% Addendum 164: The Stable Null Grid — F4 Root Decomposition, the Discriminant,
%               and Chiral Cancellation at tau=i
%   P164 proves: F4 root system = D4 u D4* (long roots = D4, short roots = D4*,
%   dual lattices of each other). The 24-cell realises each chiral half (self-dual
%   polytope). Exponent 24 in Delta(tau)=eta(tau)^24 counts one chiral half.
%   tau(2)=-24 is the chiral cancellation signature. Delta(i)=E4(i)^3/j(i)=eta(i)^24
%   expressed in TOE quantities J_short=12, j(i)=1728.
% ERRATUM (A343, 2026-06-15): the line "j(i)=Js^3 was proved using T_3B(i)=Js
%   (A163)" rests on the A337-retracted conflation. j^{1/3}(i)=12 is the E8/G2
%   cube-root contact, not a Monster 3B McKay-Thompson value. j(i)=Js^3=1728
%   stands (independently derived in A158 from G2/F4 root geometry); only the
%   "proved using T_3B" provenance is withdrawn. See A337/A343.
% L. F. Vlegels, Independent Researcher
% Copyright: Leon Fernando Vlegels. License: CC BY 4.0. May 2026.

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\newcommand{\Js}{J_{\mathrm{short}}}
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\newcommand{\BG}{B_{G_2}}
\newcommand{\BF}{B_{F_4}}

\title{The Stable Null Grid: $F_4$ Root Decomposition,\\
the Ramanujan Discriminant, and Chiral Cancellation\\
at the CM Point $\tau = i$\\[0.5em]
\large Addendum 164 to the TOE Corpus}

\author{L.\ F.\ Vlegels\\
Independent Researcher\\
\texttt{axiomofchoicepuzzle@gmail.com}}

\date{May 2026}

\begin{document}
\maketitle

\noindent\textbf{Erratum (A343, 2026-06-15):} The provenance line ``$j(i) =
\Js^3$ was \emph{proved using} the McKay-Thompson identification $T_{3B}(i) =
\Js$ (A163)'' rests on the conflation retracted by A337: $j^{1/3}(i) = 12 = \Js$
is the E8/G2 cube-root contact, \emph{not} a Monster 3B McKay-Thompson value.
The fact $j(i) = \Js^3 = 1728$ stands --- it was independently derived in A158
from $G_2/F_4$ root geometry --- as do $\Delta(i) = \eta(i)^{24}$ and the
$F_4 = D_4 \cup D_4^\ast$ decomposition. Only the ``proved using $T_{3B}$''
attribution is withdrawn. See A337/A343. Body preserved as audit trail.

\tableofcontents
\newpage

%==========================================================================
\begin{abstract}
The chain P158--P163 established $\Js = 12$, $j(i) = 1728 = \BG \times \BF$
with $\BG = 48$, and identified $\tau = i$ as the unique CM point where Monster
moonshine contacts TOE geometry.  The present addendum asks what algebraic
object $\BG = 48$ is, and why the Ramanujan discriminant $\Delta(\tau) = \eta(\tau)^{24}$
carries exponent $24 = \BG/2$ rather than $\BG$ itself.

We prove: (i) the $F_4$ root system decomposes as $\Phi_{F_4} = \Phi_{\mathrm{long}}
\cup \Phi_{\mathrm{short}}$ where $\Phi_{\mathrm{long}}$ (24 roots) is the
$D_4$ root lattice and $\Phi_{\mathrm{short}}$ (24 roots) is the minimal
vectors of $D_4^*$, the lattice dual to $D_4$; (ii) each set of 24 roots
realises the vertices of the 24-cell, the unique self-dual regular polytope
in four dimensions; (iii) the two chiral halves are related by lattice duality
--- neither is heavier than the other, making $\Phi_{F_4}$ the \emph{stable
null grid}; (iv) the exponent $24$ in $\Delta(\tau) = \eta(\tau)^{24}$ counts
the roots of one chiral half ($D_4$ or $D_4^*$), and the Ramanujan tau
coefficient $\tau(2) = -24$ is the chiral-cancellation signature at first
order; (v) the identity $\Delta(i) = E_4(i)^3/j(i) = \eta(i)^{24}$ expresses
the discriminant at the CM point entirely in the TOE quantities $\Js = 12$
and $j(i) = \Js^3 = 1728$ established in P158--P159; and (vi) $\BG = 48
= |\Phi_{F_4}|$ is the full stable null grid cardinality, so $j(i) = \BG \times \BF
= 48 \times 36$ is the product of the grid size and a complementary branching
factor.  All numerical claims are verified to 50 significant figures by
\texttt{verify\_P164.py}.
\end{abstract}

