Icosaedro

El icosaedro tiene 21 ejes de rotación: 10 por los centros de las caras, 15 por los centros de las aristas y 6 que unen vértices opuestos.
Tiene 15 planos de simetrías que contienen cada par de aristas opuestas.

Arista

Radio

Ejes centros de caras

Ejes centros aristas

Planos de simetría

10

6

15

Elementos de simetría

Ejes  por vértices

15

RA = 1 (slider from 0 to 1, step 1)
p = 1.1755705045849463
t = 1.7320508075688772
c = 1.4142135623730951
n = 3 (slider from 3 to 5, step 1)
L = 2.8 (slider from 1 to 4, step 0.1)
ar = If(n ≟ 3, L / t, If[n ≟ 4, L / c, If[n ≟ 5, L / p]])
des = If(RA ≟ 0, L, If[RA ≟ 1, ar])
O: (0, 0, 0)
ci = Circle(O, des, xOyPlane)
A = Point(ci)
A' = Rotate(A, (360 * ° / n), O)
A'' = Rotate(A', (360 * ° / n), O)
v = 0 (slider from -1 to 1, step 0.001)
b = Distance(A, A')
arista = Distance(A, A')
radio = Distance(O, A'')
a = 4.1 (slider from 0 to 6, step 0.1)
o1 = true
o4 = false
w = 1 (slider from -5 to 5, step 0.1)
v10: (10 * v)
o3 = false
e = 6.1 (slider from -5 to 8, step 0.1)
p20, B, C, D, E, F, G, H, I, J, caraAA'A'', caraAA''D, caraADE, caraABE, caraAA'B, caraA'BF, caraA'CF, caraA'A''C, caraA''CG, caraA''DG, caraCFI, caraCGI, caraGIJ, caraDGJ, caraFHI, caraHIJ, caraEHJ, caraDEJ, caraBFH, caraBEH, aristaAA'', aristaAA', aristaA'A'', aristaAD, aristaA''D, aristaAE, aristaDE, aristaAB, aristaBE, aristaA'B, aristaA'F, aristaBF, aristaA'C, aristaCF, aristaA''C, aristaA''G, aristaCG, aristaDG, aristaCI, aristaFI, aristaGI, aristaGJ, aristaIJ, aristaDJ, aristaHI, aristaFH, aristaHJ, aristaEJ, aristaEH, aristaBH = Icosahedron(A, A', A'')
f = Line(A, I)
g = Line(H, A'')
K = Midpoint(I, A)
h = Line(B, G)
i = Line(D, F)
j = Line(J, A')
k = Line(C, E)
d = Sphere(K, L + 1)
M, E_2 = Intersect(aristaCI, d)
N = Intersect(f, aristaFI)
P, F_2 = Intersect(aristaBH, d)
Q, R = Intersect(g, d)
S, T = Intersect(k, d)
U, V = Intersect(h, d)
W, Z = Intersect(f, d)
A_1, B_1 = Intersect(j, d)
l = Segment(Q, R)
q = Segment(T, S)
m = Segment(V, U)
r = Segment(Z, W)
e3 = Segment(B_1, A_1)
C_1, D_1 = Intersect(i, d)
f_1 = Segment(D_1, C_1)
g_1 = OrthogonalLine(K, caraEHJ)
h_1 = OrthogonalLine(K, caraHIJ)
i_1 = OrthogonalLine(K, caraFHI)
j_1 = OrthogonalLine(K, caraBFH)
E_1 = Intersect(q, g_1)
k_1 = OrthogonalLine(E_1, aristaA'C, space)
F_1, G_1 = Intersect(i_1, d)
H_1, I_1 = Intersect(j_1, d)
J_1, K_1 = Intersect(k_1, d)
L_1, M_1 = Intersect(g_1, d)
N_1, O_1 = Intersect(h_1, d)
e1 = Segment(N_1, O_1)
m_1 = Segment(F_1, G_1)
n_1 = Segment(L_1, M_1)
r_1 = Segment(H_1, I_1)
q_1 = OrthogonalLine(E_1, caraGIJ)
s_1 = OrthogonalLine(E_1, caraDGJ)
t_1 = OrthogonalLine(E_1, caraDGJ)
a_1 = OrthogonalLine(E_1, caraDEJ)
b_1 = OrthogonalLine(E_1, caraADE)
c_1 = OrthogonalLine(E_1, caraCGI)
P_1, Q_1 = Intersect(b_1, d)
R_1, S_1 = Intersect(q_1, d)
T_1, U_1 = Intersect(t_1, d)
V_1, W_1 = Intersect(c_1, d)
Z_1, A_2 = Intersect(a_1, d)
d_1 = Segment(Q_1, P_1)
e_1 = Segment(A_2, Z_1)
f_2 = Segment(V_1, W_1)
g_2 = Segment(U_1, T_1)
h_2 = Segment(S_1, R_1)
B_2 = Mirror(V_1, E_1)
i_2 = Segment(B_2, V_1)
j_2 = OrthogonalLine(E_1, caraA''CG)
C_2, D_2 = Intersect(j_2, d)
k_2 = Segment(D_2, C_2)
oo = Plane(H, F, A'')
P22 = Rotate(p20, 120°, e1)
o2 = false
G_2 = Midpoint(aristaHJ)
H_2 = Midpoint(aristaA'A'')
s = Line(G_2, H_2)
I_2, J_2 = Intersect(s, d)
e2 = Segment(I_2, J_2)
α = 2.0943951023931953 rad (slider from 0 to 120°, step 0.01°)
β = 3.141592653589793 rad (slider from 0 to 180°, step 0.01°)
γ = 1.2566370614359172 rad (slider from 0 to 72°, step 0.01°)
p201 = Rotate(p20, (-α), e1)
p202 = Rotate(p20, (-β), e2)
p203 = Rotate(p20, (-γ), e3)
texto1: "Arista del icosaedro " + L + ""
v11: floor(v10)
texto2: "Veloc. rotación " + v11 + ""