Eigenschaften des Ikosaeders

Der Oberflächeninhalt ergibt sich aus der Summe aller vier gleichschenklicher Dreiecke.

Da \; jedes \; Dreieck \; einen \; Flächeninhalt \; von \; \frac{ a² *  \sqrt{ 3 } }{ 2 } \; besitzt \; ,ergibt \; sich \; hieraus \; der \; Oberflächeninhalt \; unter \; der \; Formel:

4 * \frac{ a² *  \sqrt{ 3 } }{ 2 } = a² *  \sqrt{ 3 } \; bzw. \; für \; a=1 \; => \; O=\sqrt{ 3 }

Flächenwinkel unter folgender Formel ergibt: cos(α) = 1 / 3

Da der Ikosaeder aus (gleichseitigen) Dreiecken besteht, beträgt die Anzahl der Ecken 3 ,woraus sich der 

Der Flächenwinkel ergibt sich aus dem Quotient aus 1 durch die Anzahl der Ecken einer Fläche des Ikosaeders:

Die Anzahl der Flächen des Ikosaeders ergibt sich entweder aus der Anzahl der Kanten oder durch die Anzahl der Ecken.

Da der Ikosaeder aus 20 gleichseitigen Dreiecken besteht, ergibt sich die Anzahl der Flächen unter den Formeln

Im Raum:                n (Ecken) = n(Flächen)               n (Kanten) = 1,5 * n(Flächen)       =>     4 Flächen

Die Anzahl der Kanten des Ikosaeders ergibt sich entweder aus der Anzahl der Flächen oder durch die Anzahl der Ecken.

Da der Ikosaeder aus 20 gleichseitigen Dreiecken besteht, ergibt sich die Anzahl der Kanten unter den Formeln

Im Raum:                 1,5 * n (Ecken) = n(Kanten)     1,5 * n (Flächen) = n(Kanten)      =>     9 Kanten

Im Raum:                 n (Kanten) = 1,5 * n(Ecken)     n (Flächen) = n(Ecken)              =>     4 Ecken

DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

Die Anzahl der Ecken des Tetraeders ergibt sich entweder aus der Anzahl der Flächen oder durch die Anzahl der Kanten.

DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

Da der Tetraeder aus 4 gleichseitigen Dreiecken besteht, ergibt sich die Anzahl der Ecken unter den Formeln

DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

In der Ebene:          n (Ecken) = 1,5 * n(Flächen)     n (Kanten) = 2,25 * n(Flächen)     =>     4 Flächen

In der Ebene:          1,5 * n (Ecken) = n(Kanten)     1,5 * n (Flächen) = n(Kanten)     =>     9 Kanten

In der Ebene:          n (Kanten) = 1,5 * n(Ecken)     1,5 * n(Flächen) = n(Ecken)     =>     6 Ecken

