text:
\section{Introduction}
\label{introduction.sec}

\paragraph{Historical Background.}
In 1968, Gabriele Veneziano \cite{veneziano1968} noticed that an
amazing number of abstract properties required by the relativistic
scattering amplitude for four colliding spinless particles were
embodied in the classical Euler Beta function,
$B(\alpha_{1},\alpha_{2})$, which can be
defined by the integral representation
\begin{equation}
B(\alpha_{1},\alpha_{2}) =
\int_{0}^{1}\,x^{\alpha_{1}-1}(1-x)^{\alpha_{2}-1}
\, dx  \ , \ \ {\rm Re}\,\alpha_{1} >0, \, {\rm Re}\,\alpha_{2} >0   \ .
\label{B4.eq}
\end{equation}
This observation served as the implausible origin of modern string
theory (see, e.g., \cite{polchinksyI,polchinksyII}
for more details), which grew
from the discovery that the Beta function could be
related to the vibration modes of a relativistic string sweeping
out a surface in spacetime \cite{nambu1970,goto1971}.

Almost immediately following Veneziano's discovery, a
function  with a two-dimensional integral representation was found that
could be related to the relativistic scattering amplitude of {\it
five\/} spinless particles \cite{BardakciRuegg1968,Virasoro1969}.
This function, the dual five-point  function $B_{5}$, can be written in various
representations such as the following integral over a
triangular region
\begin{eqnarray}
\lefteqn{B_5(\alpha_1,\alpha_2,\alpha_3,\alpha_4,\alpha_5)  = } \nonumber\\
&=&\!\! \iint\limits_{0<y<x<1} x^{\alpha_1-\alpha_2-\alpha_5}
   y^{\alpha_2-1}(1-x)^{\alpha_3-1}
   (x-y)^{\alpha_5-1}(1-y)^{\alpha_4-\alpha_3-\alpha_5}\,dx\,dy \,
\label{triB5.eq}
\end{eqnarray}
for suitably restricted values of the arguments
$(\alpha_1,\alpha_2,\alpha_3,\alpha_4,\alpha_5)$.
The discovery of this function
indicated that the Euler Beta function was not alone: the Euler Beta function,
which would now be written as $B_{4}(\alpha_{1},\alpha_{2})$, was henceforth
to be regarded as  the first member of the family of $N$-point functions
$B_{N}$ that might be expected to have interesting properties in
analysis as well as in the quantum theory of relativistic
elementary particles.

\paragraph{Cross-Ratio Coordinates.}
A very rapid series of steps subsequently led to what became the
standard Koba-Nielsen representation \cite{KobaNielsen1969b} for the
$N$-point function $B_{N} (\alpha_{13}, \ldots, \alpha_{N-2,N})$,
which can be written as an $(N-3)$-dimensional integral
\begin{equation}
B_{N} (\alpha_{13}, \ldots, \alpha_{N-2,N})
 =  \idotsint\limits_{0<t_1<\cdots<t_{N-3}<1}
 \prod_{i,j} u_{ij}^{\;\;\;\alpha_{ij}-1} d\mu_{N} \ ,
\label{BNintegral.eq}
\end{equation}
where $d\mu_{N} = {dt_{1} \cdots d t_{N-3}}/
         {\prod_{i=1}^{N-4} t_{i} \prod_{j=2}^{N-3} (1 - t_{j})}$
and the $u_{ij}$ are the $N$-point cross-ratios parameterized by
$t_1,\ldots,t_{N-3}$ as described in detail in Section
 \ref{sec:crossratios}.  The formulas (\ref{B4.eq}) and
(\ref{triB5.eq}) correspond to (\ref{BNintegral.eq}) for the cases
         $N=4$ and $N=5$, respectively.

A variety of methods have been employed to study the properties of the
$B_{N}$ integrands as functions of complex variables.  For example,
Koba and Nielson \cite{KobaNielsen1969b} expressed
(\ref{BNintegral.eq}) as an integral  in a space
that was essentially a product of $(N-3)$ copies of $\CP{1}$.  As
noted by one of the current authors in \cite{hansonPGL72}, one can
alternatively express the complex integrand by employing
$\CP{N-3}$ cross-ratios (with a much larger symmetry group) in place of
the product of $(N-3)$ complex projective lines with the single shared
linear fractional transformation symmetry characterizing the
Koba-Nielsen framework.

