\[
  \Phi \longrightarrow \mathfrak{P}(S)
\]

\[
  \operatorname*{Sup}_{a<f} a \;=\; \operatorname*{Sup}_{a<f}\,
  \operatorname*{Sup}_{s<a} s \;=\; \operatorname*{Sup}_{s<f} s \;=\; f
  \qquad \text{cqfd}
\]

\[
  S = \{o\} \amalg \Pi \amalg \Pi' \amalg \{o'\}
\]

\[
  A = \begin{cases}
    \{o, s\}, & s \in \Pi \\
    \{o', s'\}, & s' \in \Pi'
  \end{cases} \quad 10
  \qquad
  A(\Pi),\ A(\Pi') \quad 10
  \qquad
  \{s, t\},\ s \in \Pi,\ t \in \Pi' \quad 10
\]

\[
  S = T \amalg T_1 \amalg T'_1 \amalg T'
\]

\[
  \begin{array}{ccccc}
    t & \boxed{T \xrightarrow{\ \sim\ } T'} & t' \\
    \wr & \wr \quad\ \wr & \wr \\
    t_1 & T_1 \simeq T'_1 & t'_1
  \end{array}
\]

\[
  \begin{array}{lll}
    \mathfrak{P}_2(T), \quad \mathfrak{P}_2(T') & & 6 \\[2pt]
    \{t, u\},\ t \in T,\ u \in T_1,\ u \neq t_1 &
    \{t', u'\},\ t' \in T',\ u' \in T'_1,\ u' \neq t'_1 & 12 \\[2pt]
    \{t_1, u\},\ t_1 \in T_1,\ u \in T'_1,\ u \neq t'_1 & & 6 \\[2pt]
    \{t, t'_1\}, \quad \{t', t_1\} & & 6
  \end{array}
\]

\[
  S = \sigma \amalg \varphi \amalg \sigma' \amalg \varphi' \amalg
  \bigl[\sigma \times \varphi \amalg \sigma' \times \varphi'\bigr]
  \big/ \uncertain{\mathbf{Z}/2\mathbf{Z}}
\]

\[
  \sigma(s,u) = \bigl((\varepsilon s)', (\varepsilon u)'\bigr),
\]

\[
  \begin{array}{lll}
    \sigma, \sigma' & & \uncertain{2} \\
    \{s, u\},\ \{s', u'\} \quad (s \in \sigma,\ u \in \varphi) & & 8 \\
    \{\Pi_{s,u}, \Pi_{s,v}\},\ u \neq v & & 4 \\
    \{u, \Pi_{s,u}\},\ \{u', \Pi_{s',u'}\} & & 8 \\
    \{s, \Pi_{s,u}\},\ \{s', \Pi_{s',u'}\} & & 8 \\
    \{u, v'\},\ u \neq v & & 2
  \end{array}
\]

\[
  \operatorname{Rep}(\Pi) = \{(s, a, f) \in S \times A \times F \mid
  s < a < f\}
\]

\[
  \begin{array}{lll|l}
    0 & 1 & 2 & \text{drapeaux minimaux (de long.\ 0)} \\
    S & A & F & \text{ou « facettes »} \\[4pt]
    (0,1) & (1,2) & (0,2) & \text{drapeaux de long.\ 1} \\
    R \subset S \times A & R' \subset A \times F & R'' \subset S \times F & \\[4pt]
    (0,1,2) & & & \text{repères, ou drapeaux} \\
    \operatorname{Rep} \subset S \times A \times F & & & \text{maximaux}
  \end{array}
\]

\[
  \sigma_0\sigma_2(s, a, f) = \sigma_2\sigma_0(s, a, f) = (s', a, f')
\]

\[
  \mathfrak{S}_2 = \Bigl\{ \sigma_0, \sigma_1, \sigma_2 \Bigm|
  \begin{array}{l}
    \sigma_0^2 = \sigma_1^2 = \sigma_2^2 = 1 \\
    \sigma_0\sigma_2 = \sigma_2\sigma_0
  \end{array} \Bigr\}
\]

