\[
  x(u,v;\alpha) = u \cos\alpha + v \sin\alpha \qquad \alpha \text{ défini mod } 2\pi\mathbb{Z}.
\]

\[
  v' = v \cos\theta + w \sin\theta .
\]

\[
\begin{cases}
  \|x(u,v;\alpha) - u\|^2 \underset{\text{par raison de sym.}}{=} \|x(u,v';\alpha) - u\|^2 \\ \qquad = (\cos\alpha - 1)^2 + \sin^2\alpha = 2(1 - \cos\alpha) = 4 \sin^2 \frac{\alpha}{2} \\[1ex]
  \|x(u,v,\alpha) - x(u,v',\alpha)\|^2 = \sin^2\alpha \, \|v - v'\|^2 \\ \qquad = \sin^2\alpha \; 4 \sin^2 \frac{\theta}{2} = \Bigl(4 \sin^2 \frac{\alpha}{2} \cos^2 \frac{\alpha}{2}\Bigr)\Bigl(4 \sin^2 \frac{\theta}{2}\Bigr)
\end{cases}
\]

\[
  \text{(1)} \qquad \cos\frac{\alpha}{2} = \pm \frac{1}{2 \sin \theta/2} .
\]

\[
  \text{(1')} \qquad \cos\alpha = \frac{1}{2\sin^2\theta/2} - 1 = \frac{1 - 2\sin^2\theta/2}{2\sin^2\theta/2} = \frac{\cos\theta}{2\sin^2\theta/2} = \frac{\cos\theta}{1 - \cos\theta}
\]

\[
  \text{(2)} \qquad \theta = \frac{2\pi}{5}
\]

\[
  \text{(3)} \qquad v_i = v \cos i\theta + w \sin i\theta \qquad i \in \mathbb{Z}/5\mathbb{Z}
\]

\[
  \text{(4)} \qquad x_i = x(u, v_i, \alpha) = u\cos\alpha + v_i \sin\alpha = u\cos\alpha + v\cos i\theta \sin\alpha + w \sin i\theta \sin\alpha \qquad (i \in \mathbb{Z}/5\mathbb{Z})
\]

\[
  \text{(5)} \qquad x'_i = -x_i , \quad u' = -u \qquad (i \in \mathbb{Z}/5\mathbb{Z})
\]

\[
  \text{(4)} \qquad \ell = 2 \sin\frac{\alpha}{2}
\]

\[
  \text{(4')} \qquad \ell' = 2 \cos\frac{\alpha}{2}
\]

\[
  \gamma u = x_0 , \quad \gamma x_0 = x'_{2} , \quad \gamma x_1 = x'_{-2} , \quad \gamma x_2 = x_1 , \quad \gamma x_3 = u , \quad \gamma x_4 = x_{-1}
\]

\begin{align*}
  \gamma u &= \gamma^{w}_{\alpha}(u) = u\cos\alpha + v\sin\alpha \\
  \gamma v &= \gamma^{w}_{\alpha}\Bigl(v\cos\frac{\theta}{2} - w\sin\frac{\theta}{2}\Bigr) = (-u\sin\alpha + v\cos\alpha)\cos\frac{\theta}{2} - w\sin\frac{\theta}{2} \\
  &= -u\sin\alpha\cos\frac{\theta}{2} + v\cos\alpha\cos\frac{\theta}{2} - w\sin\frac{\theta}{2} \\
  \gamma w &= \gamma^{w}_{\alpha}\Bigl(+v\sin\frac{\theta}{2} + w\cos\frac{\theta}{2}\Bigr) = (-u\sin\alpha + v\cos\alpha)\sin\frac{\theta}{2} + w\cos\frac{\theta}{2} \\
  &= -u\sin\alpha\sin\frac{\theta}{2} + v\cos\alpha\sin\frac{\theta}{2} + w\cos\frac{\theta}{2}
\end{align*}

\[
  \gamma = \begin{pmatrix}
    \cos\alpha & -\sin\alpha\cos\frac{\theta}{2} & -\sin\alpha\sin\frac{\theta}{2} \\
    \sin\alpha & \cos\alpha\cos\frac{\theta}{2} & \cos\alpha\sin\frac{\theta}{2} \\
    0 & -\sin\frac{\theta}{2} & \cos\frac{\theta}{2}
  \end{pmatrix}
\]