 We will see in the following that, for $B_{5}$, the compactification
of the set of all five-point cross-ratios can be identified with
$\RP{2}\# 4{\RP{2}}$ as an algebraic subvariety in $\RP{5}$ with a
polyhedral structure that has 12 pentagonal faces.  This embedding of
$\RP{2}\# 4{\RP{2}}$ has a symmetry group of order 120 in $\PGL{6}$.
The double covering, which is the surface of genus four embedded in
$\Sphere{5}\subset \mathbb{R}^{6}$, has a corresponding polyhedral
structure with 24 pentagonal faces and a symmetry group of order 240
in $\OO{6}$.  The integral (\ref{triB5.eq}) is taken over one of the
12 pentagonal faces of $\RP{2}\# 4{\RP{2}}$.  This is the starting
point for the contour integral representation of $B_{5}$.

The study of such a tessellation on $\RP{2}\# 4{\RP{2}}$, the five
crosscap surface, and its symmetry group dates back to the 19th
century \cite{KleinIkosa} and is treated in detail in the work of
Brahana and Coble in 1926 \cite{Brahana1926}.  It is interesting to
see that our space of five-point cross-ratios leads naturally to the
same tessellation, and to the presentation of the symmetry group in
$\OO{6}$.

\paragraph{Contour Integral Representations.}
It is well known that the analytic continuation of the function
defined by (\ref{B4.eq}) is a meromorphic function of
$(\alpha_{1},\alpha_{2})$  on the entire complex space $\mathbb{C}^2$.
In fact,  changing variables in the integral allows the Beta function
to be rewritten in terms of the standard integral representation of
the Gamma function, leading to the explicit analytic continuation formula
\begin{equation}
B(\alpha_{1},\alpha_{2}) = \frac{\Gamma(\alpha_{1})
  \Gamma(\alpha_{2})}{\Gamma(\alpha_{1}+\alpha_{2})} \ .
\label{B4gamma.eq}
\end{equation}

In 1890, Pochhammer \cite{pochhammer1890} gave another interesting
continuation formula for $B(\alpha_{1},\alpha_{2})$ in the following form,
\begin{equation}
B(\alpha_{1},\alpha_{2})= \frac{\epsilon(\alpha_{1},\alpha_{2})}
   {(1-e^{2\pi i \alpha_{1}})(1-e^{2\pi i \alpha_{2}})} \ ,
\label{pochratio.eq}
\end{equation}
where $\epsilon(\alpha_{1},\alpha_{2})$ is a contour integral of
$\beta(z;\alpha_{1},\alpha_{2})=z^{\alpha_{1}-1}(1-z)^{\alpha_{2}-1}$
along a properly immersed loop in $\mathbb{C} \setminus \{0,1\}$,
and hence is a holomorphic function of $(\alpha_{1},\alpha_{2})$.

Our observation that the $B_{5}$ function can be expressed by an
integral over one pentagonal face of $\RP{2}\# 4{\RP{2}}$
leads to a contour integral representation analogous to Pochhammer's
classic representation of $B_4$.  We obtain the following
two-dimensional contour integral representation of $B_{5}$:
\begin{eqnarray}
\lefteqn{B_{5}(\alpha_1,\alpha_2,\alpha_3,\alpha_4,\alpha_5) = }\nonumber\\
&=& \frac{\epsilon(\alpha_1,\alpha_2,\alpha_3,\alpha_4,\alpha_5)}
{ (1-e^{2\pi i \alpha_1})(1-e^{2\pi i \alpha_2})(1-e^{2\pi i \alpha_3})
(1-e^{2\pi i \alpha_4})(1-e^{2\pi i \alpha_5})} \ .
\label{B5ratio.eq}
\end{eqnarray}
Here $\epsilon(\alpha_1,\alpha_2,\alpha_3,\alpha_4,\alpha_5)$ is a holomorphic function
expressed as an integral of a holomorphic 2-form along a closed
oriented surface of genus 5 properly immersed in $\CP{2} \# 4\overline{\CP{2}}$.
Note that, unlike the representation (\ref{triB5.eq}), where
$(\alpha_1, \alpha_2, \alpha_3, \alpha_4, \alpha_5)$ must be properly
restricted for the integral
to be convergent, the representation (\ref{B5ratio.eq}) of $B_{5}$ is
a meromorphic function of
$(\alpha_1,\alpha_2,\alpha_3,\alpha_4,\alpha_5)$ and is defined on
the entire space $\mathbb{C}^5$. Hence the formula
(\ref{B5ratio.eq}) is an explicit analytic continuation formula for
the five-point function $B_{5}$ originally defined by
(\ref{triB5.eq}).

