# Physically realizable kaleidocycles

The easiest way to understand which kaleidocycles are physically realizable is to play with the visualization. Enable "show linkage", reduce the opacity and vary the hinge length. As long as the hinges do not intersect in the middle all parts are movable. The critical position is when three hinges touch in the middle. Then also neighbouring faces of the disphenoids touch each other. If If this happens it is impossible to increase the hinge angle of the other three hinges which only is no problem if the hinge angle is maximal in this position. Else the kaleidocycle blocks.

The upper left slider leads in position β =0° and 180° to maximal hinge angles ±tmax for three of the hinges. If in this position the other three hinges do not intersect the kaleidocycle never blocks.

 In this extremal position the hinges do not intersect, so they also will not intersect in any other position. In this example with regular tetrahedra we are in a position where three hinges touch in the middle. But since this is not an extremal position the kaleidocycle blocks in this position. A kaleidocyle with regular tetrahedra is physically not realizable.
 In this example the hinges also touch in the position shown..But since this is an extremal position no blocking occurs. In the visualization the slider "relative hinge length" has the value 1 if touching happens in the extremal position. This is the maximal possible hinge length. For values >1 hinges and adjacent disphenoids intersect in some positions.
 From this perspective we look in direction of the upper right hinge axis which now is unvisible. We show the situation when the hinges touch in the extremal position. The dotted line splits the dihedral angle D into two equal parts w due to the symmetry of the disphenoid. The triangles themselves are not isosceles, only from this perspective. We see that tmax +w (grey area) + w (red area) add to π, tmax+2*w = tmax+D = π Another way to argue is that during a complete cycle the angle of a hinge moves between −tmax and +tmax and hence needs 2*tmax angle space for its movement. But 2*D already is occupied by the 2 disphenoids connected to the hinge and the available angle space for the hinge movement is at most 2π−2*D. It follows that no blocking occurs if 2π−2*D>= 2*tmax or D+tmax<=π In case of equality the kaleidocycle has no hole in its 4 extremal positions and neighbouring faces touch each other. For D+tmax>π the kaleidocycles blocks at some point during the movement.

With these formulas we can find the relation between a,b and c from D+tmax<=π.

tmax+D<=π means

maxAngle[skewAngleABtoCD[a, b, c]]+dihedralAngleAB[a, b, c] <= Pi

It took me quite a while until I found a beautiful way to simplify this relation. I realized that
ArcCos[x]+ArcCos[y]<=Pi is equivalent to x+y>=0. For the simplification with Mathematica we also use the fact that the triangles are acute.

x=3-(a+b)/(a+b-c)+(-a+b)/(a-b+c)+(a-b)/(-a+b+c)-(a+b)/(a+b+c);
y=1-3/(2 (1-(a^2-b^2)^2/c^4));
FullSimplify[x+y>=0,Assumptions->{a>0,b>0,c>0,a^2+b^2>c^2,b^2+c^2>a^2,c^2+a^2>b^2}]

 Proposition: A kaleidocycle made of 3 disphenoids with edge lengths a, b and c and 3 corresponding chiral partners is physically realizable with the hinges at edge c if and only if In case of equality the kaleidocycles neigbouring faces touch 4 times during a cycle and the kaleidocycle has no hole in these 4 positions.

There still is room for some extra considerations.

• Are there disphenoids where hinges at two different edges allow kaleidocycles?
• Which disphenoids have the most "irregular" faces?
• How many allowed disphenoids a(n) are there with integer lengths <= n?
• How many allowed disphenoids b(n) with integer lengths are there with perimeter n? .......

We will give some answers on the last page.

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.
4. J. Leech, Some properties of the isosceles tetrahedron,Math. Gazette34(1950)269–27
5.Safsten C, Fillmore T, Logan A, Halverson D, Howell L (2016) Analyzing the stability properties of kaleidocycles. J Appl Mech 83:051001.