The Polyhedra of M.C. Escher
amazingly original mathematical artist Maurits Cornelis
Escher (1898-1972) created imaginative images which give a unique perspective
on this world and others. While most famous for his tessellations, Escher
also had a strong affinity for polyhedra. He shared this interest with
his brother, B.G. Escher, who was a geologist and wrote a text on crystallography.
For example, this 1961 lithograph Waterfall features the compound
of three cubes and the first
stellation of the rhombic dodecahedron as ornaments atop the two towers.
The following works of Escher's feature strong polyhedral content and
are worth looking up in your favorite books
of his work:
Sphere with Fish, beech, 1940, is a spherical carved tessellation
with tetrahedral symmetry.
Sphere with Angels and Devils, maple, 1942, is another spherical
carved tessellation with tetrahedral symmetry.
Reptiles, lithograph, 1943, features a dodecahedron.
Crystal, mezzotint, 1947, features the compound
of the cube and octahedron.
Stars, wood engraving, 1948, illustrated below, features the compound
of three octahedra, a compound of two
cubes, and the stella
octangula, all in both outline and solid form, along with dozens of
Double Planetoid, wood engraving, 1949, is based on the stella
Order and Chaos, lithograph, 1950, features a small
stellated dodecahedron penetrating a sphere.
Gravity, lithograph, 1952, features a perforated small
Tetrahedral Planetoid, woodcut, 1954, is based on the tetrahedron.
Polyhedron with Flowers, maple, 1958, is a carving with icosahedral
Flatworms, lithograph, 1959, incorporates the space-filling properties
of the tetrahedron and octahedron.
Waterfall, lithograph, 1961, illustrated above, features the compound
of three cubes and the first
stellation of the rhombic dodecahedron.
stellation of the rhombic dodecahedron, pictured at right, seems to
have been a favorite polyhedron of Escher's. It shows up in at least three
of his works:
it decorates the top of one of the towers in Waterfall,
it was the subject of a wooden take-apart puzzle he designed,
it is featured in a study for Stars (illustrated in the paper by
Arthur Loeb listed in the references) instead
of the compound of three octahedra
which appears in the final version, below.
Exercise: Name all the polyhedra and polyhedral
compounds in Escher's Stars:
compound of three octahedra, (both
a solid and an edge model)
compound of two cubes with common 3-fold
axis, (both a solid and an edge model)
octangula (compound of two tetrahedra), (both a solid and an edge model)
compound of cube and octahedron,
all five platonic solids.
Note that some of these are a bit small to make out clearly in this
reproduction. The three edge models follow the style of Leonardo
Waterfall and Stars are copyright 1996
by M.C. Escher / Cordon Art - Baarn - Holland. All rights reserved. Used