We point out that, to produce the required contour for $B_{5}$, not
only is the two-complex-variable environment supplied by the
Koba-Nielson product of two projective lines, $\CP{1} \times \CP{1}$,
inadequate, but the  richer alternative
$\CP{2}$ framework of \cite{hansonPGL72} is {\it also\/} inadequate.
 The contour lies instead in $\CP{2} \# 4\overline{\CP{2}}$,
 which is the complexification of the
above-mentioned five-crosscap surface $\RP{2} \# 4{\RP{2}}$
in $\RP{5}$ considered as the real part of $\CP{5}$.

We begin in Section \ref{sec:crossratios} by introducing the $N$-point
cross-ratio, which gives rise to the subvarieties upon which our
analysis is based.  Section \ref{sec:b5closure} constructs the
12-pentagon tessellation of the five-crosscap surface as the
compactification of the set of 5-point cross-ratios; its symmetries
and the genus-four double cover are given in Section
\ref{sec:symmetries}.  Then, in Section \ref{sec:B4Poch}, we review
Pochhammer's classical construction for the contour integral
representation of the Euler Beta function.  The framework for studying
$B_{5}$ is set up in Section \ref{sec:B5function} where the
representation (\ref{B5ratio.eq}) is proven.  Selected constructions
are applied to visualizations and computer graphics representations of
the relevant structures in Section \ref{sec:Vis}.
Remarks on the extension to general $N$ are presented in Section
\ref{sec:BNfcn}.

\section{Cross-Ratios}
\label{sec:crossratios}

Recall that the cross-ratio of four distinct ordered numbers
 $\{w,x,y,z\}\subset\mathbb{R}\cup\{\infty\}$  is defined as
\begin{equation}
u(w,x,y,z) = {\frac{(w-y)}{(w-z)}}{\Bigg \slash}{\frac{(x-y)}{(x-z)} } =
\frac{(w-y)(x-z)}{(w-z)(x-y)} \ .
\label{crossratiodef.eq}
\end{equation}
 For any integer $N\geq 4$, we define the
{\it $N$-point cross-ratio\/} of a cyclically-ordered set of $N$ distinct numbers
$\{x_1,\ldots,x_{N}\} \subset \mathbb{R} \cup \{\infty\}$  as the
ordered set of $N(N-3)/2$ numbers
$(u_{13}, u_{14}, \ldots, u_{N-2,N} )$, where
\begin{eqnarray}
u_{ij} &= & u(x_{i},x_{i+1},x_{j+1},x_{j}) =
  \frac{(x_{i}-x_{j+1})(x_{i+1}-x_{j})}{(x_{i}-x_{j})(x_{i+1}-x_{j+1})}
\label{crossratioxi.eq}
\end{eqnarray}
and $ 1\leq i < j \leq N$, $2\leq j-i \leq N-2 $.

The set of all $N$-point cross-ratios can be considered as a subset of
$\mathbb{R}^{N(N-3)/2}$, which we denote by $\mathcal{C}$.  From the
well-known fact that the cross-ratio is invariant under linear
fractional transformations of $\mathbb{R}\cup\{\infty\}$, it is clear
that $\mathcal{C}$ can be parameterized by $(N-3)$ variables, which we
denote as $(t_{1},\ldots,t_{N-3})$.  That is, each point of
$\mathcal{C}$ is the $N$-point cross-ratio of the $N$ cyclically
ordered numbers
\begin{equation}
\{0,\infty,1,t_{1},\ldots,t_{N-3}\},
\label{Nvarlist.eq}
\end{equation}
for a unique  $(t_{1},\ldots,t_{N-3})\in \mathbb{R}^{N-3}$, where
$ t_{1},\ldots,t_{N-3}$ are distinct and not equal to 0 or 1.