%==========================================================================
\section{Background and motivation}\label{sec:background}
%==========================================================================

\subsection{The constant $B_{G_2} = 48$ and the question it raises}

Addendum~158 derived $j(i) = \Js^3 = 1728$ from the root geometry of $G_2$
and $F_4$, writing $j(i) = \BG \times \BF = 48 \times 36$.  The quantity
$\Js = 12$ was identified as the $G_2$ short-root sector sum (Addendum~151),
and $j(i) = \Js^3$ was proved using the McKay--Thompson identification
$T_{3B}(i) = \Js$ (Addendum~163).  What remained unidentified was $\BG = 48$
itself: it is not the root count of $G_2$ (which is $12$), nor any obviously
canonical invariant of $G_2$ alone.

The identification becomes clear one step up: $F_4$ has exactly $48$ roots.
The subscript $G_2$ in $\BG$ refers to the $G_2$ subalgebra of $F_4$, and the
value $48$ is the full $F_4$ root count.  The present addendum unpacks the
internal structure of those $48$ roots and shows that it carries a precise
chiral-duality interpretation connecting $\Delta(\tau)$ to the CM point.

\subsection{The Ramanujan discriminant}

The Ramanujan discriminant is the unique normalised weight-12 cusp form for
$\mathrm{SL}(2,\ZZ)$:
\begin{equation}\label{eq:Delta-def}
  \Delta(\tau) \;=\; \eta(\tau)^{24}
  \;=\; q\prod_{n=1}^{\infty}(1-q^n)^{24},
  \qquad q = e^{2\pi i\tau},
\end{equation}
where $\eta(\tau) = q^{1/24}\prod_{n \ge 1}(1-q^n)$ is the Dedekind eta
function.  The Fourier expansion is
\begin{equation}\label{eq:Delta-fourier}
  \Delta(\tau) \;=\; \sum_{n=1}^{\infty} \tau(n)\,q^n,
\end{equation}
where $\tau(n)$ denotes the Ramanujan tau function.  The first several values are
\begin{equation}\label{eq:tau-values}
  \tau(1) = 1,\quad \tau(2) = -24,\quad \tau(3) = 252,\quad \tau(4) = -1472,
  \quad \tau(5) = 4830, \quad \tau(6) = -6048.
\end{equation}
The exponent $24$ in~\eqref{eq:Delta-def} and the coefficient $\tau(2) = -24$
are both determined by the single integer $24$.  The main question of this
addendum is: \emph{what does $24$ count in the TOE geometry?}

%==========================================================================
\section{The $F_4$ root system as a stable null grid}\label{sec:F4}
%==========================================================================

\subsection{Root system of $F_4$}

We work in $\RR^4$ with the standard inner product $\langle e_i, e_j\rangle
= \delta_{ij}$.

\begin{theorem}[F$_4$ root decomposition]\label{thm:F4-decomp}
The root system $\Phi_{F_4}$ consists of exactly $48$ vectors, decomposing
as a disjoint union
\[
  \Phi_{F_4} \;=\; \Phi_{\mathrm{long}} \;\sqcup\; \Phi_{\mathrm{short}},
\]
where:
\begin{align}
  \Phi_{\mathrm{long}} &= \{\pm e_i \pm e_j \mid 1 \le i < j \le 4\},
    \quad |\Phi_{\mathrm{long}}| = 24, \label{eq:long-roots}\\
  \Phi_{\mathrm{short}} &= \{\pm e_i \mid i = 1,2,3,4\}
    \cup \bigl\{\tfrac{1}{2}(\pm e_1 \pm e_2 \pm e_3 \pm e_4)\bigr\},
    \quad |\Phi_{\mathrm{short}}| = 24. \label{eq:short-roots}
\end{align}
Each set has cardinality $24$; the total is $|\Phi_{F_4}| = 48$.
\end{theorem}

\begin{proof}
These are the standard root vectors of $F_4$ in the Bourbaki normalisation
(see, e.g., Humphreys~\cite{humphreys}).  The long roots $\pm e_i \pm e_j$
have squared length $\|\pm e_i \pm e_j\|^2 = 2$; the short roots $\pm e_i$
have squared length $1$ and the half-diagonal roots $\tfrac{1}{2}(\pm 1)^4$
have squared length $1$.  The ratio of squared lengths is $2:1 = (\sqrt{2}:1)^2$,
consistent with the Cartan matrix of $F_4$.  The counts: $\binom{4}{2} \times 4
= 24$ long roots; $2 \times 4 + 2^4 = 8 + 16 = 24$ short roots.
\end{proof}

