. The cube, faceted with squares at right angles to each other, is the most familiar form of all.
Its name comes from the Greek word for “dice” – cubos. The dodecahedron is faceted with pentagons; it is perhaps the most admired since it is the least obvious of the solids timagine.
It seems that Pythagoras had already discovered the properties of ‘cosmic figures’ while Plato associated them with the fundamental elements. In “Timaeus” (360 BC) he connects the tetrahedron to fire, the cube to earth, the octahedron to air and the icosahedron to water.
The dodecahedron, on the other hand, is a symbol of the universe. This special attribution derives from the presence of the pentagon, a figure which – containing within itself the golden section – contained the idea of harmony and mathematical perfection more than any other.
Escher’s little dragon, at the pinnacle of his fanciful journey, gives a snort of power atop a dodecahedron in the print Reptiles.
If just one of the requirements for a solid to be regular is removed, a large collection of other highly symmetric forms can be discovered. The Archimedean (or semiregular) polyhedral allow two or more types of regular polygons as faces but fulfill all other requirements of regular polyhedra.
Thus, all sides of all faces are equal; every corner of the solid is the same (surrounded in the same manner by the faces that meet there). There are thirteen semiregular solids; we have included just one in our collection.
The cuboctahedron (cube+octahedron) has as its faces six squares and eight equilateral triangles. Each of its twelve vertices (corners) is surrounded by two triangles and two squares arranged so that the two squares separate the two triangles.
The name of this solid suggests its derivation – it can arise either by slicing off corners of a cube or by slicing off corners of an octahedron. It can also arise as the solid that is intersection of an octahedron with a cube as shown in Escher’s print Crystal.
Drawing is really a deception. We are used to thinking of the world as something three-dimensional, but drawing is only two-dimensional. Escher regarded this as a situation of conflict, which, if carried to its extreme consequences, generated illusions, and paradoxical situations.