For example, for $N=4$, if we set $x_{1}=0$, $x_{2}=\infty$,
 $x_{3}=1$, and $x_{4}=t$, then, according to (\ref{crossratioxi.eq}),
the set of $4$-point cross-ratios in $\mathbb{R}^{2}$ is given by the
following parameterized curve:
  \begin{eqnarray*}
u_{13} & = & u(x_1,x_2,x_4,x_3) = u(0,\infty,t,1) = t \\
u_{24} & = & u(x_2,x_3,x_1,x_4) = u(\infty,1,0,t) = 1-t  \ .
\end{eqnarray*}
Notice that there are three connected components for the domain of
$t$, as shown in Figure \ref{b4-3conn.fig}.

\begin{figure}[!htb]
\centering
  \mbox{\psfig{width=4in, figure=figures/b4-3conn.eps}}
  \caption[]{The $3$ connected components of the domain of the
    parameters for the set of 4-point cross-ratios.
   }
  \label{b4-3conn.fig}
\end{figure}

For $N=5$, the  set of $5$-point cross-ratios is a surface in
$\mathbb{R}^{5}$  parameterized by $(s,t)$ as
\begin{eqnarray}
u_{13} & = & u(x_1,x_2,x_4,x_3) = u(0,\infty,s,1) = s \nonumber \\
u_{14} & = & u(x_1,x_2,x_5,x_4) = u(0,\infty,t,s) = \frac{t}{s}
\nonumber \\
u_{24} & = & u(x_2,x_3,x_5,x_4) = u(\infty,1,t,s) = \frac{1-s}{1-t} \label{ust5.eq}\\
u_{25} & = & u(x_2,x_3,x_1,x_5) = u(\infty,1,0,t) = 1-t \nonumber \\
u_{35} & = & u(x_3,x_4,x_1,x_5) = u(1,s,0,t) = \frac{s-t}{s(1-t)} \nonumber \ .
\end{eqnarray}
The domain of $(s,t)$ has twelve connected components, as shown in
Figure \ref{12st.fig}.

\begin{figure}[!b]
\centering
  \mbox{\psfig{width=2.8in, figure=figures/b5-12st.eps}}
  \caption[]{The $12$ connected components of the domain of parameters
 for the set of 5-point cross-ratios.}
  \label{12st.fig}
\end{figure}

  One can  verify that the cross-ratios $u_{ij}$
defined by (\ref{crossratioxi.eq}) satisfy
\begin{equation}
u_{ij} = 1 - \prod_{m=i+1}^{j-1} \, \prod _{n=j+1}^{i-1} u_{mn}
\label{uijconstr.eq}
\end{equation}
with the convention that $u_{mn} = u_{nm}$ and $u_{m,n+N} =
u_{m,N}$ for all $1 \leq m,n \leq N$.  In fact, the affine algebraic
subvariety in $\mathbb{R}^{N(N-3)/2}$ defined by (\ref{uijconstr.eq}),
minus a set of measure zero, is  precisely the set
$\mathcal{C}$ of $N$-point cross-ratios.

 In particular, for $N=4$, the constraint (\ref{uijconstr.eq})  becomes
\begin{equation}
 u_{13} = 1 -  u_{24}  \ .
\label{b4uij.eq}
\end{equation}
The set $\mathcal{C}$  is the affine algebraic
subvariety in $\mathbb{R}^{2}$ with coordinates $(z_{1},\,z_{2})$ given
by the linear equation
\begin{equation}
1 - z_1 - z_2  = 0 \ .
\label{b4zi.eq}
\end{equation}

