Author: 藍邦偉　老師
Date: 2 3 月 2014

$O$

$β=%v$

$α=%v$

單位向量

座標軸

x

y

z

$ O $

$A_2$

$A_3$

$A_4$

$A_1$

$A_5$

$A_6$

$A_7$

$A_8$

$θ_1=%v$

$C_5$

$C_6$

$M_1$

$S_1$

$S_a$

$M_2$

$θ_2=%v$

$C_1$

$C_2$

$S_2$

$S_b$

$θ_3=%v$

$C_8$

$C_7$

$M_3$

$S_c$

$θ_4=%v$

$C_4$

$C_3$

$M_4$

$S_d$

$S_e$

$S_f$

$E'$

$G_1$

O

$S_3$

$S_4$

$θ_1$

$θ_4$

$θ_2$

$θ_3$

$Step=%v$

$S_e$

$S_f$

$t=%v$

$S_1$

$S_2$

$S_3$

$S_4$

$正六面體的史坦納樹$

$1-b=%v$

$c=%v$

$1-t=%v$

$S_1$

$S_2$

$S_3$

$S_4$

$S_5$

$S_6$

$t$

$b$

$c$

1-b=

c=

1-t=

c

b

t

t=

t

$設S_1(0,1-t,0),S_3(0,1-b,c)$

O = (14.907592935771799, 3.9961067657971383)
β = 1.361356816555577 rad (slider from 0 to 6.283185307179586, step 1*°)
Z_{unit}: O + (0, sin(β))
w = Vector(O, Z_{unit})
α = 4.276056667386108 rad (slider from 0 to 6.283185307179586, step 1*°)
X_{unit}: O + (cos(α), (sin(α) * cos(β)))
u = Vector(O, X_{unit})
Y_{unit}: O + ((-1 * sin(α)), (cos(α) * cos(β)))
v = Vector(O, Y_{unit})
L = 2.3 (slider from 1 to 10, step 1)
yy = Vector(O - (L * v), O + (L * v))
zz = Vector(O - (L * w), O + (L * w))
xx = Vector(O - (L * u), O + (L * u))
c_z = false
d = false
Q_2: (0.3 * Corner[2]) + (0.7 * Corner[1])
Q_3: (x(Q_2), y(Corner[3]))
q_0 = Segment(Q_2, Q_3)
poly0, q_2, q_1, q_4, q_3 = Polygon(Corner[1], Q_2, Q_3, Corner[4])
A_2: O - u + v + w
A_3: O - u - v + w
A_4: O + u - v + w
A_1: O + u + v + w
A_5: O + u + v - w
A_6: O - u + v - w
A_7: O - u - v - w
A_8: O + u - v - w
t0 = 0 (slider from 0 to 1, step 0.01)
B_0: O + ((5 - (5 * t0)) * u)
B_1: B_0 + v + w
B_2: B_0 - v + w
B_3: B_0 - v - w
B_4: B_0 + v - w
B_5: B_0 + ((1 - 1 / sqrt(3)) * v)
B_6: B_0 - ((1 - 1 / sqrt(3)) * v)
b_1 = Segment(B_1, B_2)
b_2 = Segment(B_3, B_2)
b_3 = Segment(B_3, B_4)
b_4 = Segment(B_1, B_4)
b_5 = Segment(B_1, B_5)
b_6 = Segment(B_4, B_5)
b_7 = Segment(B_6, B_5)
b_8 = Segment(B_6, B_2)
b_9 = Segment(B_6, B_3)
θ_1 = 0 rad (slider from 0 to 45*°, step 0.017453292519943295)
C_5: A_4 + (((2 * cos(θ_1)) - (2 * sin(θ_1))) * v) - (((2 * cos(θ_1)) + (2 * sin(θ_1))) * w)
C_6: A_3 + (((2 * cos(θ_1)) - (2 * sin(θ_1))) * v) - (((2 * cos(θ_1)) + (2 * sin(θ_1))) * w)
