# The 8-kaleidocycle polyhedron

I observed by accident that two kaleidocycles I had built, one in a position with a hole and the other in a "compact" position could be inserted into each other more or less accurate. I wondered if there exists a kaleidocycle with the following properties:

 1. There is a position of the kaleidocycle where the two red faces and the two green faces are simultanous coplanar. We call this the "ring"-position. If the red faces are not coplanar the inserted kaleidocycle in the "cap"-position will leave a gap. If the green faces are not coplanar you cannot glue two kaleidocycles in ring-position together at the green faces without gap.
 2. Moreover, the face normals should allow to glue together not only two kaleidocyles in "ring"-position but four of them without collisions or gaps.

Yes, a kaleidocycle with these properties exists and the disphenoids used in the kaleidocycle are unique up to similarity. If c is the name of the triangle side which serves as the hinge, the calculations give for the both other triangle sides of the disphenoid

a= c ≈1.3963094 c and

b= c ≈ 1.2618531 c

With the function dihedralAngleAB developed on one of the previous pages we compute a dihedral angle of exactly 45° at hinge edge c and about 99.13° and 63.16° at edges a and b. With the function skewAngleABtoCD we compute the angle between the two hinges of a disphenoid to

skew = ≈ 69.059°

Now we can fit 4 of the kaleidocycles bent into the "cap"-position exactly into the four holes and we get a polyhedron made of 48 congruent disphenoids.

Is this kaleidocycle physically realizable? We showed that for the triangle lengths a,b and c (hinge c) the following unequality must hold:

−8 (a2− b2)2 (a2 + b2) − 5 c6 + 11 (a2 − b2)2 c2 + 2 (a2 + b2) c4 ≥ 0

With the values computet above we get for the left side

8/289 (-33+20 Sqrt[2]) c6 ≈ − 0.13 c6

So theoretically the kaleidocycle blocks at some point. But practically - if built by paper or cardboard - no blocking will be observed because the blocking range is very small.
In the interactive visualization. the kaleidocycle can be simulated by using the skew angle 69.1° (exact value ArcCos[Sqrt[1/17 (5-2 Sqrt[2])]]) and a relative hinge length of 1.03 (exact value Sqrt[3/7 (-23+18 Sqrt[2])]).

A closer analysis show the blocking happens between the "cap"-position used for filling the holes

and a second position not far away where the hinge angle in front has changed by about 34°

The overlapping happens only between these two positions and the maximum overlap angle is 1.03° (exact value ArcCos[1/8 (-10+3 Sqrt[2])]) − 3π/4). This maximum overlap position is displayed below.

I have built models of the 8 kaleidocycles, each kaleidocycle with a little magnet for each face - so I used 192 magnets in total. Here is a little video which shows the rotation process.

There are many ways to put the 8 kaleidocycles together because each kaleidocycle has 4 possible orientations and additionally can be used in "ring"-position or in "cap"-position. Below are a few examples.

All 8 parts are identical, seen here from different perspectives.

We can use four "rings" to build a nonconvex polygon with tetrahedral symmetry and four holes..

If we insert 4 "caps" we get the convex polyhedron with the 48 octahedral symmetries.

Here another example of the polyhedron with a blue outer surface and a quarter removed.

I observed that alternatively 4 "caps" fit exactly together to build one half of the polyhedron above. So we can build the complete polededron also with eight "caps".

References

1. Doris Schattschneider and Wallace Walker, M.C. Escher Kaleidocycles, 1977. ISBN 0-906212-28-6
2. Baker, J.E., 1980. An analysis of Bricard linkages. Mechanism and Machine Theory 15, 267–286.
3. Chen, Y., You, Z., and Tarnai, T., 2005, “Threefold-Symmetric Bricard Linkages for Deployable Structures,”Int. J. Solids Struct.,42(8), pp. 2287–2300.