The polyhedron above is the rhombic enneacontahedron. Sixty of its faces are wide (yellow) rhombi, while the other thirty are narrow (red) rhombi. The wider rhombi are arranged in twelve panels of five rhombi each. If those panels are moved outward from the center by just the right amount, the narrower rhombi have room to expand, becoming equilateral octagons:
Both of these rotating images were created using Stella 4d, a program you can try, for free, at http://www.software3d.com/Stella.php.