Cleopatra's Pyramid

Abstract

each end of two of the strips and completing octahedral surfaces with the triangles at the ends of the longer strip. This deltahedron is a pile of octahedra. On a pile of six octahedra, each strip twists through 3600. Now, the inradius of a regular octahedron is e/V6; and in FIGURE 3, OD = OA/2 = e/2V3. Hence, in a pile of k octahedra a right circular cylinder of radius e/2V3 and length 2ke/V6 can be inscribed. The HHL and LLH forms are mirror images of each other, and are bona fide deltahedra. The HHL form twists to the right; and the LLH form twists to the left. When each strip contains 12 triangles, the twist is through 3600. If each strip contains the same odd number of triangles, the model has an open face at one end which may be closed by adding an additional triangle to one strip. If each strip contains 2n triangles, the spiral deltahedron consists of a pile of (n 1) concave octahedra capped at each end with a regular tetrahedron. (A concave octahedron with equilateral triangular faces consists of three regular tetrahedra with a common edge.) It follows that the volume of a spiral deltahedron is given by (V2e'/12)[2+ 3(n 1)] or \/2e3(3n 1)/12. The dihedral angles of spiral deltahedra are variously 70?32', 141?4', and the reentrant 211?36'. These spiral deltahedra may also be considered to be a special piling of regular tetrahedra or of regular tetrahedra and triangular dipyramids. A fuller appreciation of these deltahedra can be gained by actually constructing models.