This goal of this assignment is to learn about key features of spherical geometry by investigating in depth a couple of important figures on the sphere. For the basic assignment, each figure will be studied from several points of view and with several tools – Lenart sphere, stereographic projection, and vector geometry. One positive feature of this assignment is that it is possible to cross-check your work by the various approaches to ensure that that your answers are correct. Beyond that, the quality of the explanations, drawings, will be important.
There will also be a model-building option and other extensions for high honors and/or extra credit.
We have studied the regular polyhedra (Platonic solids) and some of the semiregular (Archimedean) polyhedra in 444. Each of these polyhedra can be inscribed in a sphere. This means that the vertices of one of these polyhedra are points on a sphere. When two vertices are connected by an edge of the polyhedron (a Euclidean segment is space), we can connect the vertices on the sphere by the arc of a great circle so that the Euclidean segment is the chord of the great circle that cuts off this arc. The collection of these vertices and these arcs form spherical polygons that tile, or tessellate the sphere. This figure is called the spherical form of the polyhedron, or usually just a spherical polyhedron.
If a convex polyhedron is not already inscribed in a sphere, it can be placed inside a sphere with center O and then projected centrally to the sphere. Specifically, the vertices are projected to the sphere, then the edges project to great circle arcs and the faces project to spherical polygons.
This is also another way of thinking about the spherical form of a polyhedron that can be inscribed in a sphere. For good examples of this, see the Kaleidatile software that was used in 444 Lab 10.
The polyhedra that will be studied for this assignment are the cube and the midpoint figure of the octahedron.
The cube is very familiar. It may be helpful to recall that the cube is the dual of the octahedron, so that if one is given an octahedron (space version or spherical version), the centers of the faces form the vertices of a cube.
We learned in Math 444 that one can choose 4 of the vertices of the cube to obtain the vertices of a regular tetrahedron.
An octahedron has 8 faces that are equilateral triangles. As Euclidean plane triangles in space, the midpoints of the edges of each triangle can be joined to divide the triangle into 4 congruent equilateral triangles. Thus the faces of this polyhedron can be divided into 32 such triangles overall.
The octahedron can be inscribed in a sphere. The faces of the spherical octahedron are 8 spherical triangles – they are each 90-90-90 triangles. The midpoints of the edges of each of these triangles can be joined to divide the original triangle into 4 smaller spherical triangles (these are the central projections of the triangles of the previous paragraph).
The tessellation of the sphere by these midpoint triangles is the figure to be studied. The space (Euclidean) polyhedron whose vertices are the midpoint figure vertices on the sphere is the simplest example of a geodesic dome.
In space the octahedron can be truncated by slicing off part of the solid at each vertex by the plane through the midpoints of the edges that end in that vertex. The vertices of the new polyhedron are exactly the midpoints of the edges that appear in the previous figure. This polyhedron is called the cuboctahedron (pronounced "cube-octahedron").
The last spherical figure to be investigated is the spherical cuboctahedron. This is just a minor extension of the previous figure, since the vertices of this figure are the midpoints of the previous one (but without the vertices of the original octahedron).