Author: 藍邦偉　老師
Date: 8 3 月 2014

$β=%v$

$O$

$α=%v$

單位向量

座標軸

x

y

z

$ O $

$A$

$B$

$C$

$D$

$E$

$F$

$S_1$

$S_2$

$S_3$

$S_4$

$S_5$

$S_6$

O

$S_3$

$Step=%v$

$正八面體的史坦納樹$

$t=%v$

$S_a$

$S_b$

$S_f$

$S_c$

$S_d$

$S_e$

t=

t

$t≈0.206$

$將正八面體的每個頂點與正六邊$
$形的頂點的兩條連線刪去一條$

$將造成迴路的正六邊形中的某一$
$邊拆掉$

$將\overline{AS_5}連到\overline{AS_1}$
$將\overline{CS_4}連到\overline{CS_2}$
$將\overline{BS_1}連到\overline{BS_3}$

$將多餘的S_4與\overline{S_2S_4}拆掉$

$將S_5與\overline{S_5S_1},\overline{S_5S_6}拆掉$

$連接\overline{S_1S_6}$

$S_7$

Q_2: 0.31 * Corner[2] + 0.69 * Corner[1]
β = 1.3439035240356338 rad (slider from 0 to 6.283185307179586, step 0.1*°)
O: 0.5 * (Q_2 + Corner[3])
Z_{unit}: O + (0, sin(β))
w = Vector(O, Z_{unit})
α = 4.276056667386108 rad (slider from 0 to 6.283185307179586, step 0.1*°)
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 = 1.7 (slider from 1 to 2, step 0.1)
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 = false
d = false
Q_3: (x(Q_2), y(Corner[3]))
q_2 = Segment(Q_2, Q_3)
poly1, q_3, q_1, q_4, q_0 = Polygon(Corner[1], Q_2, Q_3, Corner[4])
A: O + u + v
B: O - u + v
C: O - u - v
D: O + u - v
E: O + sqrt(2) * w
F: O - sqrt(2) * w
S_1: O + (1 - sqrt(3) / 3) * v
S_2: O + (-1 + sqrt(3) / 3) * v
S_3: O - (3 - sqrt(3)) / 6 * u + (3 - sqrt(3)) / 6 * v + (3 * sqrt(2) - sqrt(6)) / 6 * w
S_4: O - (3 - sqrt(3)) / 6 * u - (3 - sqrt(3)) / 6 * v + (3 * sqrt(2) - sqrt(6)) / 6 * w
S_5: O + (3 - sqrt(3)) / 6 * u + (3 - sqrt(3)) / 6 * v + (-3 * sqrt(2) + sqrt(6)) / 6 * w
S_6: O + (3 - sqrt(3)) / 6 * u - (3 - sqrt(3)) / 6 * v + (-3 * sqrt(2) + sqrt(6)) / 6 * w
b_0 = Segment(E, S_4)
b_1 = Segment(E, S_3)
b_4 = Segment(A, S_1)
b_a = Segment(D, S_2)
b_d = Segment(S_1, S_3)
b_e = Segment(S_4, S_3)
b_f = Segment(S_4, S_2)
b_c = Segment(S_2, S_6)
b_g = Segment(S_5, S_6)
b_h = Segment(S_5, S_1)
poly2, s_1, s_3, s_4, s_2, s_6, s_5 = Polygon(S_1, S_3, S_4, S_2, S_6, S_5)
a = Segment(S_3, S_5)
c_1 = Segment(S_1, A)
c_2 = Segment(S_1, B)
c_0 = Segment(S_1, S_2)
c_3 = Segment(C, S_2)
c_4 = Segment(D, S_2)
d_1 = Segment(S_3, B)
d_2 = Segment(S_3, E)
d_0 = Segment(S_3, S_6)
d_3 = Segment(D, S_6)
d_4 = Segment(F, S_6)
e_2 = Segment(F, S_5)
e_0 = Segment(S_4, S_5)
e_4 = Segment(S_4, E)
e_1 = Segment(A, S_5)
e_3 = Segment(S_4, C)
l = false
a_1, a_2 = SDLine(E, A, D, B)
a_3, a_4 = SDLine(A, F, D, B)
a_5, a_6 = SDLine(E, B, A, C)
a_7, a_8 = SDLine(B, F, A, C)
a_9, a_{10} = SDLine(E, C, B, D)
a_{11}, a_{12} = SDLine(C, F, B, D)
a_{13}, a_{14} = SDLine(E, D, C, A)
a_{15}, a_{16} = SDLine(D, F, C, A)
a_{17}, a_{18} = SDLine(C, D, F, E)
a_{19}, a_{20} = SDLine(D, A, F, E)
a_{21}, a_{22} = SDLine(A, B, F, E)
a_{23}, a_{24} = SDLine(B, C, F, E)
