the same type that are less symmetrically placed. The lengths of the five simple closed geodesics on a snub disphenoid with unit-length edges are (for the equatorial geodesic), , (for the geodesic through the midpoints of opposite edges), , and . Except for the tetrahedron, which has infinitely many types of simple closed geodesics, the snub disphenoid has the most types of geodesics of any deltahedron. Construction The snub disphenoid is constructed, as its name suggests, as the snub polyhedron formed from a tetragonal disphenoid, a lower symmetry form of a regular tetrahedron. The snub operation produces a single cyclic band of triangles separating two opposite edges (red in the figure) and their adjacent triangles. The snub antiprisms are analogous in having a single cyclic band of triangles, but in the snub antiprisms these bands separate two opposite faces and their adjacent triangles rather than two opposite edges. The snub disphenoid can also constructed from the square antiprism by replacing the two square faces by pairs of equilateral triangles. However, it is one of the elementary Johnson solids that do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids. A physical model of the snub disphenoid can be formed by folding a net formed by 12 equilateral triangles (a 12-iamond), shown. An alternative net suggested by John Montroll has fewer concave vertices on its boundary, making it more convenient for origami construction. Cartesian coordinates Let be the positive real root of the cubic polynomial Furthermore, let and The eight vertices of the snub disphenoid may then be given Cartesian coordinates Because this construction involves the solution to a cubic equation, the snub disphenoid cannot be constructed with a compass and straightedge, unlike the other seven deltahedra. With these coordinates, it's possible to calculate the volume of a snub disphenoid with edge length as , where , is the positive root of the polynomial The exact form of can be expressed as, where is the imaginary unit. Related polyhedra Another construction of the snub disphenoid is as a digonal gyrobianticupola. It has the same topology and symmetry, but without equilateral triangles. It has 4 vertices in a square on a center plane as two anticupolae attached with rotational symmetry. Its dual has right-angled pentagons and can self-tessellate space. References External links Johnson solids Deltahedra