Polytopes Building blocks We weld steel struts to erect a jungle-gym. We stitch squares of fabric to sew a quilt. We mortar bricks to construct a house. These kinds of building blocks in dimensions one, two, and three are called polytopes. The boundary of a high dimensional polytope is made out of low dimensional ones. We weld four struts to make the bound- ary of a square. We stitch six rectangles to make the bound- ary of a brick. As we shall see, we mortar eight cubes to make the boundary of a hypercube. The most symmetric polytopes are called regular. In dimen- sion three these are the Platonic solids: the tetrahedron, cube (hexahedron), octahedron, dodecahedron, and icosahedron. To the right is Leonardo da Vinci’s illustration of the edges of the dodecahedron, drawn for the manuscript version of Luca Pacioli’s De divina proportione (1509). In dimension four there are six regular polytopes: the five-cell, the eight-cell (the hypercube), the 16–cell, the 24–cell, the 120–cell, and the 600-cell. Seeing the fourth dimension It is impossible to see in four dimensions. Instead, we must project four-dimensional objects into three-dimensional space. As a warm up consider the same problem one dimension lower. Start with the cube and radially project its edges to the two-sphere S2. Thicken the edges to strips to get the white “beach-ball cube” shown to the right. Now place a point light source at the north pole; this stereo- graphically projects the strips down into the plane. (Challenge: Draw what you get when you start with an octahedron.) Hypercube The same procedure, applied one dimension higher, projects the edges of the hypercube into three-space. The result is shown to the right. Since the thickening happens in the three- sphere S3 the edges become tubes instead of strips. The south pole of the three-sphere is sent to the exact center; there is a small cube surrounding it. Six distorted cubes share a face with this central cube, and a single “infinite” cube lies outside the sculpture. Together these eight cubical cells form the boundary of the hypercube. See our paper Sculptures in S3 (2012) for further discussion of polytopes and other objects in the three-sphere. Saul Schleimer University of Warwick [email protected] Henry Segerman Oklahoma State University [email protected] This work is in the public domain..