The space diagonal is 5 times the square root of 2929. Peterson and Jordan focus on other three-dimensional figures with integer edges and diagonals, particularly pyramids and prisms. A triangular pyramid, or tetrahedron, has a triangular base and four faces, counting the bottom. An integer polyhedron is one in which the distance between each pair of vertices is an integer. Because the faces of an integer polyhedron must themselves be integer polygons, it's natural to use integer polygons as the building blocks of integer polyhedra, Peterson and Jordan remark.
Octahedral pyramids are a good starting point.
An octahedron has eight faces. In its most familiar form as one of the Platonic solids, each face is an equilateral triangle. An octahedral pyramid has a seven-sided heptagon as its base. In this case, all of the points lying along a line through this circle's center and perpendicular to the plane of the heptagon are equidistant from the polygon's vertices. You can then choose the lateral edges of the pyramid to be 17.