The kinematics of the linkage

To analyze the movement of the kaleidocycle in a first approach we just have a look at the underlying linkage.

Due to the 3-fold rotational symmetry only two angles t1 and t2 do occur. Our goal is to find the relation between t1 and t2.
Let without loss of generality the distance between adjacent hinges be 1. If the skew angle between hinges h1 and h2 is ts, the skew angle between h2 and h3 is −ts, between h3 and h4 again ts etc. Remember that we always hinge together a piece and its mirror image. Suppose h1 is oriented along the z-axis and the connection with length 1 between h1 and h2 along the x-axis.

Now if we want to map h6 onto h1, we rotate along the z-axis by angle t1, rotate along the x-axis by angle ts and do a translation along the x-axis by 1.
To map h5 onto h6 (now in the original position of t1) we now have to rotate along the z-axis by angle t2, rotate along the x-axis by −ts and do a translation along the x-axis by 1.

If we repeat this procedure 3 times, t1 is again in its original position. This we can exploit to find the relation between t1 and t2.

We define the rotation along the z-axis by angle w in homogenous coordinates:


The rotation along the x-axis by w and the following translation in x-direction by 1 is given by


id =MatrixPower[m[-ts].rotz[t2].m[ts].rotz[t1],3];

As explained above id must be the Identity matrix.

id[[1, 1]] == 1 // FullSimplify for example gives us the relation:

(-1+Cos[t1] Cos[t2]-Cos[ts] Sin[t1] Sin[t2]) (Cos[t1] Cos[t2] (3+Cos[2 ts])+2 Cos[ts] (Cos[ts]-2 Sin[t1] Sin[t2])+2 (Cos[t1]+Cos[t2]) Sin[ts]^2)==0

id[[1, 1]] == id[[2, 2]] // FullSimplify

Cos[t1] Sin[t2/2] Sin[ts] (Cos[t1] Cos[t2] (3+Cos[2 ts])+2 Cos[ts] (Cos[ts]-2 Sin[t1] Sin[t2])+2 (Cos[t1]+Cos[t2]) Sin[ts]^2)==0

From both relations together we can conclude that

Cos[t1] Cos[t2] (3+Cos[2 ts])+2 Cos[ts] (Cos[ts]-2 Sin[t1] Sin[t2])+2 (Cos[t1]+Cos[t2]) Sin[ts]^2==0

In [3] we find the relation

Cos[ts]^2+ Sin[ts]^2 (Cos[t1]+Cos[t2])+(1+Cos[ts]^2)Cos[t1]Cos[t2]-2Cos[ts]Sin[t1]Sin[t2]=0

in [5] we find


All three relations show the same solution space for a given skew angle ts when visualized.

We use our relation for the visualization of the relation between t1 and t2.

r=Cos[t1] Cos[t2] (3+Cos[2 ts])+2 Cos[ts] (Cos[ts]-2 Sin[t1] Sin[t2])+2 (Cos[t1]+Cos[t2]) Sin[ts]^2

The smallest possible skew angle ts is π/3 and gives

ContourPlot[ r/.ts->Pi/3,{t1,-Pi,Pi},{t2,-Pi,Pi},Contours->{0}]

The hinge angles periodically change from −π to π and back to -π.

ts = π/2 can be seen as the largest possible skew angle because ts and π − ts just belong to mirrored versions.With a skew angle ts = π/2 the faces of the diphenoid are isosceles triangles and this arrangement is used in almost all the models in Schattschneider [1]. Here we get:

ContourPlot[ r/.ts->Pi/2,{t1,-Pi,Pi},{t2,-Pi,Pi},Contours->{0}]

We will show soon that the hinge angles t1 and t2 are in the range −2 π/3 and +2 π/3 in this case.

And finally here is a plot of the general case for values between the two extremes:

During a complete cycle of the kaleidocycle, the point P(t1|t2) performs a complete revolution on the "ellipse". The maximum of the hinge angle is somewhere between 2/3π and π. The knowlegde of this maximum angle is important to decide if the kaleidocycle is physically realizible if disphenoids are used instead of the linkage bars.


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.

< Home >
<1. Kaleidocycles with 6 disphenoids>
<2. Some formulas for disphenoids>
<4. Some exceptional points in the solution space >
<5. Physically realizable kaleidocycles >
<6. A closer look at the solutions >
<7. The 8-kaleidocycle polyhedron>

© 2020  Herbert Kociemba