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

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