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Mathematica code:
v1[t_] := {Cos[t], 0, Sin[t]}v2[t_, a_] :=  1/Sqrt[1 + Sin[t]^2 Tan[a]^2] {-Sin[t], -Sin[t] Tan[a], Cos[t]}v3[t_, a_] :=  1/Sqrt[1 + Sin[t]^2 Tan[a]^2] {-Sin[t]^2 Tan[a], 1, Cos[t] Sin[t] Tan[a]}P[t_, a_] := {v3[t, a][[2]]/Tan[a] - v3[t, a][[1]], 0, -v3[t, a][[3]]/2}Q[t_, a_] := {v3[t, a][[2]]/Tan[a], v3[t, a][[2]], v3[t, a][[3]]/2}vertices[t_, a_, A_, B_, C_, D_] := {P[t, a] - A*Sqrt[2]/2 v1[t],   P[t, a] + B*Sqrt[2]/2 v1[t],  Q[t, a] - C*Sqrt[2]/2 v2[t, a],   Q[t, a] + D*Sqrt[2]/2 v2[t, a]}Tetrahedron[T_, t_, a_, o_, A_, B_, C_, D_] := Table[  {FaceForm[White], Opacity[o], EdgeForm[Thick],   Polygon[    Table[     T[vertices[t, a, A, B, C, D][[1 + Mod[i + j, 4]]]], {i, 1, 3, 1}]]},  {j, 0, 3, 1}]Kaleidocycle[t_, n_, o_, R_, A_, B_, C1_, D1_, x_, y_, z_, S_] := Translate[  Scale[   Rotate[    Table[     Rotate[      Table[       Tetrahedron[T, t, 2 Pi/n, o, A/Tan[2 Pi/n], B* A/Tan[2 Pi/n],         C1/Tan[2 Pi/n], D1*C1/Tan[2 Pi/n]],       {T, {TransformationFunction[IdentityMatrix[4]],          ReflectionTransform[{-Sin[2 Pi/n], Cos[2 Pi/n], 0}]}}],      r*4 Pi/n, {0, 0, 1}],     {r, 0, n - 1, 1}],    R*Sin[t], {0, 1, 0}],   S],  {x, y, z}]rr[q_] := (SeedRandom[q]; RandomReal[])K[t_, pr_, Q_, w_] := Graphics3D[  Table[   Kaleidocycle[t + 2 Pi*rr[w*q], 5 + 2 Floor[10 rr[2 w*q]], 1, 0,     1.5, 1, 1.5, 1, 1.2 pr*rr[3 w*q], 1.2*6/5 pr*rr[4 w*q],     10 pr*(1 rr[5 w*q] - 1), 1],   {q, 1, Q, 1}],  PlotRange -> {{0, pr}, {0, 6/5 pr}, {-10 pr, 10 pr}},   ImageSize -> 500, Axes -> False, Boxed -> False,   Lighting -> "Neutral", ViewPoint -> {0, 0, Infinity} ,   Background -> White ]Manipulate[ K[t, 15.5, 18, 5], {t, 0, 2Pi}]

Mathematica code:

v1[t_] := 
{Cos[t], 0, Sin[t]}

v2[t_, a_] :=
1/Sqrt[1 + Sin[t]^2 Tan[a]^2] {-Sin[t], -Sin[t] Tan[a], Cos[t]}

v3[t_, a_] :=
1/Sqrt[1 + Sin[t]^2 Tan[a]^2] {-Sin[t]^2 Tan[a], 1, Cos[t] Sin[t] Tan[a]}

P[t_, a_] :=
{v3[t, a][[2]]/Tan[a] - v3[t, a][[1]], 0, -v3[t, a][[3]]/2}

Q[t_, a_] :=
{v3[t, a][[2]]/Tan[a], v3[t, a][[2]], v3[t, a][[3]]/2}

vertices[t_, a_, A_, B_, C_, D_] :=
{P[t, a] - A*Sqrt[2]/2 v1[t],
P[t, a] + B*Sqrt[2]/2 v1[t],
Q[t, a] - C*Sqrt[2]/2 v2[t, a],
Q[t, a] + D*Sqrt[2]/2 v2[t, a]}

Tetrahedron[T_, t_, a_, o_, A_, B_, C_, D_] :=
Table[
{FaceForm[White], Opacity[o], EdgeForm[Thick],
Polygon[
Table[
T[vertices[t, a, A, B, C, D][[1 + Mod[i + j, 4]]]], {i, 1, 3, 1}]]},
{j, 0, 3, 1}]

Kaleidocycle[t_, n_, o_, R_, A_, B_, C1_, D1_, x_, y_, z_, S_] :=
Translate[
Scale[
Rotate[
Table[
Rotate[
Table[
Tetrahedron[T, t, 2 Pi/n, o, A/Tan[2 Pi/n], B* A/Tan[2 Pi/n],
C1/Tan[2 Pi/n], D1*C1/Tan[2 Pi/n]],
{T, {TransformationFunction[IdentityMatrix[4]],
ReflectionTransform[{-Sin[2 Pi/n], Cos[2 Pi/n], 0}]}}],
r*4 Pi/n, {0, 0, 1}],
{r, 0, n - 1, 1}],
R*Sin[t], {0, 1, 0}],
S],
{x, y, z}]

rr[q_] := (SeedRandom[q]; RandomReal[])

K[t_, pr_, Q_, w_] :=
Graphics3D[
Table[
Kaleidocycle[t + 2 Pi*rr[w*q], 5 + 2 Floor[10 rr[2 w*q]], 1, 0,
1.5, 1, 1.5, 1, 1.2 pr*rr[3 w*q], 1.2*6/5 pr*rr[4 w*q],
10 pr*(1 rr[5 w*q] - 1), 1],
{q, 1, Q, 1}],
PlotRange -> {{0, pr}, {0, 6/5 pr}, {-10 pr, 10 pr}},
ImageSize -> 500, Axes -> False, Boxed -> False,
Lighting -> "Neutral", ViewPoint -> {0, 0, Infinity} ,
Background -> White ]

Manipulate[
K[t, 15.5, 18, 5],
{t, 0, 2Pi}]
 
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