\[
  \begin{array}{c}
    \text{Géom.\ d'incidence} \\ \text{(avec iso)}
  \end{array}
  \longrightarrow
  \begin{array}{l}
    \mathfrak{S}_2\text{-ens.\ où } \sigma_0, \sigma_1, \sigma_2 \\
    \text{et } \sigma_0\sigma_2 \text{ opèrent sans pts} \\
    \text{fixes (avec iso)}
  \end{array}
\]

\begin{align*}
  E/(\sigma_1, \sigma_2) &\longrightarrow S = \Phi_0 \\
  E/(\sigma_0, \sigma_2) &\longrightarrow A = \Phi_1 \\
  E/(\sigma_0, \sigma_1) &\longrightarrow F = \Phi_2
\end{align*}

\[
  g \cdot r = r \Longrightarrow g = \mathrm{id}
\]

\[
  \operatorname{Isom}(\Pi_0, \Pi) \xrightarrow{\ \sim\ }
  \operatorname{Rep}(\Pi), \qquad \varphi \longmapsto \varphi(r_0)
\]

\[
    G_a \longrightarrow \mathfrak{S}_\sigma \times \mathfrak{S}_\varphi
    = \mathbf{Z}/2\mathbf{Z} \times \mathbf{Z}/2\mathbf{Z}
  \]

\[
  \bar A = A/\underline{a} \quad \text{ens.\ des paires « directions
  d'arêtes »} \ (\text{\uncertain{termes} \ill{} arêtes parallèles})
\]

\[
  \rho : \bar A \longrightarrow \bar A
\]

\[
  A \longrightarrow \bar A
\]

\[
  \bar A \longrightarrow \bar A
\]

\[
  \rho^3(\bar a) = \bar a
\]

\[
  \rho^3 = \mathrm{id}_{\bar A}
\]

\[
  \beta = \rho\alpha \quad \text{ou} \quad \alpha = \rho\beta
\]

\[
  P(s) \longrightarrow T
\]

\[
  G \longrightarrow \operatorname{Alt}_T
\]

\[
  \omega_{u(s)} = u(\omega_s)
\]

\[
  T \simeq f \amalg \operatorname{Or}(f)
\]

\[
  a \times \varphi \longrightarrow A
\]

\[
  T \simeq \{\operatorname{tr}\bar{a}\} \amalg a \times \varphi ,
\]

\[
  \underset{\text{icosaèdre}}{G = \operatorname{Aut}\Pi}
  \longrightarrow
  \mathfrak{S}_{\underset{\text{ens des trièdres distingués}}{T(\Pi)}}
\]

\[
  G \simeq \mathfrak{z} \times G^{+}
\]

\[
  G^{+} \xrightarrow{\sim} \mathfrak{A}_5
\]

\[
  S/\underline{a} \longrightarrow \operatorname{Pent}^{+}(T, \varpi)
\]

\[
  S/\underline{a} \xrightarrow{\sim} \operatorname{Pent}^{+}(T, \varpi) .
\]

\[
  \mathbb{A}'(T) = \left\{ (t, \gamma, \alpha) \;\middle|\;
  \begin{array}{l}
    t \in T \\
    \gamma \in \operatorname{Pol}(T \smallsetminus \{t\}) \\
    \alpha \in \operatorname{Ar}(\gamma)/\text{antipod.}
  \end{array}
  \right\}
\]

\[
  5 \cdot \tfrac12 (4 - 1)! \cdot 2 = 30
\]

\[
  T \simeq \{t\} \amalg a \times \varphi
  \qquad (\text{où } \varphi = \operatorname{Ar}(\pi) \smallsetminus a),
\]

\[
  A/\underline{a} \longrightarrow \mathbb{A}^{+}(T, \varpi) ,
\]

\[
  F/\underline{a} \longrightarrow \mathfrak{P}_2(T)
\]

\begin{align*}
  \bar{s} \in \bar{S} \quad &\longleftrightarrow \quad \pi \in
    \operatorname{Pent}^{+}(T, \varpi) \\
  \bar{a} \in \bar{A} \quad &\longleftrightarrow \quad \{t, \gamma, \alpha\}
    \in \mathbb{A}^{+}(T, \varpi) \\
  \bar{f} \in \bar{F} \quad &\longleftrightarrow \quad \Omega \in
    \mathfrak{P}_2(T)
\end{align*}