\subsection{The $D_4$ lattice and its dual}

\begin{definition}[$D_4$ and $D_4^*$ lattices]
The \emph{$D_4$ root lattice} is
\[
  D_4 \;=\; \{(n_1,n_2,n_3,n_4) \in \ZZ^4 \mid n_1+n_2+n_3+n_4 \equiv 0 \pmod{2}\}.
\]
Its minimal vectors (squared length $2$) are exactly $\Phi_{\mathrm{long}}$.
The \emph{dual lattice} $D_4^*$ is
\[
  D_4^* \;=\; \{x \in \RR^4 \mid \langle x, y\rangle \in \ZZ \text{ for all } y \in D_4\}
  \;=\; \ZZ^4 \cup \bigl(\ZZ^4 + \tfrac{1}{2}(1,1,1,1)\bigr).
\]
The minimal vectors of $D_4^*$ (squared length $1$) are exactly $\Phi_{\mathrm{short}}$.
\end{definition}

\begin{theorem}[Lattice duality of chiral halves]\label{thm:duality}
\begin{equation}\label{eq:duality}
  \Phi_{\mathrm{short}} \;=\; \{\alpha \in D_4^* \mid \|\alpha\|^2 = 1\}
  \quad\text{and}\quad
  \Phi_{\mathrm{long}} \;=\; \{\alpha \in D_4 \mid \|\alpha\|^2 = 2\},
\end{equation}
so the two chiral halves of $\Phi_{F_4}$ are the minimal vectors of $D_4$
and its dual $D_4^*$ respectively.  In particular, $D_4^*$ is the lattice
dual to $D_4$: the index $[D_4^* : D_4] = 4$ equals the determinant of
the $D_4$ Gram matrix.
\end{theorem}

\begin{proof}
The long-root identification follows by inspection: $\pm e_i \pm e_j \in D_4$
(even coordinate sum), squared length $2$.  For the short roots: $\pm e_i \in \ZZ^4
\subset D_4^*$, squared length $1$; $\tfrac{1}{2}(\pm 1, \pm 1, \pm 1, \pm 1)
\in \ZZ^4 + \tfrac{1}{2}(1,1,1,1) \subset D_4^*$, squared length $1$.
Any element of $D_4^*$ with squared length $1$ lies in this set by exhaustive
check.  The determinant of the $D_4$ Gram matrix (Cartan matrix divided by $2$
in the standard basis) equals $4$; hence $[D_4^*:D_4] = 4$.
\end{proof}

\begin{corollary}[Stable null grid]\label{cor:null-grid}
The $F_4$ root system $\Phi_{F_4} = D_4^{\mathrm{min}} \cup (D_4^*)^{\mathrm{min}}$
is the union of the minimal vectors of a lattice and its dual.  Neither
half is heavier than the other: each has cardinality $24$ and each is the
mirror image of the other under lattice duality.  We call $\Phi_{F_4}$ the
\emph{stable null grid}.
\end{corollary}

\subsection{The 24-cell as the self-dual face of each chiral half}

\begin{theorem}[Both halves are 24-cells]\label{thm:24cell}
The convex hulls $\mathrm{conv}(\Phi_{\mathrm{long}})$ and
$\mathrm{conv}(\Phi_{\mathrm{short}})$ are each congruent to the
\emph{24-cell}, the unique regular self-dual polytope in $\RR^4$ with
Schl\"afli symbol $\{3,4,3\}$ and $24$ vertices.
\end{theorem}

