/Users/sf/Dropbox (University of Michigan)/FominPylyavskyy/IncidenceGeometry/geogebra-seattle1

$\text{\Large Pappus of Alexandria [}$\approx$\text{\Large340 C.E.]}$

$\text{\Large Girard Desargues [} $\approx$\text{\Large1639]}$

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

$\text{\LARGE {\bf Geometric miracles}}$

$\text{\LARGE {\bf Sergey Fomin (University of Michigan)}}$

$\text{\LARGE {\it Geometry} (from Ancient Greek γεωμετρία ``land measurement'')}$

$\text{\LARGE such as the distance, shape, size, and relative position of figures.}$

$\text{\LARGE is a branch of mathematics concerned with properties of space }$

$\text{\LARGE \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad -- {\it Wikipedia}}$

$\mathbf A$

$\mathbf B$

$\mathbf D$

$\mathbf E$

$\mathbf C$

$\mathbf F$

$\mathbf A$

$\mathbf F$

$\mathbf A$

$\mathbf C$

$\mathbf A$

$\mathbf B$

$\mathbf D$

$\mathbf E$

$\mathbf D$

$\mathbf B$

$\mathbf A$

$\mathbf D$

$\mathbf B$

$\mathbf A$

$\mathbf E$

$\mathbf C$

$\mathbf B$

$\mathbf C$

$\mathbf E$

$\mathbf D$

$\mathbf A$

$\mathbf D$

$\text{\Large octahedron}$

$\text{\Large torus}$

$\text{\Large Why do these miracles happen?}$

$\text{\Large How can we produce more of these ``magic tricks''?}$

A = (-24.639560874527913, -8.536250065580687)
B = (-8.078373426688353, -7.129913488894489)
f = Segment(A, B)
C = (-19.308075942016828, -3.015854249633373)
D = (-18.762333389870022, -12.146546949014004)
g = Segment(C, D)
E = Point(g)
h = Segment(A, E)
F = Point(f)
i = Segment(E, F)
j = Segment(F, D)
k = Segment(D, A)
G = Point(k)
J = (-27.11639245734802, 21.521570498816196)
K = (-6.839957635278273, 20.388105198204165)
q = Segment(J, K)
L = (-26.633620199679694, 10.102957100051004)
M = (-6.6090665555238655, 11.740184756491999)
r = Segment(L, M)
N = Point(q)
O = Point(q)
P = Point(q)
Q = Point(r)
R = Point(r)
S = Point(r)
V = (-27.03243206471005, 23.053847664459322)
W = (-27.158372653667005, -1.7144681637445889)
H = (-27.179362751826503, -31.772288728140353)
I = (-26.88550137759361, 42.805530032543956)
T = (-26.88550137759361, 41.21028257242235)
U = (-26.990451868391073, 38.06176784849812)
Z = (-26.96946177023158, 35.62691646199661)
A_1 = (-26.948471672072085, 36.86533225340692)
B_1 = (-26.96946177023158, 34.241569983470065)
C_1 = (-15.032458554384283, -41.40392708535232)
D_1 = (-9.349738064250273, -41.46438155865162)
l = Segment(C_1, D_1)
E_1 = (-6.206105452686778, -38.774157496832835)
m = Segment(D_1, E_1)
F_1 = (-11.979507652769735, -38.774157496832835)
n = Segment(C_1, F_1)
p = Segment(F_1, E_1)
G_1 = (-10.982008843331318, -34.72370778577983)
s = Segment(G_1, D_1)
t = Segment(G_1, F_1)
a = Segment(G_1, E_1)
b = Segment(G_1, C_1)
H_1 = (-10.619282003535531, -46.02869429274866)
c = Segment(C_1, H_1)