\begin{align*}
  \gamma u &= u\cos\alpha + v\sin\alpha = x_0 \qquad \text{OK} \\
  \gamma x_0 &= \gamma u \cos\alpha + \gamma v \sin\alpha = u\Bigl(\cos^2\alpha - \sin^2\alpha\cos\frac{\theta}{2}\Bigr) + v\Bigl(\sin\alpha\cos\alpha + \sin\alpha\cos\alpha\cos\frac{\theta}{2}\Bigr) \\
  &\qquad - w\Bigl(\sin\alpha\sin\frac{\theta}{2}\Bigr)
\end{align*}

\[
\begin{cases}
  -\cos\alpha = \cos^2\alpha - \sin^2\alpha\cos\frac{\theta}{2} \\
  -\cos 2\theta \struck{\sin\alpha} = \struck{\sin\alpha}\cos\alpha\Bigl(1 + \cos\frac{\theta}{2}\Bigr) \\
  \struck{\ill{}} \sin 2\theta = \struck{\sin\alpha} \sin\frac{\theta}{2}
\end{cases}
\]

\[
  \gamma x_1 = \gamma u \cos\alpha + \gamma v \sin\alpha\cos\theta + w\sin\alpha\sin\theta
\]

\[
  \Gamma \simeq \mu_2 \times \Gamma'
\]

\[
\begin{cases}
  (u, x_i), & \struck{\ill{}} \; (u', x'_i) \\
  (x_i, x_{i+1}), & (x'_i, x'_{i+1}) \qquad i \in \mathbb{Z}/5\mathbb{Z} \\
  (x_i, x'_{i-2}) & (x'_i, x_{i-2})
\end{cases}
\]

\[
\begin{array}{lll}
  (u, x_i, x_{i+1}) & (u', x'_i, x'_{i+1}) & i \in \mathbb{Z}/5\mathbb{Z} \\
  (x_i, x'_{i-2}, \underset{\textstyle x_{i+1}}{\underset{\shortparallel}{x_{i-4}}}) & (x'_i, x_{i-2}, \underset{\textstyle x'_{i+1}}{\underset{\shortparallel}{x'_{i-4}}}) &
\end{array}
\]

\[
  \zeta^{3} + \zeta^{-1} + \zeta + \zeta^{-3} = \zeta + \zeta^{2} + \zeta^{3} + \zeta^{4} = -1
\]

\[
  \boxed{\gamma^{2} + \gamma - 1 = 0}
\]

\[
  \uncertain{\mathfrak{D}}_5 = \mathbb{Z}[\tfrac{1}{5}][t]/(t^{2} + t - 1)
\]

\[
  \boxed{\gamma = (-1 \pm \sqrt{5})/2}
\]

\[
  2\cos\frac{2\pi}{5} = (\sqrt{5} - 1)/2 \qquad (2\gamma + 1)^{2} = 5 \qquad \struck{4\gamma^{2} + 4\gamma = 4}
\]

\[
  \cos\frac{2\pi}{5} = \frac{\sqrt{5} - 1}{4} \qquad \cos\frac{4\pi}{5} = \frac{-\sqrt{5} - 1}{4} \qquad c^{2} + c - 1
\]

\[
  (-1 + \sqrt{5})/2 = \bigl((-1 - \sqrt{5})/2\bigr)^{2} - 2 = \frac{1 + 5 + 2\sqrt{5}}{4} - 2 = -\frac{1}{2} + \frac{1}{2}\sqrt{5}
\]

\[
  \zeta \longmapsto \zeta^{2} \qquad \zeta \longmapsto \zeta^{3} \qquad \text{si } \zeta + \zeta^{-1} = t
\]

\[
  \struck{\cos' D, D} \qquad \cos'_{\ill{}}(D, \zeta D) = \operatorname{Tr} \zeta^{2} = \zeta^{2} + \zeta^{-2} = t^{2} - 2
\]

\[
  t', t'' \qquad t'' = t'^{2} - 2 = (t''^{2} - 2)^{2} - 2 \qquad t' = t''^{2} - 2
\]

\[
\begin{array}{cccc}
  \zeta & \zeta^{2} & \zeta^{3} & \zeta^{4} \\
  & & \shortparallel & \shortparallel \\
  & & \zeta^{-2} & \zeta^{-1} \\[1ex]
  \zeta + \zeta^{-1} & \zeta^{2} + \zeta^{-2} & \zeta^{3} + \zeta^{-3} & \zeta^{4} + \zeta^{-4} \\
  \shortparallel & \shortparallel & \shortparallel & \shortparallel \\
  \eta = \eta_1 & \eta_2 & \uncertain{\eta_2} & \eta_1
\end{array}
\]