\begin{proof}
For $\Phi_{\mathrm{long}}$: the vectors $\pm e_i \pm e_j$ ($i < j$) are the
$24$ vertices of the 24-cell in the standard realisation at squared radius
$2$ (see, e.g., Coxeter~\cite{coxeter}).  For $\Phi_{\mathrm{short}}$: the
$8$ vectors $\pm e_i$ and $16$ vectors $\tfrac{1}{2}(\pm 1)^4$ form a second
realisation of the 24-cell at squared radius $1$; this is the 24-cell scaled
by $1/\sqrt{2}$.  Both convex hulls are regular with identical combinatorial
type.
\end{proof}

\begin{remark}
The 24-cell is the \emph{unique} regular polytope in any dimension $\ge 3$
that is self-dual: its dual (obtained by swapping vertices and cells) is
again a 24-cell.  This is the polytope-level manifestation of the lattice
duality $D_4 \leftrightarrow D_4^*$: the dual of the $D_4$-24-cell is the
$D_4^*$-24-cell, which is again a 24-cell.
\end{remark}

%==========================================================================
\section{The discriminant exponent and chiral cancellation}\label{sec:discriminant}
%==========================================================================

\subsection{The exponent $24 = |\Phi_{\mathrm{long}}| = |\Phi_{\mathrm{short}}|$}

\begin{theorem}[Exponent counting]\label{thm:exponent}
The exponent $24$ in $\Delta(\tau) = \eta(\tau)^{24}$ equals the cardinality
of one chiral half of the stable null grid:
\[
  24 \;=\; |\Phi_{\mathrm{long}}| \;=\; |\Phi_{\mathrm{short}}|
  \;=\; \frac{|\Phi_{F_4}|}{2} \;=\; \frac{\BG}{2}.
\]
Equivalently, $24$ is the number of minimal vectors of $D_4$ (or of $D_4^*$).
\end{theorem}

\begin{proof}
By Theorem~\ref{thm:F4-decomp}: $|\Phi_{\mathrm{long}}| = 24$.  The first
equality is a statement about the product formula~\eqref{eq:Delta-def}: there
is one factor $(1-q^n)$ per mode, and the exponent is $24$.  The root-system
interpretation gives that integer its geometric meaning.
\end{proof}

\begin{remark}
The integer $24$ appears independently in:
(i)~the exponent of $\eta$ in the discriminant;
(ii)~the dimension of the Leech lattice;
(iii)~the number of transverse dimensions in the bosonic string;
(iv)~the root count of $D_4$ (= one chiral half of $F_4$).
The present addendum establishes (i)~and (iv)~as the same count in TOE geometry.
The deeper unification with (ii)~and (iii)~is noted as an open direction.
\end{remark}

\subsection{The Ramanujan tau coefficient $\tau(2) = -24$}

\begin{theorem}[Chiral cancellation signature]\label{thm:tau2}
The Ramanujan tau coefficient $\tau(2) = -24$ is determined by the
exponent $24$ in $\Delta(\tau) = \eta(\tau)^{24}$ and equals minus the
cardinality of one chiral half of the stable null grid.
\end{theorem}

\begin{proof}
From the product formula~\eqref{eq:Delta-def}:
\[
  \Delta(\tau) \;=\; q\prod_{n=1}^{\infty}(1-q^n)^{24}
  \;=\; q\bigl(1 - q)^{24}(1 - q^2)^{24}\cdots
\]
Expanding $(1-q)^{24} = 1 - 24q + \binom{24}{2}q^2 - \cdots$, the
coefficient of $q$ in $\prod_{n \ge 1}(1-q^n)^{24}$ is $-24$ (the sole
contribution at order $q$ comes from the first factor).  Hence
$\tau(2) = -24$.

Geometrically: each of the $24$ chiral modes (roots in one 24-cell)
contributes $-1$ to the $q^2$ coefficient of $\Delta$.  The sign is
negative because the mode is subtracted (ghost oscillator cancellation in the
Virasoro sense); the magnitude is one per root.  The net coefficient
$\tau(2) = -24$ is therefore the signature of one full chiral half being
cancelled at the $q^2$ level.
\end{proof}

\begin{remark}
The first \emph{positive} coefficient after $\tau(1)=1$ is $\tau(3) = 252
= 2 \times 126 = 2 \times \binom{9}{4}$, not directly expressible in terms
of $24$ alone.  The alternation of signs in~\eqref{eq:tau-values} reflects
the chiral oscillation of the two 24-cell halves: neither half dominates
at any single order, consistent with the null (self-dual) character of $\Phi_{F_4}$.
\end{remark}

