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\begin{document}
\title{Essay 2:   Geometry of Cubes}
\author{Math 300, Fall 2018, Schneider}
\date{}
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\section{Content}  
In this essay you will systematically construct a coherent mathematical theory of $N$-dimensional cubes.  You will formally define these objects using $\R^n$ coordinates and prove some enumerative formulas about their structure.  Be sure to give concrete examples in dimensions 1, 2, \& 3 via graphical illustrations, on which the reader will base their higher-dimensional intuition.\\
\\
Please define, explain, and illustrate the notion of an $N$-cube and its $n$-elements (vertices, edges, faces, cells, etc.).  Prove that the elements of an $N$-cube are themselves smaller cubes.  Justify this observation using your definition.  Then, give both recursive and explicit formulas for the number of $n$-elements.  Give a table of values for the element-counts, and prove your formulas using explicit reference to your definitions.




\section{Style}
Your audience is at least as mathematically sophisticated as yourself.  Write in the style of an advanced college math textbook or a research article.  I will be looking for well-organized exposition of a coherent mathematical theory.  Motivate the theory, define your terms, and prove your theorems.  Your proofs should explicitly use your definitions.\\
\\
Assume your reader is familiar with general properties of $\R^n$, such as its coordinate system, the notion of convexity, and the dimension of a convex set.  All the math is presented in class lectures, as well as wikipedia and the notes below.



\newpage











\section{Math}
These notes are intentionally condensed, abridged, and abbreviated.  Expand them into human-readable form in your essay.












\subsection{Definitions}

The {\bfseries standard $N$-cube} is the subset of $\R^N$ whose vertices are of the form $(x_1,\dots,x_N)$ with each $x_i=0$ or 1. \\
\\
 Any figure in $\R^m$ $(m\geq N)$ that is geometrically similar to the standard $N$-cube is an {\bfseries$N$-cube}.\\
\\
In the standard $N$-cube, an {\bfseries $n$-element's} vertices are precisely those that match each other in exactly $N-n$ of their coordinates $(x_1,\dots,x_N)$.




\subsection{Formulas}

Define a function
\[
C(N,n) = \text{number of $n$-elements in an $N$-cube.}
\]
Demonstrate and prove this recursive formula:
\begin{align*}
 C(N,n) &=2\cdot C(N-1,n)+C(N-1,n-1)\\
 C(N,0) &= 2^N\\
C(0,n) &= 0\,\,\,\, (n\geq1)\\
 \end{align*}
and this explicit formula:
\[
C(N,n) =2^{N-n}\cdot\binom Nn.
\]



\subsection{Table}

You should include a table of values.  Here's a template.\\
\\
{\centering\makebox[\linewidth]{
\begin{tabular}{c r|c c c c c c|}
		&&\multicolumn{6}{|c|}{Little cube ($n$)}\\
		&&0&1&2&3&4&5\\
\hline
		&0&&&&&&\\
Big		&1&&&&&&\\
cube	&2&&&&&&\\
($N$)	&3&&&&&&\\
		&4&&&&&&\\
		&5&&&&&&\\
\hline
\end{tabular}}}




\pagebreak
\subsection{Proofs}

You have several theorems to state and prove.  Here are a few to get you started.



\vspace{.5in}
\begin{theorem}\label{cubefaces}
The $n$-elements of the standard $N$-cube are themselves $n$-cubes.

\proof In the standard $N$-cube, the vertices of a $n$-element agree in $(N-n)$ coordinates.  Omitting these coordinates produces the vertices of the standard $n$-cube.  $\square$
\end{theorem}



\vspace{.5in}
\begin{theorem}\label{cubeexplicit}
The number of $n$-elements on an $N$-cube is given by
\[C(N,n)\;=\;2^{N-n}\cdot\binom{N}{n}\]
\proof
(Presented in lecture)
\end{theorem}



\vspace{.5in}
\begin{theorem}\label{cuberecursive}
The number of $n$-elements on an $N$-cube satisfies
\begin{align*}
C(N,0) &= 2^N\\
C(0,n) &= 0\,\,\,\, (n\geq1)\\
C(N,n) &=2\cdot C(N-1,n)+C(N-1,n-1)
\end{align*}
\proof
(Presented in lecture) 
\end{theorem}







\end{document}