By symmetrically gluing together opposite faces of a dodecahedron together, one of three spaces can be obtained, depending on the angle the faces are rotated by before twisting. In fact, this can be done with any of the regular polyhedra besides the tetrahedron, and in that case the stella octangula can be used instead. However, it seems only the case of the dodecahedron (or equivalently, its conjugate, the great stellated dodecahedron) has been investigated. What are the other spaces? Or rather, what are the spaces not already named by the following list?

• Stella octangula, 180 degrees (this is the 3-sphere)
• Stella octangula, 60 degrees
• Octahedron, 180 degrees (this is real projective space)
• Octahedron, 60 degrees
• Cube, 180 degrees (this is real projective space)
• Cube, 90 degrees
• Cube, 0 degrees (this is the 3-torus)
• Icosahedron, 180 degrees or great icosahedron, 180 degrees (this is real projective space)
• Icosahedron, 60 degrees or great icosahedron, 60 degrees
• Dodecahedron, 180 degrees or great stellated dodecahedron, 180 degrees (this is real projective space)
• Dodecahedron, 108 degrees or great stellated dodecahedron, 36 degrees (this is the Seifert-Weber space)
• Dodecahedron, 36 degrees or great stellated dodecahedron, 108 degrees (this is the Poincaré homology sphere)
• Great dodecahedron, 180 degrees or small stellated dodecahedron, 180 degrees
• Great dodecahedron, 108 degrees or small stellated dodecahedron, 36 degrees
• Great dodecahedron, 36 degrees or small stellated dodecahedron, 108 degrees