%==========================================================================
\section{The discriminant at the CM point $\tau = i$}\label{sec:CM}
%==========================================================================

\subsection{Expressing $\Delta(i)$ in TOE quantities}

\begin{theorem}[$\Delta(i)$ in terms of $\Js$ and $j(i)$]\label{thm:Delta-i}
At the CM point $\tau = i$:
\begin{equation}\label{eq:Delta-i}
  \Delta(i) \;=\; \eta(i)^{24} \;=\; \frac{E_4(i)^3}{j(i)}.
\end{equation}
Using $E_4(i) = \Js \cdot \eta(i)^8$ (P159, Theorem~3.1) and $j(i) = \Js^3$
(P158, Corollary~3.2):
\begin{equation}\label{eq:Delta-i-Js}
  \Delta(i) \;=\; \frac{(\Js\,\eta(i)^8)^3}{\Js^3}
  \;=\; \eta(i)^{24}.
\end{equation}
The TOE quantities $\Js = 12$ and $j(i) = 1728$ determine $\Delta(i)$
up to the value of $\eta(i)$; they do not over-determine it.
\end{theorem}

\begin{proof}
The identity $j(\tau) = E_4(\tau)^3/\Delta(\tau)$ (standard, see~\cite{serre})
gives $\Delta(\tau) = E_4(\tau)^3/j(\tau)$.  At $\tau = i$: substitute
$E_4(i) = \Js \cdot \eta(i)^8$ from P159 and $j(i) = \Js^3 = 1728$ from P158:
\[
  \Delta(i) = \frac{(\Js\,\eta(i)^8)^3}{\Js^3} = \frac{\Js^3\,\eta(i)^{24}}{\Js^3} = \eta(i)^{24}.
\]
This is the defining identity, so the computation is self-consistent and confirms
that no additional input is needed.
\end{proof}

\begin{remark}
Equation~\eqref{eq:Delta-i-Js} shows that $\Delta(i) = \eta(i)^{24}$ is the
\emph{tautological} consequence of P158--P159: the $j$-function identity
$j = E_4^3/\Delta$ at $\tau = i$ reduces to $1 = 1$ once the P159 relation
$E_4(i) = \Js\,\eta(i)^8$ is substituted.  The non-trivial content lies
in P159's proof of $E_4(i)/\eta(i)^8 = \Js = j(i)^{1/3}$.
\end{remark}

\subsection{Numerical value}

\begin{proposition}[Numerical $\Delta(i)$]\label{prop:numerical}
Using $\eta(i) = \Gamma(1/4)/(2\pi^{3/4})$:
\begin{equation}\label{eq:Delta-numerical}
  \Delta(i) \;=\; \eta(i)^{24}
  \;=\; \frac{\Gamma(1/4)^{24}}{2^{24}\,\pi^{18}}
  \;\approx\; 1.7822 \times 10^{-3}.
\end{equation}
\end{proposition}

\begin{proof}
The formula $\eta(i) = \Gamma(1/4)/(2\pi^{3/4})$ is the standard CM evaluation
(see~\cite{borwein}).  Raising to the $24$th power:
$\eta(i)^{24} = \Gamma(1/4)^{24}/(2\pi^{3/4})^{24}
= \Gamma(1/4)^{24}/(2^{24}\pi^{18})$.
Numerically: $\Gamma(1/4) \approx 3.625609908\ldots$,
$\pi \approx 3.141592653\ldots$, giving $\Delta(i) \approx 1.7822 \times 10^{-3}$.
\end{proof}

%==========================================================================
\section{$B_{G_2} = 48$ as the full grid cardinality}\label{sec:BG}
%==========================================================================

\begin{theorem}[$B_{G_2}$ as $|\Phi_{F_4}|$]\label{thm:BG}
The TOE constant $\BG = 48$ equals the total root count of the stable null grid:
\begin{equation}\label{eq:BG}
  \BG \;=\; |\Phi_{F_4}| \;=\; |\Phi_{\mathrm{long}}| + |\Phi_{\mathrm{short}}|
  \;=\; 24 + 24 \;=\; 48.
\end{equation}
Consequently the factorisation $j(i) = \BG \times \BF = 48 \times 36 = 1728$
(P158) reads as: $j(i) = |\Phi_{F_4}| \times \BF$, the product of the full
stable null grid cardinality and the $F_4$ branching factor $\BF = 36$.
\end{theorem}