A = (6, 3)
B = (6, 0)
Vieleck1, a, a_1, b, C = Polygon(A, B, 3.0)
D = Midpoint(B, C)
E = Midpoint(C, A)
F = Midpoint(B, A)
c = Segment(B, E)
d = Segment(A, D)
e = Segment(F, C)
G = Intersect(c, d)
f = Circle(G, C)
g = Line(B, E)
h = Line(C, F)
i = Line(A, D)
H = Intersect(f, i, 2.0)
I = Intersect(f, g, 2.0)
J = Intersect(f, h, 2.0)
Vieleck2, h_1, j, i_1 = Polygon(J, I, H)
K = Intersect(a, h_1)
L = Intersect(a, i_1)
M = Intersect(a_1, j)
N = Intersect(b, j)
k = Circle(B, 2.5)
p = Circle(A, 2.5)
O, P = Intersect(k, p)
l = Segment(O, B)
m = Segment(O, A)
n = Segment(O, J)
q = Circle(C, 2.5)
r = Circle(A, 2.5)
Q = Intersect(p, r, 1.0)
R, S = Intersect(q, p)
T, U = Intersect(q, k)
s = Segment(T, B)
t = Segment(T, C)
b_1 = Segment(C, S)
c_1 = Segment(S, A)
d_1 = Segment(H, T)
e_1 = Segment(I, S)
f_1 = Segment(S, T)
g_1 = Segment(O, S)
j_1 = Segment(O, T)
k_1 = OrthogonalLine(T, f_1)
l_1 = Line(I, B)
V = Intersect(k_1, l_1)
m_1 = OrthogonalLine(O, g_1)
n_1 = OrthogonalLine(S, g_1)
p_1 = OrthogonalLine(S, f_1)
q_1 = OrthogonalLine(T, j_1)
r_1 = OrthogonalLine(O, j_1)
W = Intersect(p_1, r_1)
Z = Intersect(n_1, q_1)
s_1 = Segment(O, V)
t_1 = Segment(V, T)
a_2 = Segment(T, Z)
b_2 = Segment(Z, S)
c_2 = Segment(S, W)
d_2 = Segment(W, O)
e_2 = Segment(J, W)
f_2 = Segment(A, W)
g_2 = Segment(A, O)
h_2 = Segment(A, S)
i_2 = Segment(I, W)
j_2 = Segment(I, Z)
k_2 = Segment(H, Z)
l_2 = Segment(C, T)
m_2 = Segment(C, Z)
n_2 = Segment(C, S)
p_2 = Segment(B, T)
q_2 = Segment(B, V)
r_2 = Segment(B, O)
s_2 = Segment(J, V)
t_2 = Segment(V, H)
A_1 = Midpoint(Z, I)
B_1 = Midpoint(S, I)
a_3 = Segment(Z, B_1)
b_3 = Segment(S, A_1)
C_1 = Intersect(a_3, b_3)
c_3 = Circle(G, C_1)
D_1 = Midpoint(Z, S)
d_3 = Circle(G, D_1)
e_3 = Circle(G, S)
Oberfächeninhalt = false
Flächenwinkel = false
Flächen = false
Kanten = false
Ecken = false
Höhe = false
Volumen = false
Inkugel = false
Kantenkugel = false
Umkugel = false
O' = Mirror(O, yAxis)
Z' = Mirror(Z, yAxis)
i_5 = Segment(O', Z')
p_5 = Circle(O', 1.09)
E_2 = Intersect(p_5, i_5)
Vieleck4, l_5, e_6, o', F_2 = Polygon(E_2, O', 3.0)
Vieleck5, m_5, e_7, f_6, G_2 = Polygon(E_2, F_2, 3.0)
Vieleck6, n_5, e_8, g_6, H_2 = Polygon(E_2, G_2, 3.0)
Vieleck7, q_5, h_6, g_7, I_2 = Polygon(H_2, G_2, 3.0)
Vieleck8, r_5, h_7, i_6, J_2 = Polygon(H_2, I_2, 3.0)
Vieleck9, s_5, j_6, i_7, K_2 = Polygon(J_2, I_2, 3.0)
Vieleck10, t_5, j_7, k_6, L_2 = Polygon(J_2, K_2, 3.0)
Vieleck11, a_6, l_6, k_7, M_2 = Polygon(L_2, K_2, 3.0)