M_1 = Midpoint(A_3, A_4)
S_1: O + ((-1 + 1 / sqrt(6)) * v) + ((1 - 1 / sqrt(6)) * w)
S_a: M_1 + ((sqrt(6) / 6 * (cos(θ_1) - sin(θ_1))) * v) + ((sqrt(6) / 6 * ((-cos(θ_1)) - sin(θ_1))) * w)
c_1 = Segment(A_4, S_a)
c_2 = Segment(A_3, S_a)
c_3 = Segment(Midpoint[C_5, C_6], S_a)
Rectangle1, c_7, c_4, c_5, c_6 = Polygon(A_3, A_4, C_5, C_6)
M_2 = Midpoint(A_7, A_8)
θ_2 = 0 rad (slider from 0 to 45*°, step 0.017453292519943295)
C_1: A_8 + (((2 * cos(θ_2)) - (2 * sin(θ_2))) * v) + (((2 * cos(θ_2)) + (2 * sin(θ_2))) * w)
C_2: A_7 + (((2 * cos(θ_2)) - (2 * sin(θ_2))) * v) + (((2 * cos(θ_2)) + (2 * sin(θ_2))) * w)
S_2: O + ((-1 + 1 / sqrt(6)) * v) - ((1 - 1 / sqrt(6)) * w)
S_b: M_2 + ((sqrt(6) / 6 * (cos(θ_2) - sin(θ_2))) * v) + ((sqrt(6) / 6 * (cos(θ_2) + sin(θ_2))) * w)
Rectangle2, d_3, d_4, d_1, d_2 = Polygon(A_7, A_8, C_1, C_2)
d_5 = Segment(A_8, S_b)
d_6 = Segment(A_7, S_b)
d_7 = Segment(Midpoint[C_1, C_2], S_b)
θ_3 = 0 rad (slider from 0 to 45*°, step 0.017453292519943295)
C_8: A_1 + ((((-2) * cos(θ_3)) + (2 * sin(θ_3))) * v) + ((((-2) * cos(θ_3)) - (2 * sin(θ_3))) * w)
C_7: A_2 + ((((-2) * cos(θ_3)) + (2 * sin(θ_3))) * v) + ((((-2) * cos(θ_3)) - (2 * sin(θ_3))) * w)
M_3 = Midpoint(A_2, A_1)
S_c: M_3 + ((sqrt(6) / 6 * ((-cos(θ_3)) + sin(θ_3))) * v) - ((sqrt(6) / 6 * (cos(θ_3) + sin(θ_3))) * w)
Rectangle3, e_3, e_4, e_2, e_1 = Polygon(A_1, A_2, C_7, C_8)
e_5 = Segment(A_1, S_c)
e_6 = Segment(A_2, S_c)
e_7 = Segment(Midpoint[C_7, C_8], S_c)
θ_4 = 0 rad (slider from 0 to 45*°, step 0.017453292519943295)
C_4: A_5 + ((((-2) * cos(θ_4)) + (2 * sin(θ_4))) * v) + (((2 * cos(θ_4)) + (2 * sin(θ_4))) * w)
C_3: A_6 + ((((-2) * cos(θ_4)) + (2 * sin(θ_4))) * v) + (((2 * cos(θ_4)) + (2 * sin(θ_4))) * w)
M_4 = Midpoint(A_5, A_6)
S_d: M_4 + ((sqrt(6) / 6 * ((-cos(θ_4)) + sin(θ_4))) * v) + ((sqrt(6) / 6 * (cos(θ_4) + sin(θ_4))) * w)
Rectangle4, f_1, f_2, f_4, f_3 = Polygon(A_5, A_6, C_3, C_4)
f_5 = Segment(A_5, S_d)
f_6 = Segment(A_6, S_d)
f_7 = Segment(Midpoint[C_3, C_4], S_d)
s_1 = Segment(S_a, A_4)
s_0 = Segment(S_a, A_3)
s_2 = Segment(S_b, A_7)
s_3 = Segment(S_b, A_8)
s_4 = Segment(S_c, A_1)
s_5 = Segment(S_c, A_2)