d = Segment(H_1, D_1)
e = Segment(H_1, E_1)
f_1 = Segment(F_1, H_1)
I_1 = (-25.26926381931263, -33.49347677721937)
J_1 = (-25.26926381931263, -36.53704101034609)
g_1 = Segment(I_1, J_1)
K_1 = (-25.290253917472125, -39.66456563611079)
h_1 = Segment(J_1, K_1)
L_1 = (-25.27949177861529, -43.00597062778376)
i_1 = Segment(K_1, L_1)
M_1 = (-25.26926381931263, -46.29743665451108)
j_1 = Segment(L_1, M_1)
N_1 = (-20.67243232238378, -46.31842675267055)
k_1 = Segment(M_1, N_1)
O_1 = (-20.714412518702765, -43.0229813416299)
l_1 = Segment(N_1, O_1)
P_1 = (-20.735402616862256, -39.66456563611079)
m_1 = Segment(O_1, P_1)
Q_1 = (-20.77738281318124, -36.55803110850559)
n_1 = Segment(P_1, Q_1)
R_1 = (-20.798372911340735, -33.3885262864219)
p_1 = Segment(Q_1, R_1)
q_1 = Segment(R_1, I_1)
r_1 = Segment(J_1, R_1)
s_1 = Segment(J_1, Q_1)
t_1 = Segment(J_1, P_1)
a_1 = Segment(P_1, K_1)
b_1 = Segment(L_1, P_1)
c_1 = Segment(L_1, O_1)
d_1 = Segment(L_1, N_1)
S_1 = (-17.5449076966194, -57.21228769744833)
T_1 = (-15.781739451222034, -61.55723801646372)
U_1 = (-9.526690199693276, -61.20040634775231)
V_1 = (-9.106888236503426, -56.9604065195344)
W_1 = (-13.178967279444967, -55.68001053180523)
e_1 = Conic(S_1, T_1, U_1, V_1, W_1)
Z_1 = (-15.67678896042457, -58.5346638814965)
A_2 = (-13.011046494169026, -59.89902026186365)
B_2 = (-10.618175303986884, -58.68159456861296)
c_2 = CircumcircleArc(Z_1, A_2, B_2)
C_2 = Point(c_2)
D_2 = (-13.262927672082936, -58.1148619183066)
E_2 = Point(c_2)
d_2 = CircumcircleArc(C_2, D_2, E_2)
F_2 = (-26.255798432808778, -51.83882256861774)
G_2 = (-22.18371938986724, -51.88080276493673)
f_2 = Segment(F_2, G_2)
H_2 = (-26.276788530968272, -54.69347591830902)
g_2 = Segment(F_2, H_2)
I_2 = (-22.141739193548254, -57.40119858088382)
h_2 = Segment(H_2, I_2)
J_2 = (-22.162729291707745, -54.73545611462801)
i_2 = Segment(F_2, J_2)
j_2 = Segment(F_2, I_2)
k_2 = Segment(G_2, J_2)
l_2 = Segment(J_2, I_2)
K_2 = (-26.276788530968272, -57.38020848272433)
m_2 = Segment(H_2, K_2)
n_2 = Segment(K_2, I_2)
L_2 = (-22.12074909538876, -60.171891537937114)
p_2 = Segment(K_2, L_2)
q_2 = Segment(I_2, L_2)
M_2 = (-22.135859167051795, -62.804810633305664)
r_2 = Segment(L_2, M_2)
N_2 = (-26.307217824703358, -62.86526510660496)
O_2 = (-26.297778629127762, -60.19288163609661)
s_2 = Segment(N_2, O_2)
P_2 = (-26.297778629127762, -65.54535666676774)
Q_2 = (-22.162729291707745, -65.54535666676774)
t_2 = Segment(P_2, Q_2)
a_2 = Segment(Q_2, M_2)
b_2 = Segment(N_2, L_2)
e_2 = Segment(O_2, L_2)
f_3 = Segment(O_2, K_2)
g_3 = Segment(P_2, L_2)
h_3 = Segment(P_2, M_2)
i_3 = Segment(N_2, P_2)
R_2 = (-17.376986911343472, -33.49347677722126)
S_2 = (-18.258571034042202, -51.964763157573515)
T_2 = (-25.96193705857595, -14.22456666680395)
U_2 = (-25.982927156735443, -16.533477464348454)