Vieleck12, b_6, l_7, m_6, N_2 = Polygon(L_2, M_2, 3.0)
Vieleck13, c_6, n_6, m_7, O_2 = Polygon(N_2, M_2, 3.0)
Vieleck14, d_6, o'_1, e_9, P_2 = Polygon(O', E_2, 3.0)
Vieleck15, p_6, e_{10}, h_8, Q_2 = Polygon(E_2, H_2, 3.0)
Vieleck16, q_6, h_9, j_8, R_2 = Polygon(H_2, J_2, 3.0)
Vieleck17, r_6, j_9, l_8, S_2 = Polygon(J_2, L_2, 3.0)
Vieleck18, s_6, l_9, n_7, T_2 = Polygon(L_2, N_2, 3.0)
Vieleck19, t_6, o_2, m_8, U_2 = Polygon(O_2, M_2, 3.0)
Vieleck20, a_7, m_9, k_8, V_2 = Polygon(M_2, K_2, 3.0)
Vieleck21, b_7, k_9, i_8, W_2 = Polygon(K_2, I_2, 3.0)
Vieleck22, c_7, i_9, g_8, Z_2 = Polygon(I_2, G_2, 3.0)
Vieleck23, d_7, g_9, f_7, A_3 = Polygon(G_2, F_2, 3.0)
E_1 = (-11, 7)
F_1 = (11, 7)
G_1 = (11, -5)
H_1 = (-11, -5)
f_3 = Segment(E_1, F_1)
g_3 = Segment(F_1, G_1)
h_3 = Segment(G_1, H_1)
i_3 = Segment(H_1, E_1)
j_3 = Segment(O', A_3)
k_3 = Segment(A_3, R_2)
l_3 = Segment(P_2, Z_2)
m_3 = Segment(Q_2, W_2)
n_3 = Segment(R_2, V_2)
p_3 = Segment(S_2, U_2)
q_3 = Segment(T_2, O_2)
r_3 = Segment(N_2, V_2)
s_3 = Segment(T_2, W_2)
t_3 = Segment(O_2, U_2)
a_4 = Segment(S_2, Z_2)
b_4 = Segment(Q_2, F_2)
c_4 = Segment(P_2, O')
d_4 = Segment(O', N_2)
e_4 = Segment(F_2, O_2)
Vieleck3, v, o, w, s_4, z, t_4 = Polygon(V, O, W, S, Z, T)
I_1 = (-8, -2)
J_1 = (-8, -3)
K_1 = (-8, -4)
L_1 = Midpoint(H, I)
M_1 = Midpoint(H, J)
f_4 = CircleSector(H, L_1, M_1)
g_4 = CircleSector(H, L_1, M_1)
Vieleck24, o_1, v_1, j_4 = Polygon(V, J, O)
Vieleck25, w_1, j_5, i_4 = Polygon(J, I, W)
Vieleck26, w_2, j_{10}, o_3 = Polygon(J, O, W)
Vieleck27, w_3, i_{10}, s_7 = Polygon(I, S, W)
Vieleck28, z_1, i_{11}, s_8 = Polygon(I, S, Z)
Vieleck29, h_4, i_{12}, z_2 = Polygon(I, Z, H)
Vieleck30, h_5, t_7, z_3 = Polygon(T, Z, H)
Vieleck31, h_{10}, t_8, v_2 = Polygon(T, V, H)
Vieleck32, j_{11}, h_{11}, v_3 = Polygon(H, V, J)
Vieleck33, j_{12}, h_{12}, i_{13} = Polygon(H, I, J)
k_4 = Segment(W, T)
l_4 = Segment(V, O)
m_4 = Segment(O, W)
n_4 = Segment(W, S)
p_4 = Segment(S, Z)
q_4 = Segment(Z, T)
r_4 = Segment(T, C)
a_5 = Segment(C, Z)
b_5 = Segment(C, S)
c_5 = Segment(C, A)
d_5 = Segment(A, B)
e_5 = Segment(C, B)
f_5 = Segment(T, B)
g_5 = Segment(B, V)
k_5 = Segment(B, O)
p_7 = Segment(O, A)
q_7 = Segment(A, W)
r_7 = Segment(A, S)
a_8 = Segment(W, J)
b_8 = Segment(J, O)
c_8 = Segment(J, V)
d_8 = Segment(J, I)
f_8 = Segment(I, H)
n_8 = Segment(H, J)
p_8 = Segment(I, S)
q_8 = Segment(I, Z)
r_8 = Segment(H, Z)
a_9 = Segment(H, T)
b_9 = Segment(H, V)
c_9 = Segment(T, V)
d_9 = Segment(P_2, O')
