\documentclass[11pt]{article}
%\geometry{landscape}                % Activate for for rotated page geometry
\usepackage{amsmath}
\usepackage{amssymb}

%my commands
\newcommand{\reals} {\mathbb{R}}
\newcommand{\riesphere} {\mathbb{C}}
\newcommand{\nil}{\mathbf{Nil}}
\newcommand{\sol}{\mathbf{Sol}}
\newcommand{\tang}{\text{T}}


\title{ALICE on the Eight-fold Way}
%\author{The Author}
%\date{}

\begin{document}
%\maketitle
\section{2D Geometries}
\subsection{What is geometry?}
%Brief history of geometry - ie, preGreek, Pythagoras, Classic Greek, Euclid, postGreek, attempts to prove parallel postulate, invention of nonEuclidean geometries, Riemann, Klien, Einstien, modern?
\begin{itemize}
\item Game Landscapes - this is not geometry!
\item Euclidean Geometry
\item Symmetry
\end{itemize}

\subsection{Classical Approach}
Euclidean vs. Non-Euclidean

\begin{table}[htbp]
  \centering
  \begin{tabular}{|c|l|l|l|}
    \hline
    & Spherical/Elliptic & Euclidean & Hyperbolic \\ 
    \hline
    Model Pictures & & &\\
    Model/Example & $S^2$, $\riesphere$ & $\reals^2$ & $H^2$ \\ 
    Angle Sum & $> \pi$ & $= \pi$ & $< \pi$ \\ 
    Perimeter, $C_r$ & $< 2\pi r$ & $ = 2\pi r$ & $ > 2\pi r$ \\ 
    Curvature, $K$ & $ > 0$ & $ = 0$ & $ < 0$ \\
    \hline
  \end{tabular}
  \caption{Comparison of 2D Geometries}
  \label{tab:2dgeom}
\end{table}

%Insert pictures of modeling with 2D geometries here
These 3 geometries can be used to model various compact 2D spaces as follows: spherical geometry models only spherical geometry (in the obvious way;) Euclidean geometry models the 1-torus by identifying edges; hyperbolic geometry models $n$-torus for $n > 1$ in a similar way.

\section{3D Geometries}
%Insert picture of the 3 isotropic 3D geometries here?

\subsection{Classification Theorem for 3D Manifolds}
Thurston - Geometrization Conjecture (1982)\\
Perelman - Proof using Ricci Flows (2004?)

Every three dimensional manifold can be locally modeled with one of the following geometries:

\begin{table}[htbp]
  \centering
  \begin{tabular}{|c|c c c|}
    \hline
    Isotropic & $S^3$ & $\reals^3$ & $H^3$ \\ 
    \hline
    & $S^2 \times \reals$&  & $H^2 \times \reals$\\
    Anisotropic & & $\nil$ & $\widetilde{SL_2}(\reals)$ \\
    & & $\sol$ & \\
    \hline
  \end{tabular}
  \caption{List of 3D Geometries}
  \label{tab:3dgeom}
\end{table}

\begin{table}[htbp]
  \centering
  \begin{tabular}{|c|c c c|}
    \hline
    Symmetry Group & $E(3)$ & $O(3,\reals)$ & $SO(3,\reals)$ \\ 
    \hline
    & $S^2 \times \reals$&  & $H^2 \times \reals$\\
    & & $\nil$ & $\widetilde{SL_2}(\reals)$ \\
    & & $\sol$ & \\
    \hline
  \end{tabular}
  \caption{List of 3D Geometries}
  \label{tab:3dgeom}
\end{table}

\section{Nil Geometry}
The $\nil$ geometry is also known as Heisenberg geometry. It is the geometry of the Heisenberg group. This geometry can be thought of as both a Riemannian manifold and sub-Riemannian manifold.
\subsection{Manifolds}
A manifold, $M$, is a structure that locally resembles some Euclidean space. At every point $p$ on the manifold there is a tangent space, $\tang_{p}(M)$, describing the possible directions of movement. A Riemannian manifold is one in which $\tang_p(M) = S^n$ for some sphere $n$ --- that is, every possible direction is allowed; a sub-Riemannian manifold is one where some directions are disallowed.
%This is wrong(?) --- it may be that all directions are present in the tangent space, and only it's simply that movement is only allowed in a certain subspace(subset of directions?) of the tangent space

\subsection{$\nil$ as a sub-Riemannian Manifold}
$M = (x,y,t)$\\
Nil Geometry as a sub-Riemannian manifold is c ompletely nonintegrable - whatever that means.
%columns not rows
$$\tang_p(M) = H_p = \text{span} \left \{ (1,0,2y), (0,1, -2x)  \right \} $$

%do I mean this?
On the $t$ axis, with points vertically separated, the space behaves as if it has infinite positive curvature. 

Off the $t$ axis, with points without purely vertical separations, the space behaves as if it has infinite negative curvature.

%is this true off axis?
Points differing only in the $t$ dimension behave like antipodal points.
                                      
Geodesics minimize the area of the figure projected down onto $(x,y)$. 
%Insert a drawing here - it makes this part much more clear!

%True?
The space is complete and connected.

\subsection{Relationship between the Riemannian and sub-Riemannian Picture}
The Riemannian picture of $\nil$ geometry has $\reals^3$ as the underlying space with a metric setting the length of the vectors $(1,0,2y), (0,1,-2x),(0,0,1)$ to 1. All directions of movement are allowed.
   
Tyson:
In fact, one can consider a whole family of Riemannian metrics on Nil by fixing $\epsilon$ positive and declaring $(1,0,2y)$, $(0,1,-2x)$ and (0,0,$\epsilon$) to be unit vectors. The case $\epsilon = 1$ is the traditional Riemannian metric on Nil. Philosophically, as $\epsilon \rightarrow 0$ this sequence of Riemannian manifolds converges to the sub-Riemannian geometry of the Heisenberg group. This vague statement can be made precise using the language of "Gromov-Hausdorff convergence" (don't worry about what this means). Intuitively, what we are doing is penalizing motion in the t-direction, making it more and more expensive to travel in this direction. In the limit, it becomes impossible to travel (infinitesimally) in this direction and we are forced to take the "long way around" along the sub-Riemannian geodesics.

It turns out that these $\epsilon$ Riemannian metrics are related to each other by scaling. Nil equipped with one of these metrics is *not* a self-similar space: it "looks different depending on how far away you're standing". In fact, as one moves farther and farther away, the effect is to decrease the value of $\epsilon$. So another perspective is that the sub-Riemannian geometry of Nil is what you see when you look at the Riemannian Nil from "infinitely far away". It would be very interesting to have some visual clarification / evidence for this statement.                                                                                                                                                                                                                                                                                                                                                      
\section{Project Ideas}
There are currently implementations of all isotropic three dimensional geometries in OpenGL; further, there are semi-complete implementations of $S^2 \times \reals$ and $H^2 \times \reals$. 
\begin{itemize}
\item Implement $\nil$ in OpenGL!
\item Driving on geodesics with a steering wheel
\item Moving a pointer, with its tangent space attached, in $\nil$ and seeing the tangent space deform
\item Above, with movement of pointer constrained to directions in tangent space
\end{itemize}
\end{document}  
