/Users/sf/Dropbox (University of Michigan)/FominPylyavskyy/IncidenceGeometry/geogebra-seattle2

$(\mathbf A\mathbf C)$

$(\mathbf A\mathbf B)$

$\mathbf R$

$\mathbf P$

$\mathbf B$

$\mathbf N$

$\mathbf C$

$\mathbf M$

$\mathbf D$

$\mathbf A$

2

1

$\mathbf C$

$\mathbf D$

$\mathbf A$

$\mathbf B$

$\mathbf E$

$\mathbf P$

2

$\mathbf Q$

$\mathbf R$

1

$\mathbf A$

$\mathbf B$

$\mathbf C$

$\mathbf D$

$\mathbf A$

$\mathbf B$

1

2

3

4

1

$\mathbf M$

$\mathbf M$

$\mathbf N$

$\mathbf N$

$\mathbf A$

$\mathbf C$

$\mathbf D$

$\mathbf B$

$\text{\Large tetrahedron $\longrightarrow$\Large Desargues' theorem}$

$\text{\Large torus $\longrightarrow$\Large Pappus' theorem}$

$\mathbf A$

$\mathbf B$

$\mathbf B$

$\mathbf C$

$\mathbf C$

$\mathbf A$

$\mathbf A$

$\mathbf B$

$\mathbf R$

$\mathbf Q$

$\mathbf P$

$\mathbf Q$

$\mathbf R$

$\mathbf P$

1

2

3

4

5

6

1

$(\mathbf B\mathbf C)$

$\mathbf Q$

1

$\text{\Large triangular bipyramid}$

$\mathbf A$

$\mathbf C$

$\mathbf D$

$\mathbf C$

$\mathbf A$

$\mathbf E$

$\mathbf{B}$

$\mathbf B$

1

2

3

4

5

6

1

$\mathbf P$

$\mathbf Q$

$\mathbf Q$

$\mathbf P$

$\mathbf R$

$\mathbf R$

$\mathbf C$

$\mathbf B$

$\mathbf E$

$\mathbf D$

$\mathbf A$

$\text{\Large S.~Fomin and P.~Pylyavskyy, Incidences and tilings, \texttt{arXiv:2305.07728}, Section~8}$