\[
\begin{aligned}
  &= \sup_{\ill{}} [\bar\alpha(U) \wedge \bar\beta(V)] \wedge [\bar\alpha(U') \wedge \bar\beta(V')] \\
  &= \bigl(\sup \bar\alpha(U) \wedge \bar\beta(V)\bigr) \wedge \bigl(\sup \bar\alpha(U') \wedge \bar\beta(V')\bigr) \\
  &= \bar\gamma(W) \wedge \bar\gamma(W') , \qquad \text{ok}
\end{aligned}
\]

\[
  \bar\gamma(\underbrace{\struck{\ill{}}\sup W_i}_{W}) \geq \sup\bigl(\bar\gamma(W_i)\bigr) \quad \text{clair, prouvons } \leq
\]

\[
  \bar\gamma(W) = \sup_{\gamma(U \times V) \subset \sup_i W_i} \bar\alpha(U) \wedge \bar\beta(V)
\]

\[
  \begin{array}{l} U = \sup U_j \\ V = \sup V_k \end{array} \;\Big|\; \forall\, j, k \ \exists\, i \text{ avec } \gamma(U_j \times V_k) \leq W_i
\]

\[
  \bar\alpha(U) \wedge \bar\beta(V) = \bigl(\sup_j \bar\alpha(U_j)\bigr) \wedge \bigl(\sup_k \bar\beta(V_k)\bigr) = \sup_{j,k} \bar\alpha(U_j) \wedge \bar\beta(V_k) \qquad \Bigl(\overset{?}{\leq} \sup_i \bar\gamma(W_i)\Bigr)
\]

\[
  \bar\alpha(U_j) \wedge \bar\beta(V_k) \leq \bar\gamma(W_i) \leq \sup_{i'} \bar\gamma(W_{i'}) \qquad \text{qfd}
\]

\[
  (I_\lambda)_{\lambda \in \Lambda} \qquad I_\lambda \in \operatorname{Ob}(\uncertain{\mathrm{OM}})
\]

\[
  \alpha^{0}_{\mu} : I_\mu \longrightarrow \prod{}' I_\lambda \qquad \alpha^{0}_{\mu}(x)_\lambda = \begin{cases} x & \text{si } \lambda = \mu \\ 1_{I_\lambda} & \text{si } \lambda \neq \mu \end{cases} \qquad \text{appl. cr.}
\]

\[
  \gamma : \prod{}' I_\lambda \longrightarrow K \quad \text{flèche dans } (\uncertain{\mathrm{OM}})
\]

\[
  \alpha_\mu = \gamma \circ \alpha^{0}_{\mu} : I_\mu \to K \quad \text{appl. cr.}
\]

\[
  \Gamma = (\mathbb{Z}/5\mathbb{Z})^{*} \simeq \mathbb{Z}/4\mathbb{Z}
  \qquad
  \begin{array}{ccc}
    \mu_5 & \supset & \mu_5^{*} \\
    | & & | \\
    \operatorname{Spec} \mathbb{Z} & \longleftarrow & \mathbb{Z}[\frac{1}{5}]
  \end{array}
\]

\[
  \struck{\ill{}}\; \tau = \mu_5^{*}/\beta \quad \text{engendré par } \struck{\zeta}\, \zeta + \bar\zeta = \zeta + \zeta^{-1} = \eta
\]

\[
  \zeta \mapsto \zeta^{2} \text{ ou } \zeta \mapsto \zeta^{-2} = \zeta^{3} \text{ resp.} \qquad \zeta \mapsto \zeta^{-1} = \zeta^{4} \qquad \zeta \mapsto \zeta
\]

\[
  \begin{array}{c} B \supset B^{\Gamma} \\ | \\ A \end{array}
  \qquad
  B = A[\zeta] \struck{\ill{}} = A[T]/(1 + T + T^{2} + T^{3} + T^{4})
  \qquad T \longmapsto T^{4}
\]

\[
\begin{aligned}
  \beta(T) &= T^{-1} = T^{4} = -(1 + T + T^{2} + T^{3}) \\
  \beta(T^{2}) &= T^{-2} = T^{3} \\
  \beta(T^{3}) &= T^{-3} = T^{2} \\
  \beta(T^{4}) &-
\end{aligned}
\qquad\qquad 1 = T^{0} \mid T \quad T^{2} \quad T^{3}
\]

