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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.

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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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© 2020  Herbert Kociemba