Spherical Geometry
From Prep08Wiki
Spherical geometry is as old as Euclidean geometry. Eratosthenes measured the circumference of the Earth in 300BC. Or even older - these Neolithic stone spheres date to c2000BC.
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Why Study non-Euclidean Geometry
Many M.C. Escher works involve spherical geometry (Sphere with Angels and Devils, Sphere with Fish, Concentric Rinds, ...) and hyperbolic geometry(Circle Limit I, Circle Limit II,..). Most spherical geometry in art enters through the use of spherical tessellations or polyhedra, especially the Platonic solids.
It's a beautiful topic. The classification of regular tessellations in the three geometries is a mathematical highlight of the Escher course. Students can use the material as fodder for creating their own artwork.
Spherical Geometry
Geometry takes place on the surface of a sphere, such as the Earth. The word geometry means 'Earth measurement'.
Geodesics play the role of straight lines. In spherical geometry, geodesics are great circles. From geodesics come geodesic segments, and then triangles and polygons. On a sphere, polygons "bulge" relative to flat polygons with the same vertices. This is a simple (if slightly incorrect) argument that the angle sums of spherical polygons are always larger than their flat counterparts. In particular, the angle sum of a spherical triangle is always more than 180°. The amount in excess of 180° is called the defect of the triangle. There is a surprising relationship that angle sum determines area. In particular, the fraction of the sphere's area covered by a polygon is its defect divided by 720°.
Look at Introducing Spherical Geometry to see ways to introduce these ideas into the classroom.
Spherical Tessellations
Use defect and area fraction to count faces of polyhedra. Example: escherwiki:Concentric Rinds. Concentric Rinds has 45°-90°-60° triangles, so each has angle sum 195°, defect of 15°, and area fraction 15/720 = 1/48, so there are 48 triangles in each rind.
Students can discover the five regular tessellations of the sphere by considering corner angles of polygons. The polygons must fit around a vertex and have angle sum greater than their Euclidean conterpart, so a process of elimination gives the five regular spherical tessellations, which correspond to the five Platonic solids.
Polyhedra
Plato, in The Timaeus, associates four of the Platonic solids with the four elements, Earth, Air, Water, and Fire, as well as a less clear fifth element for the dodecahedron that relates to the universe. Johannes Kepler illustrated Plato's association, and also related Platonic solids to the orbits of the planets (unsuccessfully).
Duality is a theme throughout Escher's work. Every plane tessellation has a dual tessellation, as do spherical tessellations and hence polyhedra. The Platonic solids come in dual pairs, with the tetrahedron being self-dual. Kepler gives some insight into the gender equity of his time with this quote on duality:
However, there are, as it were, two noteworthy weddings of these figures, made from different classes: the males, the cubes and the dodecahedron, among the primary; the females, the octahedron and the icosahedron, among the secondary, to which is added one, as it were, bachelor or hermaphrodite, the tetrahedron, because it is inscribed in itself, just as those female solids are inscribed in the males and are, as it were, subject to them, and have the signs of the feminine sex, opposite the masculine, namely, angles opposite planes.
Some Escher artwork with themes of duality:
One can go further with combinatorics of polyhedra. Students can count edges by this process:
- Count the faces
- Count the edges per face
- Multiply to get the total edges, but then divide by 2 because each edge was counted twice.
To go further, introduce (or allow students to discover) the Euler characteristic V - E + F, which is always 2 for polyhedra.
Resources
- escherwiki:Spherical Geometry
- Polyhedra and Art (through history), by George Hart.
- Geometry in Art and Architecture, by Paul Calter has a section on the Platonic solids.