\[
\begin{aligned}
  \beta(c_0 + c_1 T + c_2 T^{2} + c_3 T^{3}) \struck{\ill{}}
  &= c_0 \mathbin{\ill{}} c_1(\uncertain{-1} + T + T^{2} + T^{3}) + c_2 T^{3} + c_3 T^{2} \\
  &= (c_0 - c_1) - c_1 T + (c_3 - c_1) T^{2} + (c_2 - c_1) T^{3} \\
  &\overset{?}{=} c_0 + c_1 T + c_2 T^{2} + c_3 T^{3}
\end{aligned}
\]

\[
  c_1 = 0 \qquad c_2 = c_3 \qquad\qquad c_0 + c_2 (T^{2} + T^{3})
\]

\[
  \begin{array}{c} \mu_n \\ \cup \\ \mu_n^{*} \\ \downarrow \\ S \end{array}
  \quad (\mathbb{Z}/n\mathbb{Z})^{*} \qquad \zeta \longmapsto \zeta^{-1}
\]

\[
  \varphi(n) = \prod \bigl(p_i^{\alpha_i} - p_i^{\alpha_i - 1}\bigr) \qquad \text{pas gros} \qquad p^{n} - p^{n-1} = p^{n-1}(p - 1) \qquad (\mathbb{Z}/n\mathbb{Z})^{*} \ni -1
\]

\[
\begin{array}{ccccc}
  \mu_n^{T} & \subset & \mathcal{U} & & \\
  \cup & & & & \\
  \mu_n^{*T} & \subset & \mathcal{U} & & \zeta + \bar\zeta \\
  \Big\downarrow{\scriptstyle \text{étale rang } 2} & & \Big\downarrow{\scriptstyle \operatorname{Tr}_{K/\mathbb{Q}}} & & \\
  \vartheta_n^{T} = \mathcal{V}_n & \subset & \mathcal{O} & & \\
  \Big\downarrow{\scriptstyle \text{étale rang } \uncertain{\varphi(n)/2}} & & & & \\
  S & & & &
\end{array}
\]

\[
  \begin{array}{ccc}
    \mu_n & \subset & \mathbb{G}_m \\[1ex]
    \mu_n^{*} & \subset & \mathbb{G}_m \\
    \downarrow & & \downarrow{\scriptstyle \rho} \\
    \vartheta_n & \subset & \mathbb{E}^{1}
  \end{array}
  \qquad \rho(\lambda) = \lambda + \lambda^{-1}
\]

\[
  \zeta + \zeta^{-1} \;\Big|\; \begin{array}{l} \frac{p - 1}{2} = 2 \\ \cos \end{array}
\]

\[
  1 + \zeta + \zeta^{2} + \zeta^{3} + \zeta^{4} = 0 \qquad \zeta + \zeta^{-1} = \gamma
\]

\[
\begin{aligned}
  \operatorname{Tr} \gamma &= \gamma + \gamma^{\alpha} = \zeta + \zeta^{-1} + \zeta^{2} + \zeta^{-2} = \zeta + \zeta^{2} + \zeta^{3} + \zeta^{4} = -1 \\
  \operatorname{N} \gamma &= \gamma\gamma^{\alpha} = (\zeta + \zeta^{-1})(\zeta^{2} + \zeta^{-2}) =
\end{aligned}
\]

\[
  \struck{\ill{}} \qquad \operatorname{Hom}_{\mathrm{OM}}(K, L) \longrightarrow \operatorname{Hom}_{\mathrm{OM}}(I, L) \times \operatorname{Hom}_{\mathrm{OM}}(J, L) \qquad \varphi \longmapsto (\varphi \circ \alpha, \varphi \circ \beta)
\]

\[
  \bar\alpha : I \longrightarrow L \qquad \bar\beta : J \longrightarrow L \qquad \text{dans } (\mathrm{OM})
\]

\[
  \bar\gamma : K \longrightarrow L
\]

\[
  \bar\gamma(W) = \sup_{\substack{U \in I \\ V \in J \\ \gamma(U \times V) \subset W}} \bar\alpha(U) \wedge \bar\beta(V)
\]

\[
  \bar\gamma \circ \alpha = \bar\alpha , \qquad \bar\gamma \circ \beta = \bar\beta
\]