2

A = (-2.555566563590378, 3.3265301883026255)
B = (8.191582439055933, 3.255238984304982)
f = Segment(A, B)
C = (-2.5020981605921375, -2.572816942503144)
D = (8.22722804105476, -2.038132912520748)
g = Segment(C, D)
E = Point(f)
F = Point(g)
I = (1.632791671271783, 9.404105329102633)
J = (1.2050444472858601, 18.262037425811098)
p = Segment(I, J)
K = (-1.5040213046249844, 14.590540419931804)
L = (7.122214379091126, 14.697477225928417)
q = Segment(K, L)
M = Point(p)
r = Segment(M, K)
s = Segment(K, I)
N = Point(q)
t = Segment(I, N)
a = Segment(N, M)
O = Point(s)
b = Segment(L, O)
Q = Intersect(t, b)
R = (0.6508823416264866, -16.021215712004338)
S = (3.450717373211955, -14.69232162210416)
f_1 = Line(R, S)
T = (3.9559174644731576, -18.238517756445766)
g_1 = Line(R, T)
U = (3.509667996409472, -16.592972842960933)
V = (5.685134153219939, -15.881762753234435)
h_1 = Line(U, V)
i_1 = Segment(R, U)
j_1 = Segment(U, S)
k_1 = Segment(U, T)
l_1 = Segment(R, V)
m_1 = Segment(T, V)
W = Point(g_1)
Z = Point(i_1)
p_1 = Segment(W, Z)
A_1 = Point(f_1)
C_1 = Point(h_1)
F_1 = Intersect(k_1, p_1)
b_1 = Segment(U, C_1)
c_1 = Segment(R, W)
d_1 = Segment(R, A_1)
n_1 = Segment(S, V)
B_1 = (-7.36772283343201, 13.41423555397053)
D_1 = (-4.26655545953407, 13.36076715097229)
q_1 = Segment(B_1, D_1)
E_1 = (-7.409498675273832, 15.983238325687811)
r_1 = Segment(D_1, E_1)
s_1 = Segment(E_1, B_1)
G_1 = (-4.284378260533483, 15.94507329588721)
t_1 = Segment(E_1, G_1)
a_1 = Segment(G_1, D_1)
H_1 = (-7.42119123643025, 18.547202241801543)
e_1 = Segment(E_1, H_1)
f_2 = Segment(H_1, G_1)
I_1 = (-4.284378260533483, 18.565025042800958)
g_2 = Segment(G_1, I_1)
h_2 = Segment(H_1, I_1)
J_1 = Point(q_1)
K_1 = Point(r_1)
L_1 = Point(t_1)
M_1 = Point(f_2)
N_1 = Point(h_2)
O_1 = Point(s_1)
P_1 = Point(e_1)
Q_1 = Point(a_1)
R_1 = Point(g_2)
P = (-7.242963226436116, 9.19023171710959)
S_1 = (-6.102303962473655, 11.88147466802099)
c = Segment(P, S_1)
T_1 = (-3.5179978175587046, 10.77646100605737)
d = Segment(S_1, T_1)
U_1 = (-4.106150250539348, 9.172408916110177)
e = Segment(T_1, U_1)
i_2 = Segment(U_1, P)
j_2 = Segment(P, T_1)
k_2 = Segment(S_1, U_1)
V_1 = (-7.395553379396843, 19.60901525370534)
G = (-8.829192515383912, 4.73453146725631)
H = (-7.493170450535773, 3.5440344034127476)
W_1 = (-4.690165979260748, 3.5858702910437175)
h = Segment(H, W_1)
Z_1 = (-7.4746596394284905, 1.3481992773677627)
i = Segment(H, Z_1)
A_2 = (-4.658657081521166, 1.3481992773677627)
j = Segment(Z_1, A_2)
k = Segment(A_2, W_1)
B_2 = (-7.456836838429077, -0.7905368425618269)
l = Segment(Z_1, B_2)
C_2 = (-4.623011479522338, -0.8261824445606534)
m = Segment(B_2, C_2)
n = Segment(C_2, A_2)
D_2 = (-7.42119123643025, -2.982741365489656)
l_2 = Segment(B_2, D_2)
E_2 = (-4.587365877523512, -3.0005641664890694)
m_2 = Segment(D_2, E_2)
n_2 = Segment(E_2, C_2)
p_2 = Segment(H, A_2)
q_2 = Segment(Z_1, C_2)
r_2 = Segment(B_2, E_2)
F_2 = Point(l_2)
G_2 = Point(l)
H_2 = Point(i)
I_2 = Point(k)
J_2 = Point(n)
K_2 = Point(n_2)
L_2 = Point(m_2)
M_2 = Point(r_2)
N_2 = Point(m)
O_2 = Point(q_2)
P_2 = Point(j)
Q_2 = Point(p_2)
R_2 = Point(h)
U_2 = (-2.4842753595927243, 0.9917432573794962)
V_2 = (8.191582439055933, 0.082780406409423)
b_2 = Segment(U_2, V_2)
W_2 = Point(b_2)
Z_2 = Point(g)
A_3 = (-7.813912255706549, -12.548837038633797)
s_2 = Segment(Z_2, E)
S_2 = (-7.813292858417346, -21.32240359388539)
T_2 = (-7.7954700574179325, -19.254958677953503)
t_2 = Segment(S_2, T_2)
B_3 = (-7.777647256418519, -17.34791897101633)
a_2 = Segment(T_2, B_3)
C_3 = (-7.7581310721985846, -15.449458581047626)
c_2 = Segment(B_3, C_3)
D_3 = (-7.730240480444604, -13.650515412915915)
d_2 = Segment(C_3, D_3)
E_3 = (-4.229971215320073, -17.415745299703218)
e_2 = Segment(B_3, E_3)
F_3 = (-4.202080623566092, -13.622624821161935)
f_3 = Segment(E_3, F_3)
g_3 = Segment(F_3, D_3)
G_3 = (-4.230909857535242, -21.32240359388539)
h_3 = Segment(E_3, G_3)
i_3 = Segment(S_2, G_3)
j_3 = Segment(G_3, T_2)
k_3 = Segment(B_3, G_3)
l_3 = Segment(C_3, E_3)
m_3 = Segment(D_3, E_3)
H_3 = Point(i_3)
I_3 = Point(j_3)
J_3 = Point(k_3)
K_3 = Point(e_2)
L_3 = Point(l_3)
M_3 = Point(m_3)
N_3 = Point(g_3)
O_3 = Point(t_2)
P_3 = Point(a_2)
Q_3 = Point(c_2)
R_3 = Point(d_2)
S_3 = Point(f_3)
T_3 = Point(h_3)
U_3 = (-1.7616538450928134, -20.720780422550007)
V_3 = (0.2743593529477519, -21.41804521639951)
n_3 = Segment(U_3, V_3)
W_3 = (1.5015453901228872, -20.762616310180977)
p_3 = Segment(V_3, W_3)
q_3 = Segment(W_3, U_3)
Z_3 = (-0.4089601450247666, -19.005509029680223)
r_3 = Segment(U_3, Z_3)
s_3 = Segment(Z_3, V_3)
t_3 = Segment(Z_3, W_3)
A_4 = (-0.199780706869914, -22.86835598760648)
a_3 = Segment(A_4, U_3)
b_3 = Segment(A_4, V_3)
c_3 = Segment(A_4, W_3)
B_4 = (-7.381608083519851, 20.836201290880325)
C_4 = Intersect(b_2, s_2)