s_6 = Segment(S_d, A_5)
s_7 = Segment(S_d, A_6)
S_e: B_0 - ((1 - 1 / sqrt(3)) * v)
S_f: B_0 + ((1 - 1 / sqrt(3)) * v)
E': O - ((1 + sqrt(6) / 2) * v) + ((1 + sqrt(6) / 2) * w)
G_1: O - ((1 + sqrt(6) / 6) * v) + ((1 + sqrt(6) / 6) * w)
h_1 = Segment(E', A_4)
h_2 = Segment(E', A_3)
i = false
S_3: O + ((1 - 1 / sqrt(6)) * v) + ((1 - 1 / sqrt(6)) * w)
S_4: O + ((1 - 1 / sqrt(6)) * v) - ((1 - 1 / sqrt(6)) * w)
Poly_a, a_{26}, a_{27}, a_{28}, a_{29} = Polygon(A_3, A_4, A_5, A_6)
Poly_b, a_{30}, a_{31}, a_{32}, a_{25} = Polygon(A_1, A_2, A_7, A_8)
γ_1 = Angle(C_5, A_4, A_5)
γ_4 = Angle(C_4, A_5, A_4)
γ_2 = Angle(A_1, A_8, C_1)
γ_3 = Angle(A_8, A_1, C_8)
a_1, a_2 = SDLine(A_1, A_5, A_4, A_2)
a_3, a_4 = SDLine(A_4, A_1, A_8, A_3)
a_7, a_8 = SDLine(A_2, A_6, A_5, A_7)
a_9, a_{10} = SDLine(A_3, A_2, A_4, A_7)
a_5, a_6 = SDLine(A_1, A_2, A_5, A_4)
a_{11}, a_{12} = SDLine(A_3, A_7, A_2, A_4)
a_{13}, a_{14} = SDLine(A_3, A_4, A_7, A_2)
a_{15}, a_{16} = SDLine(A_4, A_8, A_7, A_5)
a_{17}, a_{18} = SDLine(A_7, A_6, A_3, A_8)
a_{19}, a_{20} = SDLine(A_6, A_5, A_1, A_8)
a_{21}, a_{22} = SDLine(A_5, A_8, A_1, A_6)
a_{23}, a_{24} = SDLine(A_7, A_8, A_6, A_3)
m_a = Segment(A_3, S_1)
m_b = Segment(A_4, S_1)
m_c = Segment(M_4, S_1)
p_a = Segment(S_4, A_5)
p_b = Segment(S_4, A_6)
p_c = Segment(S_4, M_1)
o_a = Segment(S_3, A_1)
o_b = Segment(S_3, A_2)
o_c = Segment(S_3, M_2)
n_a = Segment(S_2, A_7)
n_b = Segment(S_2, A_8)
n_c = Segment(S_2, M_3)
n = 0 (slider from 0 to 12, step 1)
poly1 = Poly_G4(A_4, A_8, A_5, A_1)
poly2 = Poly_G4(A_2, A_1, A_5, A_6)
poly5 = Poly_G4(A_3, A_4, A_1, A_2)
poly3 = Poly_G4(A_3, A_2, A_6, A_7)
poly4 = Poly_G4(A_4, A_3, A_7, A_8)
poly6 = Poly_G4(A_7, A_6, A_5, A_8)
S_g: O - ((1 - 1 / sqrt(3)) * v)
S_h: O + ((1 - 1 / sqrt(3)) * v)
s_8 = Segment(S_d, S_h)
s_9 = Segment(S_c, S_h)
s_a = Segment(S_g, S_h)
s_b = Segment(S_g, S_a)
s_c = Segment(S_g, S_b)
t = 0.293 (slider from 0.1 to 1, step 0.001)
D_1: O + (t * v) + (t * w)
D_2: O - (t * v) + (t * w)
D_3: O - (t * v) - (t * w)
D_4: O + (t * v) - (t * w)
g_a = Segment(D_1, D_2)
g_b = Segment(D_3, D_2)
g_c = Segment(D_3, D_4)
g_d = Segment(D_1, D_4)
g_e = Segment(A_1, D_1)