\[
  (\bar\gamma \circ \alpha)(U') = \bar\gamma(U' \times 1) = \sup_{\gamma(U \times V) \leq \gamma(U' \times 1)} \bar\alpha(U) \wedge \bar\beta(V)
\]

\[
  \bar\gamma(1_K) \struck{\ill{}} \overset{\text{def}}{=} \bar\alpha(1_I) \wedge \bar\beta(1_J) = 1_K
\]

\[
  \bar\gamma(W \wedge W') = \sup_{\gamma(U \times V) \subset W \wedge W'} \bar\alpha(U) \wedge \bar\beta(V) = \sup_{\substack{\gamma(U \times V) \subset W \\ \gamma(U' \times V') \subset W'}} \bar\alpha(U \wedge U') \wedge \bar\beta(V \wedge V')
\]

\[
  \underline{X} \longrightarrow (\mathrm{Ens}) \qquad \underline{\hat X} \longrightarrow (\mathrm{Ens}) ,
\]

\[
  x \subset e \qquad x \quad y \qquad z \to x \cap y
\]

\[
  4 \longrightarrow 3 \longrightarrow 2 \longrightarrow 1 \longrightarrow 0
\]

\[
  X_1 \longleftarrow X_0 \qquad y \leq x \qquad U_x \subset U_y \qquad F(y) \longrightarrow F(x)
\]

\[
  \operatorname{Hom}_G(X, E) \subset \operatorname{Hom}_G(P_S, E) \simeq E
\]

\[
  \struck{\ill{}} + v \cos i\frac{2\pi}{5} + w \sin i\frac{2\pi}{5} = \rho_i \qquad \boxed{0 \leqslant i \leqslant 4}
\]

\[
\begin{cases}
  u \\
  u\cos\alpha + \rho_i \sin\alpha \\
  -u\cos\alpha - \rho_i \sin\alpha \\
  -u
\end{cases}
\]

\[
  \frac{4\pi}{5} < \frac{2\pi}{3} \qquad \frac{\pi}{3} < \qquad \frac{4\pi}{5} \to \frac{\pi}{6} \;? \qquad \frac{4\pi}{5} \leqslant \frac{5\pi}{6}
\]

\[
  \begin{array}{l} u \perp v' \\ u \perp w' \end{array} \qquad v'w' = \cos\frac{2\pi}{5}
\]

\[
\begin{aligned}
  x(\alpha) &= u\cos\alpha + v'\sin\alpha \\
  y(\alpha) &= u\cos\alpha + w'\sin\alpha
\end{aligned}
\]

\[
  x(\alpha) - u = u(\cos\alpha - 1) + v'\sin\alpha
\]

\[
\begin{aligned}
  \|x(\alpha) - u\|^{2} = \|y(\alpha) - u\|^{2} &= (\cos\alpha - 1)^{2} + \sin^{2}\alpha = 2 - 2\cos\alpha \\
  &= 4\sin^{2}\frac{\alpha}{2} \qquad \Bigl(1 - \cos\alpha = 2\sin^{2}\frac{\alpha}{2}\Bigr)
\end{aligned}
\]

\[
  \|x(\alpha) - u\| = 2\sin\frac{\alpha}{2}
\]

\[
\begin{aligned}
  \|x(\alpha) - y(\alpha)\|^{2} &= \|v'\sin\alpha - w'\sin\alpha\|^{2} \\
  &= \sin^{2}\alpha \, \|v' - w'\|^{2} = \sin^{2}\alpha \Bigl(2 - 2\cos\frac{2\pi}{5}\Bigr) \\
  &\struck{= \sin^{2}\alpha + \sin^{2}\alpha - 2\cos\frac{2\pi}{5}\sin^{2}\alpha} \\
  &= 4\sin^{2}\alpha \sin^{2}\frac{\pi}{5} \struck{= \ill{}}
\end{aligned}
\]

\[
  \|x(\alpha) - y(\alpha)\| = 2\sin\alpha \sin\frac{\pi}{5} = 4\sin\frac{\alpha}{2}\cos\frac{\alpha}{2}\sin\frac{\pi}{5}
\]

\[
  \|x(\alpha) - u\| = \|y(\alpha) - u\| = \|x(\alpha) - y(\alpha)\|
\]

