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\begin{document}
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%Slide 1
\title{{\LARGE\bf What shape is a circle?} \\
 \  \\
 \  \\
  Complex Systems \\
  Summer School,\\
  \ \\
   Santa Fe Institute \\
   \ \\
   }
\author{\ \\ {\large Tom Carter}
\newline
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\date{\today}


\maketitle



%Slide 2
\sectionhead{How do we define a circle?}
     We usually define a circle as
     $$C = \{\vec{x}\ |\ || \vec{x} || = 1 \}$$
     (i.e., the set of all vectors $\vec{x}$ of length 1).
     
     Of course, we need to make sense of $|| \vec{x} ||$, the length of a vector.
     
     Most people start with the definition
     $$|| \vec{x} || = \left(\sum_i x_i^2\right)^\frac{1}{2},$$
     but since we are mathematicians, and don't like our definitions to be too special,
     we can generalize:
     $$|| \vec{x} ||_p = \left(\sum_i |x_i|^p\right)^\frac{1}{p}.$$
     for $p > 0$.

\pagedone

    We then have the more general definition of the $p$ circle (in dimension $n$):
    $$C_p^n = \{\vec{x}\ |\ || \vec{x} ||_p = 1 \}.$$
    What does $C_p^n$ look like?  Let's work in two dimensions, and leave out the
    dimension label.
    
    $C_2$ is the familiar circle:
    
    \ \\ 
    \ \\
    
    \centerline{\includegraphics[width = 4in]{circle-2}}
    
\pagedone

    $C_1$ is a diamond:
    
    \centerline{\includegraphics[width = 4in]{circle-1}}
    
    Note that if our vector space is over $\{0, 1\}$, then a vector is just a string
    of zeros and ones, and $||\vec{x}||_1$ is just the number of ones in the string.
    
    We can convert our length measures into a distance measures:
    $$d_p(\vec{x}, \vec{y}) = ||\vec{x} - \vec{y}||_p
           = \left(\sum_i|x_i - y_i|^p\right)^\frac{1}{p}.$$
           
\pagedone

    In particular, if our vector space is over $\{0, 1\}$, then $d_1(\vec{x}, \vec{y})$
    is just the Hamming distance between the two vectors (i.e., the number of places
    in which the two strings differ).
    
    We can also define
    $$||\vec{x}||_\infty = \lim_{p \to \infty} (||\vec{x}||_p).$$
    Going through the math, we have that
    $$||\vec{x}||_\infty = \max_i(|x_i|).$$
    We then have that $C_\infty$ is a square:
    
    \ \\ 
    
    \centerline{\includegraphics[width = 3in]{circle-infty}}
    
\pagedone

    This means that for a mathematician, a circle is a circle, is a diamond, is a square
    (which may explain why I always had trouble with those ``shape matching'' tests \ldots
    :-)
    
    In general, we have the following sort of picture of various circles:
    
    \ \\ 
    
    \centerline{\includegraphics[width = 5in]{circle-many}}

    Homework exercises:
    
    What happens for $0 < p < 1$?
    
    What happens if we take the limit as $p$ goes to $0$?
    
    Show that in the limit as $p$ goes to $0$, the corresponding distance
    $$d_0(\vec{x}, \vec{y}) = ||\vec{x} - \vec{y}||_0$$
    is a generalized Hamming distance that counts the number of coordinates that are different
    from each other \ldots
    


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