g_f = Segment(A_2, D_1)
g_g = Segment(A_3, D_2)
g_h = Segment(A_4, D_2)
g_i = Segment(A_7, D_3)
g_j = Segment(A_8, D_3)
g_k = Segment(A_5, D_4)
g_l = Segment(A_6, D_4)
b = 0.7113248654051871 (slider from 0 to 1, step 0.001)
c = 0.5 (slider from 0 to 1, step 0.001)
e = 0.42264973081037416 (slider from 0 to 1, step 0.001)
H_1: O + (e * v)
H_2: O - (e * v)
H_3: O + (b * v) + (c * w)
H_4: O - (b * v) + (c * w)
H_5: O - (b * v) - (c * w)
H_6: O + (b * v) - (c * w)
r_1 = Segment(H_1, H_2)
r_2 = Segment(H_1, H_3)
r_3 = Segment(H_1, H_6)
r_4 = Segment(H_4, H_2)
r_5 = Segment(H_5, H_2)
r_6 = Segment(A_1, H_3)
r_7 = Segment(A_2, H_3)
r_8 = Segment(A_3, H_4)
r_9 = Segment(A_4, H_4)
r_a = Segment(A_5, H_6)
r_b = Segment(A_6, H_6)
r_c = Segment(A_7, H_5)
r_d = Segment(A_8, H_5)
d1: (2 * e)
d2: sqrt((b - e)^(2) + c^(2))
d3: sqrt(3 - (2 * b) - (2 * c) + b^(2) + c^(2))
text12_1: "\overline{S_1S_2}+4×\overline{S_1S_3}+8×\overline{A_1S_3}
\\=" + (LaTeX[d1]) + "+4×" + (LaTeX[d2]) + "+8" + (LaTeX[d3]) + "
\\=" + (LaTeX[d1 + (4 * d2) + (8 * d3)]) + ""
a = 0.711
r_q = Segment(O, H_1)
r_e = Segment(H_1, O + v)
r_f = Segment(H_3, O + v + (c * w))
r_g = Segment(H_3, O + (b * v))
inputBoxC = Textfield(c)
inputBoxb = Textfield(b)
inputBoxt = Textfield(e)
inputBox1 = Textfield(t)
text10_2: "4\overline{S_1S_2}+8\overline{A_1S_1}=" + (LaTeX[(8 * t) + (8 * sqrt((2 * t^(2)) - (4 * t) + 3))]) + ""
text10_3: "t=\frac{2-\sqrt{2}}{2}≈" + (LaTeX[0.2928932188134524]) + ""
text12_2: "$1-t=1-\frac{\sqrt{3}}{3}≈" + (LaTeX[0.423]) + "$
$1-b=1-\frac{\sqrt{3}}{6}≈" + (LaTeX[0.711]) + "$"
A = (16.413655675058994, 4.099576724984807)

Tool: SDLine
Input: A, B, C, D
Output: g, h

b: sgn(sin(Angle[D, B, A]))
a: sgn(sin(Angle[A, B, C]))
c: sgn(a + b + 0.5)
d: sqrt(c)
e: sqrt((-c))
E = Midpoint(A, B)
A' = Dilate(A, d, E)
B' = Dilate(B, d, E)
g = Segment(A', B')
A'_1 = Dilate(A, e, E)
B'_1 = Dilate(B, e, E)
h = Segment(A'_1, B'_1)

Tool: Poly_G4
Input: A, B, C, D
Output: poly1'

α = Angle(C, B, A)
a: sqrt(sgn(sin(α)))
A' = Dilate(A, a, B)
B' = Dilate(B, a, B)
C' = Dilate(C, a, B)
D' = Dilate(D, a, B)
poly1', a', b', c', d' = Polygon(A', B', C', D')