\[
  \Longleftrightarrow \quad 2\cos\frac{\alpha}{2}\sin\frac{\pi}{5} = 1 \quad \text{i.e.} \quad \boxed{\cos\frac{\alpha}{2} = \frac{1}{2\sin\pi/5}}
\]

\[
\begin{aligned}
  (AB, A'B') &= P \\
  (EC', CE') &= Q = \alpha(P) \\
  (SD, S'D') &= R = \alpha(Q) = \alpha^{2}(P) \\
  (AB, A'B') &= \uncertain{\alpha(R)} = \alpha^{2}(Q) = \alpha^{3}(P)
\end{aligned}
\]

\[
  \alpha D \longrightarrow D \qquad \begin{array}{cc} 30 & 15 \\ \uncertain{\alpha^{!}}? & \alpha \end{array}
\]

\[
  G/\mathfrak{F}_2 \longrightarrow G/\mathfrak{F}_4 \qquad 2, 4, 3, 1
\]

\[
  \frac{\pi}{\uncertain{3}} \leqslant \frac{2\pi}{5} \leqslant \frac{5\pi}{6}
  \qquad
  \cos\alpha = \cos^{2}\frac{\alpha}{2} - \sin^{2}\frac{\alpha}{2} = 2\cos^{2}\frac{\alpha}{2} - 1 = \frac{1}{2\sin^{2}\theta} - 1 = 1 - 2\sin^{2}\theta = \cos 2\theta
\]

\[
  u \to u\cos\alpha + v\sin\alpha \qquad v \to u(-\sin\alpha) + v\cos\alpha
\]

\[
  \frac{1\,2\,3\,4\,\struck{0}}{2,4,1,3,0} \qquad \frac{}{3,1,4,2,0} \qquad \zeta \quad \zeta^{2}
\]

\[
  u, 2u, 3u, 4u, 0 \qquad \zeta^{i} \qquad e^{i\theta} \qquad u^{2} + w^{2} = 1 \qquad \theta^{2} + \mu^{2} = 1 \qquad x^{2} - y^{2}
\]

\[
  g(\sigma . i) \qquad \rho(g)(\sigma . i) = \bigl(\rho(g)\ldots\bigr)\bigl(\rho(\ldots)\bigr) \qquad g(\sigma i) = g\sigma g^{-1}\, gi
\]

\[
  G/H = X \ni e \qquad y e = a . e = a \qquad y x = y g e = g \uncertain{u} e = g' a e = \ill{}
\]

\[
  b' = b h \qquad b^{-1} b' \in H \quad \Longrightarrow \quad (ba)^{-1} b' a \in H \quad \text{i.e.} \quad a^{-1} b^{-1} b' a \in H \quad \text{i.e.} \quad a^{-1} H a \subset H
\]

\[
  \frac{N(H')}{\bigcap \operatorname{Conj} H} \qquad
  N(H) \cap N(H') = N(\uncertain{H'}) \subset N(H) = \mathfrak{Z}_3 . \mathfrak{F}_2^{2}
  \qquad
  \begin{array}{ccc}
    N(H') & \subset & N(H) \\
    \cup & & \cup \\
    H' & \subset & H \;\text{---}\; \mathfrak{F}_2^{2} \\
    \cup & & \cup \\
    K \cap H' = K'_0 & \subset & K' = 0
  \end{array}
\]

\[
  \cos^{2}\alpha \cos\frac{2\pi}{5} = \cos\alpha \qquad 2\sin\frac{\alpha}{2}\sin\frac{2\pi}{5} = 1 \qquad \boxed{\cos\alpha = \frac{1}{2\sin\frac{2\pi}{\uncertain{10}}}}
\]

\[
  1 + x + x^{2} + x^{3} + x^{4} = 0 \qquad \Bigl(2\sin\frac{\alpha}{2}\Bigr)\Bigl(2\sin^{2}\frac{\pi}{\uncertain{4}}\Bigr) \qquad \alpha = 60\,?
\]

\[
   \cos\frac{\alpha}{2} = \frac{\sqrt{3}}{2} \qquad \sqrt{3}\sin\frac{\pi}{5} = 1 \qquad \frac{\pi}{6}
\]

\[
\begin{array}{c}
\boldsymbol{\mu}_5 \\ \vert \\ \boldsymbol{\mu}_5^{*} \\ \vert \\ \mathbb{V}_5 \\ \vert \\ \mathbb{Z}[\tfrac{1}{4}]
\end{array}
\]