poly3 = Poly_G3(E, A, B)
poly4 = Poly_G3(A, F, B)
poly5 = Poly_G3(E, B, C)
poly6 = Poly_G3(C, B, F)
poly7 = Poly_G3(E, C, D)
poly8 = Poly_G3(D, C, F)
poly9 = Poly_G3(E, D, A)
poly10 = Poly_G3(A, D, F)
p_c = PolyLine(A, F, C, E, A)
S_0: S_3
k_a = Segment(S_0, B)
k_b = Segment(S_0, E)
n = 0 (slider from 0 to 21, step 1)
p_a = PolyLine(A, B, C, D, A)
p_b = PolyLine(E, D, F, B, E)
t = 0.423 (slider from 0 to 1, step 0.0010)
S_a: O + t * v
S_b: O - t * v
S_f: O + 0.5 * t * u + 0.5 * t * v - 1 / sqrt(2) * t * w
S_c: O - 1 / 2 * t * u + 0.5 * t * v + 1 / sqrt(2) * t * w
S_d: O + 0.5 * t * u - 0.5 * t * v - 1 / sqrt(2) * t * w
S_e: O - 0.5 * t * u - 0.5 * t * v + 1 / sqrt(2) * t * w
f_1 = Segment(S_c, E)
f_7 = Segment(S_b, S_d)
f_8 = Segment(S_f, S_d)
f_0 = Segment(S_c, S_a)
f_a = Segment(S_e, S_b)
k_4 = Segment(D, S_6)
n_m: sqrt(t^(2) - 2 * t + 2)
f_6 = Segment(S_b, D)
k_3 = Segment(C, S_2)
k_2 = Segment(B, S_1)
b_8 = Segment(B, S_3)
f_5 = Segment(S_b, C)
b_3 = Segment(F, S_6)
b_6 = Segment(C, S_4)
k_1 = Segment(A, S_5)
b_2 = Segment(F, S_5)
b_i = Segment(S_1, S_6)
f_4 = Segment(S_d, F)
f_3 = Segment(S_a, A)
f_9 = Segment(S_d, S_a)
g_1 = Segment(E, S_4)
g_2 = Segment(C, S_2)
g_3 = Segment(D, S_6)
g_4 = Segment(F, S_5)
g_5 = Segment(A, S_1)
g_6 = Segment(B, S_3)
g_7 = Segment(S_4, S_3)
g_8 = Segment(S_4, C)
g_9 = Segment(S_5, A)
g_{10} = Segment(S_1, S_5)
g_{11} = Segment(S_6, S_5)
g_{12} = Segment(S_4, S_2)
iB17 = Textfield(t)
k_k: (2 + sqrt(3)) * t + 6 * n_m
G = (6.928022317189301, 4.087147536942045)
H: Corner[4] + (0.15, -0.3)
text17_1: "(2+\sqrt3)\overline{S_aS_c}+6\overline{ES_c}
\\=(2+\sqrt3)×" + (LaTeX[t]) + "+6×" + (LaTeX[n_m]) + "
\\=" + (LaTeX[If[Mod[floor(1000 * k_k), 1000] ≟ 0, k_k + ".000", If[Mod[floor(1000 * k_k), 100] ≟ 0, k_k + "00", If[Mod[floor(1000 * k_k), 10] ≟ 0, k_k + "0", If[Mod[floor(1000 * k_k), 10] ≠ 0, k_k + ""]]]]]) + ""
tri1_{12}, s_7, c_5, s_8 = Polygon(C, S_2, S_4)
tri2_{12}, s_9, a_{25}, s_{10} = Polygon(A, S_1, S_5)
h_1 = Segment(E, S_4)
h_2 = Segment(C, S_4)
h_3 = Segment(F, S_6)
h_4 = Segment(D, S_6)
h_6 = Segment(C, S_2)
h_5 = Segment(D, S_2)
h_7 = Segment(A, S_5)
h_8 = Segment(F, S_5)
h_9 = Segment(A, S_1)
h_{10} = Segment(B, S_1)
tri3_{12}, s_{11}, b, s_{12} = Polygon(B, S_1, S_3)
k_c = Segment(B, S_1)
k_d = Segment(B, S_3)
f_2 = Segment(S_c, B)
S_7: O
j_1 = Segment(S_1, S_7)
j_2 = Segment(S_4, S_7)
j_3 = Segment(S_6, S_7)
j_4 = Segment(S_1, A)
j_5 = Segment(S_1, B)
j_6 = Segment(S_6, D)
j_7 = Segment(S_6, F)
j_8 = Segment(S_4, C)
j_9 = Segment(S_4, E)

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_G3
Input: A, B, C
Output: poly1'

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