Kaleidocycles with 6 Disphenoids 

The word "kaleidocycle" was coined by Wallace Walker in the 1950s. Together with Doris Schattschneider he published a book in 1977 [1] using patterns of M.C. Escher which contributed greatly to the spread of this word. A kaleidocycle from [1]
The kaleidocycle shown here connects 6 tetrahedra on opposite edges, the faces are isosceles triangles. The object has only one degree of freedom, you can rotate the ring inside out but the ring itself does not wiggle else. This is one of the reasons for its appeal. It is not necessary for the faces of one tetrahedron to be isosceles triangles. If we allow the faces of one tetrahedron to consist of arbitrary but congruent triangles, we get a tetrahedron which is called disphenoid (or isosceles tetrahedron). Skew edges of the disphenoids have the same length and disphenoid can be constructed if and only if the 4 congruent triangles are acute. The disphenoids in general are chiral (differ from their mirror image). To build a kaleidocycle with 6 disphenoids you have to hinge together the disphenoid with its mirror image. The resulting kaleidocycle then consists of three disphenoids and three mirrored counterparts and also has only one degree of freedom.
Only in 2018 it was discovered by J. Schönke and E. Fried that there are also linkages with 7 and more chiral disphenoids with only one degree of freedom. They call the resulting objects Moebius Kaleidocycles. These kaleidocycles do not contain the mirror counterparts at all. Though extremely intruiging we will not deal with these objects here. 

The kaleidocycles we want to analyze are related to Bricard linkages which are already extensively studied, see for example [2] or [3]. I could not find an answer to the question for which edge lengths a, b and c of a disphenoid the corresponding kaleidocycle is physically realizable in these papers.. For example with 6 regular tetrahedra (a=b=c) a kaleidocycle is not possible since neighbouring faces block each other during the movement at some point. 

On the next few pages we will show the following: −8 (a^{2}− b^{2})^{2 }(a^{2 }+ b^{2}) − 5 c^{6 }+ 11 (a^{2 }− b^{2})^{2} c^{2 }+ 2 (a^{2 }+ b^{2}) c^{4} ≥ 0 The corresponding kaleidocycle necessarily has the hinges at the edges with length c. 

I also found a kaleidocycle which is unique up to similarity such that you can put 8 identical kaleidocycles together to build a polyhedron with the 48 octahedral symmetries:


The mathematics of a kaleidocycle is not really hard, but some of the fomulas are a bit lengthy, so all computations developed on the next page are done within the Mathematica system. 

References 1. Doris Schattschneider and Wallace Walker, M.C. Escher Kaleidocycles, 1977. ISBN 0906212286 

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