\[
\cos'(D_i, D_j)^2 + \cos'(D_i, D_j)
\]

\[
1 - \gamma = \gamma^2, \qquad 2 - \gamma = 1 + \gamma^2, \qquad
\gamma^2 + \gamma - 1 = 0
\]

\[
\cos\alpha = \frac{\struck{4}\gamma/2}{1 - \gamma/2} = \frac{\gamma}{2-\gamma}
= a + b\gamma = \frac{1}{5}(2\gamma + 1) = \frac{1}{\sqrt{5}}
\]

\[
\begin{aligned}
\cos 2\alpha &= \cos^2\alpha - \sin^2\alpha = 2\cos^2\alpha - 1
  = 2\,\frac{\gamma^2}{4 - 4\gamma + \gamma^2} - 1 \\
&= \frac{2(1-\gamma)}{4 - 4\gamma + 1 - \gamma} - 1
  = \frac{2 - 2\gamma}{5(1-\gamma)} - 1 = \frac{2}{5} - 1 = -\frac{3}{5}
\end{aligned}
\]

\[
\frac{2}{25}\,\underbrace{[4\gamma^2 + 4\gamma + 1]}_{4 - 4\gamma + 4\gamma + 1} - 1
\]

\[
\frac{B(x,y)^2}{Q(x)Q(y)} - 2 = -\frac{6}{5}
\qquad\qquad 2\cos 2\alpha = -\frac{6}{5}
\]

\[
\boxed{B(x,y)^2 = \frac{4}{5}\,Q(x)Q(y)}
\]

\[
2a + (2b - 1 \uncertain{-a})\gamma + b\gamma^2 = 0, \qquad
\gamma^2 + \Bigl(-2 + \frac{1}{b} + \frac{a}{b}\Bigr)\gamma - \frac{2a}{b} = 0
\]

\[
-\frac{2a}{b} = -1, \quad \boxed{b = 2a}, \qquad
-2 + \frac{1}{2a} + \frac{1}{2} = 1, \qquad a = \frac{1}{5},\ b = \frac{2}{5}
\]

\[
\cos' 2\alpha = 2\cos 2\alpha = 2(2\cos^2\alpha - 1) = (2\cos\alpha)^2 - 2,
\qquad \frac{1}{2a} = \frac{5}{2}, \qquad a = \frac{1}{5}
\]

\[
\tfrac{1}{2}B = B' \qquad\qquad
\boxed{B'(x,y)^2 = \frac{1}{5}\,Q(x)Q(y)}
\]

\[
\pi(f, g)^{*} = (u' \mapsto g^{*} \circ u' \circ f^{\circ}), \qquad
\pi(f, g)_{*} = (u' \mapsto g_{*} f_{*}^{(E)})
\]

\[
\pi(f, g) : \pi(C, E) \longrightarrow \pi(\uncertain{C'}, E')
\]

\[
\text{(1)}\qquad
\boxed{\gamma = \gamma(C, E) : \pi(C, E) \longrightarrow \hat{C} \times_{\mathrm{top}} E}
\]

\[
\begin{aligned}
\gamma_1 &= \gamma_1(C, E) : \pi(C, E) \to \hat{C} \\
\gamma_2 &= \gamma_2(C, E) : \pi(C, E) \to E
\end{aligned}
\]

\[
\gamma_1^{*} : \hat{C} \to \pi(C, E) \quad \text{défini par} \quad
F \mapsto [X \mapsto F(X)_E]
\]

\[
\gamma_2^{*} : E \to \pi(C, E), \qquad X \mapsto (i \mapsto X)
\]

\[
\omega(\varphi) = (\omega_x(\varphi))_{x \in X'_0} \in \prod_{x \in X'_0} \Omega_x(X)
\]

\[
\mathrm{Pl}_{\mathrm{isot}}(X, S) \xrightarrow{\ \omega\ } \prod_{x \in X'_0} \Omega_x(X)
\overset{\mathrm{df}}{=} \Omega(X)
\]

\[
\mathrm{Pl}^{\alpha}_{\mathrm{isot}}(X, S) \longrightarrow
\mathrm{Pl}_{\mathrm{isot}}(\mathcal{U}_{\alpha}, S)
\]

\[
I \times S^1 \qquad (I = \pi_0(\mathcal{V}_{\alpha}))
\]

\[
\mathcal{U}'_{\lambda} = S_g - \bigcup_{j \in I_{\lambda}} \mathring{D}_j - P_{\lambda}
\]

