Deutsche Version

Inversion of a Dodecahedron 2

If you start the inversion process of the almost-dodecahedron from the home position of the 6 pieces, the pieces overlap.

Home position, hinge angle 0°
hinge angle 23°
hinge angle 46°

I was able to numerically compute the angle, where the overlap is maximal, it is about 49.863°.

If we move P to P' and Q to Q' the pieces do not overlap but just touch. The lengths of the corresponding edges shrink by the factor 0.5921208162848078 (P-edge) and 0.6479222750007346 (Q-edge).

The red wirefram shows what is left if we a apply the cutting process twice. The new piece keeps the twofold rotational symmetry of the original piece and has about 56% of the volume of the original piece.

The remaining fully invertable chain Removed part


During a complete inversion cycle there are four positions, where adjacent faces touch each other.

To build a model, from a practical point of view it seems the best to dissect the removed part into one big piece

and three congruent convex wedges which can be dropped into the remainig gaps.



You can download the nets for the necessesary parts here.

The chain, the big "Riegel" and the three wedges.

The assembled almost-dodecahedron.

< Home > < Inversion of a Dodecahedron 1>

© 2020  Herbert Kociemba