f_9 = Segment(P_2, E_2)
n_9 = Segment(E_2, O')
p_9 = Segment(O', F_2)
q_9 = Segment(F_2, E_2)
r_9 = Segment(F_2, A_3)
s_9 = Segment(A_3, G_2)
t_9 = Segment(G_2, F_2)
a_{10} = Segment(E_2, G_2)
b_{10} = Segment(G_2, Z_2)
c_{10} = Segment(Z_2, I_2)
d_{10} = Segment(I_2, G_2)
f_{10} = Segment(I_2, H_2)
g_{10} = Segment(H_2, G_2)
k_{10} = Segment(H_2, E_2)
l_{10} = Segment(H_2, Q_2)
m_{10} = Segment(Q_2, E_2)
n_{10} = Segment(R_2, H_2)
p_{10} = Segment(R_2, J_2)
q_{10} = Segment(J_2, H_2)
r_{10} = Segment(J_2, I_2)
s_{10} = Segment(J_2, K_2)
t_{10} = Segment(K_2, I_2)
a_{11} = Segment(I_2, W_2)
b_{11} = Segment(K_2, W_2)
c_{11} = Segment(S_2, J_2)
d_{11} = Segment(S_2, L_2)
e_{11} = Segment(L_2, J_2)
f_{11} = Segment(L_2, K_2)
g_{11} = Segment(L_2, M_2)
k_{11} = Segment(M_2, K_2)
l_{11} = Segment(K_2, V_2)
m_{11} = Segment(M_2, V_2)
n_{11} = Segment(T_2, L_2)
p_{11} = Segment(T_2, N_2)
q_{11} = Segment(N_2, L_2)
r_{11} = Segment(N_2, M_2)
s_{11} = Segment(N_2, O_2)
t_{11} = Segment(O_2, M_2)
a_{12} = Segment(M_2, U_2)
b_{12} = Segment(O_2, U_2)
c_{12} = Segment(I, W)
N_1 = Intersect(d_4, c_4)
O_1 = Intersect(l_3, c_4)
P_1 = Intersect(b_4, d_4)
Q_1 = Intersect(b_4, j_3)
R_1 = Intersect(l_3, e_4)
S_1 = Intersect(j_3, k_3)
T_1 = Intersect(l_3, a_4)
U_1 = Intersect(m_3, s_3)
V_1 = Intersect(n_3, r_3)
W_1 = Intersect(p_3, t_3)
Z_1 = Intersect(e_4, t_3)
A_2 = Intersect(p_3, r_3)
C_2 = Intersect(e_4, s_3)
D_2 = Intersect(m_3, e_4)
B_3 = Intersect(q_3, r_3)
C_3 = Intersect(q_3, s_3)
D_3 = Intersect(p_3, a_4)
E_3 = Intersect(n_3, k_3)
F_3 = Intersect(m_3, b_4)
G_3 = Intersect(d_4, k_3)
H_3 = Intersect(m_3, n_3)
I_3 = Intersect(a_4, d_4)
J_3 = Intersect(d_4, s_3)
K_3 = Intersect(c_2, d_2)
L_3 = Intersect(s_1, d_2)
M_3 = Intersect(t_1, s_1)
N_3 = Intersect(t_1, a_2)
O_3 = Intersect(a_2, b_2)
P_3 = Intersect(b_2, c_2)
Q_3 = Intersect(e_1, j_2)
R_3 = Intersect(j, k_2)
S_3 = Intersect(i_1, e_2)
T_3 = Intersect(g_2, h_2)
U_3 = Intersect(b, m_2)
V_3 = Intersect(a_1, a)
d_{12} = Circle(G, S)
B_2 = Midpoint(Z, S)
W_3 = Midpoint(I, S)
e_{12} = Segment(W_3, Z)
f_{12} = Segment(B_2, I)
Z_3 = Intersect(e_{12}, f_{12})
g_{12} = Circle(G, B_2)
k_{12} = Circle(G, Z_3)
A_4 = Midpoint(O', F_2)
B_4 = Midpoint(O', E_2)
p_{12} = CircleSector(O', A_4, B_4)
q_{12} = CircleSector(O', A_4, B_4)
C_4 = (-8.334285714285718, -3.0019047619047625)
r_{12} = Circle(J_1, Segment[J_1, C_4])
l_{12} = Segment(K_1, I_1)
D_4 = Intersect(r_{12}, l_{12}, 2.0)
E_4 = Intersect(r_{12}, l_{12}, 1.0)