\[
\begin{cases}
\alpha \in \Omega(X) & [\text{d'où } I = I(\alpha), \text{ à expliciter combinatoirement}] \\
\pi \text{ partition de } I \text{ en } (I_{\lambda})_{\lambda \in \Lambda} & \\
(g_{\lambda})_{\lambda \in \Lambda} \in \mathbb{N}^{\Lambda} & \\
(\beta_{\lambda})_{\lambda \in \Lambda} \in \mathbb{N}^{\Lambda} &
\end{cases}
\]

\[
X \xrightarrow{\ \varphi_{\alpha, \pi, g, \beta}\ } S_{\alpha, \pi, g, \beta}
\]

\[
\mathrm{Pl}_{\mathrm{isot}}(X, S) \longrightarrow
\text{l'ensemble des systèmes } \alpha, \pi, g, \nu \text{ associés à } X
\]

\[
(*)\qquad
\mathrm{Isot}(S_{\rho}, S) \xrightarrow{\ r\ } \mathrm{Pl}_{\mathrm{isot}}(X, S; \rho)
\]

\[
G = \mathrm{Isot}_{o}(S, S) \qquad [\simeq G_{\rho} = \mathrm{Isot}_{o}(S_{\rho}, S_{\rho})]
\]

\[
G \longrightarrow \mathrm{Autext}(\pi_1(S, s))
\]

\[
h\varphi \underset{\mathrm{isot}}{\simeq} \varphi \;\Longrightarrow\;
h\varphi \underset{\mathrm{homot}}{\simeq} \varphi
\]

\[
\pi_1(h) \circ c \simeq c \quad \text{mod automorphismes intérieurs}
\]

\[
\pi_1(h \varphi_i) = \pi_1(h) \circ \pi_1(\varphi_i) \in
\struck{\ill{}}\ \mathrm{Homext}(\pi_1(X_i), \pi_1(S))
\]

\[
\mathcal{G}/\mathcal{H} \simeq \mathfrak{X} \qquad
\pi_1(\mathcal{H}) \to \pi_1(\mathcal{G}) \to \pi_1(\mathfrak{X}) \to
\pi_0(\mathcal{H}) \to \pi_0(\mathcal{G}) \to \pi_0(\mathfrak{X})
\]

\[
\mathcal{H} = \mathrm{Aut}(S / X)
\]

\[
\text{Graphes}\quad
(S, A, A' \xrightarrow{\ \varphi = (\varphi_1, \varphi_2) = (\varphi_1, \varphi_1 \circ \sigma)\ } S \times S, \sigma)
\;\;\langle\simeq\rangle\;\;
(S, R \subset A \times A, L \to A, \sigma)
\]

\[
\mathrm{Aut}\,\Gamma = \mathrm{Perm}(S) \times \underbrace{\mathrm{Aut}(A, \sigma)}
\]

\[
\varphi : A \longrightarrow S
\]

\[
\left\{
\begin{array}{l}
\boxed{G \supset H' \supset H} \quad \text{et} \\
\boxed{\dot{\sigma} \in (H' \cap \mathrm{Norm}_G(H))/H \qquad \dot{\sigma}^2 = 1,\ \dot{\sigma} \neq 1}
\end{array}
\right.
\]

\[
\bigcap_{g \in G} g H g^{-1} = \{1\}
\]

\[
\underset{a}{A} \xrightarrow[\text{source}]{\ \varphi_1\ } \underset{s}{S}, \qquad
\left\{
\begin{array}{l}
\sigma : A \to A \\
\sigma^2 = \mathrm{id},\ \sigma \text{ ss pt fixe}
\end{array}
\right.
\]

\[
\boxed{G \supset \underset{G_a}{H'} \supset \underset{G_s}{H} \qquad
\dot{\sigma} \in \mathrm{Norm}_G(H)/H \qquad \dot{\sigma}^2 = 1 \qquad
\dot{\sigma} \neq 1}
\]

\[
\boxed{\bigcap_{g \in G} g^{-1} H g = \{1\}}
\]

\[
\begin{aligned}
G/H &\longrightarrow G/H' \times G/H' \\
gH &\longmapsto gH',\ g\sigma H'
\end{aligned}
\]

\[
\boxed{H' \cap \dot{\sigma}^{-1} H' \dot{\sigma} \subset H}
\]