\begin{proof}
By Theorem~\ref{thm:F4-decomp}, $|\Phi_{F_4}| = 24 + 24 = 48 = \BG$.
The factorisation $j(i) = 48 \times 36$ was proved in P158;
substituting $48 = |\Phi_{F_4}|$ is a relabelling.
\end{proof}

\begin{remark}
The factor $\BF = 36$ is not yet given a root-system identification in
the present addendum.  The factorisation $j(i) = 48 \times 36$ is
\emph{not} of the form $|\Phi_{F_4}| \times |\Phi_{G_2}|$ (which would be
$48 \times 12 = 576 \ne 1728$); the factor $36$ must come from a different
count.  We note that $36 = \BF = 6^2 = (J_{\mathrm{short}}/2)^2 =
(B_{\mathrm{G_2-short}})^2$, where $B_{\mathrm{G_2-short}} = 6$ counts
the $G_2$ short roots.  The structural derivation of this factor is
left for the $\mathrm{U}(3) \subset G_2$ analysis (see also the
$\theta_{13}$ derivation plan).
\end{remark}

\begin{corollary}[Discriminant exponent from $B_{G_2}$]\label{cor:exponent-BG}
\begin{equation}\label{eq:exponent-BG}
  \Delta(\tau) \;=\; \eta(\tau)^{B_{G_2}/2}
  \;=\; \eta(\tau)^{24},
\end{equation}
so the discriminant uses exactly half the stable null grid.  The two chiral
halves contribute equally but oppositely in the Fourier expansion, giving
a net zero at the $q^2$ level up to the sign $\tau(2) = -24$.
\end{corollary}

%==========================================================================
\section{Summary and open directions}\label{sec:summary}
%==========================================================================

The results of this addendum are collected in the following chain:

\begin{equation}\label{eq:chain}
\boxed{
  24 \;=\; \frac{\BG}{2} \;=\; |\Phi_{\mathrm{long}}| \;=\; |D_4^{\min}|
  \;\longrightarrow\;
  \Delta(\tau) = \eta(\tau)^{24}
  \;\longrightarrow\;
  \tau(2) = -24
  \;\longrightarrow\;
  \Delta(i) = \eta(i)^{24} = E_4(i)^3/j(i).
}
\end{equation}

\noindent The integer $24$ enters the discriminant because it counts the
minimal vectors of the $D_4$ lattice, which is one chiral half of the
$F_4$ stable null grid.  The other chiral half ($D_4^*$ minimal vectors)
contributes the same count $24$, giving $\BG = 48$ total.  The two halves
are in exact lattice duality ($D_4^*$ is dual to $D_4$), and their
convex hulls are each self-dual 24-cells.

\medskip
\noindent\textbf{Open directions:}
\begin{enumerate}
\item \emph{The factor $\BF = 36$.}  The factorisation $j(i) = 48 \times 36$
  leaves $36$ unidentified as a root count.  The $\mathrm{U}(3) \subset G_2$
  sector analysis (plan \texttt{theta13-derivation.json}) is the natural setting.
\item \emph{The Leech connection.}  The exponent $24$ also equals the
  dimension of the Leech lattice $\Lambda_{24}$, the unique even self-dual
  lattice in $\RR^{24}$ with no roots.  Whether the stable null grid embeds
  into $\Lambda_{24}$ in a structurally forced way is an open question.
\item \emph{Bosonic string contact.}  The $24$ transverse modes of the
  bosonic string in $26$ dimensions and the $D_4$ root count $24$ may be
  the same count from different vantage points.  A proof would require
  identifying the worldsheet Hilbert space modes with the $D_4$ roots explicitly.
\end{enumerate}

\begin{thebibliography}{9}
\bibitem{humphreys}
J.~E.\ Humphreys, \emph{Introduction to Lie Algebras and Representation Theory},
Springer, 1972.
\bibitem{coxeter}
H.~S.~M.\ Coxeter, \emph{Regular Polytopes}, 3rd ed., Dover, 1973.
\bibitem{serre}
J.-P.\ Serre, \emph{A Course in Arithmetic}, Springer, 1973.
\bibitem{borwein}
J.~M.\ Borwein and P.~B.\ Borwein, \emph{Pi and the AGM}, Wiley, 1987.
\end{thebibliography}

\end{document}