\[
  \begin{array}{c}
    u \text{---} v \\
    \tilde{v} \text{---} \tilde{u}
  \end{array},
  \qquad \text{et} \quad \alpha = \bigl\{ \{u, v\}, \{\tilde{u}, \tilde{v}\}
  \bigr\}
\]

\[
  \bar{R} \simeq \vec{A}(\bar{\Pi}) \simeq S(\bar{\Pi}) \times T(\bar{\Pi})
  \quad ]
\]

\[
  (\tau, \gamma, \alpha) \in \mathbb{A}^{+}(T, \varpi) \ \ldots
\]

\[
  \{\pi, \tau, \Omega\}, \qquad \pi \in \operatorname{Pent}^{+}(T, \varpi),
  \quad \tau \in T, \quad \Omega \in \mathfrak{P}_2(T)
\]

\[
  T = \{t_0, t_1, t_2, t_3, t_4\} ,
\]

\[
  \Omega = \{t_1, t_3\} \in \mathfrak{P}_2(T) ,
\]

\[
  \left\{
  \begin{array}{l}
    \tilde{S}(\bar{\Pi}) \simeq S(\Pi) \\
    \tilde{A}(\bar{\Pi}) \simeq A(\Pi) \\
    \tilde{F}(\bar{\Pi}) \simeq F(\Pi)
  \end{array}
  \right.
\]

\begin{align*}
  S &\simeq \operatorname{Pent}_\omega(E) && \text{struct.\ pent.\ comp.\ avec } \omega \\
  A &\simeq \operatorname{Car}(E) && \text{partitions du type } (2, 2, 1) \\
    & && \simeq \text{structures carrées sur une partie à 4 él.\ de } E \\
  F &\simeq \operatorname{Tr}_3(E) && \text{triangles} \subset E \\
  \vec{A} &\simeq \operatorname{Pent}^{s}_\omega(E) && \omega\text{-structures pentagonales à somm.\ marqué} \\
  A^{\uparrow} &\simeq \operatorname{Tr}_3(E) && \text{triangles pointés} \\
  A^{\wedge} &\simeq \operatorname{Pent}^{a}_\omega(E) \simeq \operatorname{Pent}^{s}_\omega(E)
    && \text{structures pol.\ à arête marquée} \\
  A^{\uparrow\to} &\simeq \operatorname{Rep}_\omega(E) && \text{ordres totaux sur } E \text{ comp.\ avec } \omega \\
  \tilde{S} &\simeq \overrightarrow{\operatorname{Pent}}_\omega(E) && \text{permutations circulaires comp.\ avec } \omega \\
  \tilde{A} &\simeq \overrightarrow{\operatorname{Car}}(E) && \text{carrés orientés} \subset E \\
  \tilde{F} &\simeq \overrightarrow{\operatorname{Tr}}_3(E) && \text{triangles orientés} \subset E
\end{align*}

\[
  \left\{
  \begin{aligned}
    \sigma_0(a_1, a_2, a_3, a_4, a_5) &= (a_1, a_4, a_5, a_2, a_3) \\
    \sigma_1(a_1, a_2, a_3, a_4, a_5) &= (a_2, a_1, a_5, a_4, a_3) \\
    \sigma_2(a_1, a_2, a_3, a_4, a_5) &= (a_1, a_5, a_4, a_3, a_2)
  \end{aligned}
  \right.
\]

\begin{align*}
  \rho_f = \sigma_1 \sigma_0 &: (a_1, a_2, a_3, a_4, a_5) \longmapsto
    (a_4, a_1, a_3, a_2, a_5) \\
  \rho_s = \sigma_2 \sigma_1 &: (a_1, a_2, a_3, a_4, a_5) \longmapsto
    (a_2, a_3, a_4, a_5, a_1) \\
  \sigma = \sigma_0 \sigma_1 &: (a_1, a_2, a_3, a_4, a_5) \longmapsto
    (a_1, a_3, a_2, a_5, a_4)
\end{align*}