Wythoff's Icosahedral Kaleidoscope
This Demonstration shows the iterated reflections of segments in the mirror planes of a solid angle.[more]
On the left you see the unfolded angle; the triangle is rotated around the axis by 90° and the triangle is rotated around the axis by 120°. The points and rotate to meet above the point ; the points and also meet above the plane .
You can drag the points labeled , , and originally at , , and on the three faces , , and of the solid angle. You can reset the displaced points back to their original positions on the sides and of the angle.
Check the boxes to see the line segments. On the right, they are iteratively reflected in the three mirror planes (not shown).[less]
Thumbnail: and give the great stellated dodecahedron
Snapshot 1: gives the regular icosahedron
Snapshot 2: gives the regular dodecahedron
Snapshot 3: and give the small stellated dodecahedron
Snapshot 4: and give the great dodecahedron
If you can find placements for the segments that give the icosidodecahedron or its dual, the rhombic triacontahedron, please let me know so I can add them.
 H. S. M. Coxeter, Regular Complex Polytopes, London: Cambridge University Press, 1974 pp. 23–24.