For $N=5$, we have
\begin{eqnarray}
u_{13} & = & 1 - u_{24} \, u_{25}  \nonumber\\
u_{14} & = & 1 - u_{25} \, u_{35} \nonumber\\
u_{24} & = & 1 - u_{35} \, u_{13} \label{5uij.eq}\\
u_{25} & = & 1 - u_{13} \, u_{14} \nonumber\\
u_{35} & = & 1 - u_{14} \, u_{24} \nonumber \ ,
\end{eqnarray}
and $\mathcal{C}$ is the affine algebraic subvariety in
$\mathbb{R}^{5}$ with coordinates
$(z_{1},z_{2},z_{3},z_{4},z_{5})$ given by
\begin{eqnarray}
1 - z_1  - z_3 z_4  & = & 0 \nonumber \\
1 - z_2  - z_4 z_5  & = & 0 \nonumber \\
1 - z_3  - z_5 z_1  & = & 0 \label{b5zi.eq} \\
1 - z_4  - z_1 z_2  & = & 0 \nonumber \\
1 - z_5  - z_2 z_3  & = & 0 \nonumber \ .
\end{eqnarray}

{\it Remark.\/} It can be verified that the system (\ref{b5zi.eq})
has rank 3 at the zero locus, and therefore does actually define a
smooth algebraic subvariety of dimension 2.
\comment{ See file DoubleB5.nb; compute Jacobian, solve for zero
  locus, compute minors, determine rank.}

  Now consider the corresponding projective subvarieties.  For $N=4$,
(\ref{b4zi.eq}) becomes
\begin{equation}
z_{0} - z_1 - z_2   = 0 \ ,
\label{b4projzi.eq}
\end{equation}
which  obviously defines a projective line in $\RP{2}$
with homogeneous coordinates $[z_{0},z_{1},z_{2}]$.

Similarly, for  $N=5$, (\ref{b5zi.eq}) yields
the following homogeneous quadratic  equations
in the homogeneous coordinates $[z_0,z_1,z_2,z_3,z_4,z_5]$ of $\RP{5}$:
\begin{eqnarray}
  z_{0}^{\ 2} - z_0 z_1 - z_3 z_4 & = & 0 \nonumber\\
  z_{0}^{\ 2} - z_0 z_2 - z_4 z_5 & = & 0 \nonumber\\
  z_{0}^{\ 2} - z_0 z_3 - z_5 z_1 & = & 0 \label{b5projzi.eq} \\
  z_{0}^{\ 2} - z_0 z_4 - z_1 z_2 & = & 0 \nonumber \\
  z_{0}^{\ 2} - z_0 z_5 - z_2 z_3  & = & 0 \nonumber\ .
\end{eqnarray}
One can verify that (\ref{b5projzi.eq}) defines a smooth
two-dimensional subvariety in $\RP{5}$, which we will denote by $M$.
To see the topology of $M$, we examine the parameterization
(\ref{ust5.eq}).  As we will show in detail in Section
\ref{sec:b5closure}, the image of each
of the twelve connected components of the parameter domain has a
smooth pentagonal closure tessellating $M$ as shown in Figure
\ref{b5-12reg.fig}.
\begin{figure}[!bh]
\centering
  \mbox{\psfig{width=3.0in, figure=figures/b5-12reg.eps}}
  \caption[]{The pentagonal tessellation of the closure of the set of
 5-point cross-ratios.}
  \label{b5-12reg.fig}
\end{figure}
Extending (\ref{b5projzi.eq}) to complex variables
defines a complex algebraic variety  $M^{c}$ that is obviously the
complexification of the real manifold $M$.
$M^{c}$ is $\CP{2}$ with four points blown up
and is topologically homeomorphic to $\CP{2} \# 4\overline{\CP{2}}$.

  The tessellation represented in Figure \ref{b5-12reg.fig} has 12
pentagonal faces, $(12\times 5)/2 = 30$ edges, and $(12\times 5)/4 =
15$ vertices; the Euler number of $M$ is thus $\chi = 15-30+12 = -3$,
and therefore $M$ is the connected sum of five $\RP{2}$'s, i.e., a
sphere with five crosscaps.  Therefore, viewing the five-crosscap
surface as the set of cross-ratios yields a natural tessellation with
12 pentagonal faces, which we can call a ``dodecahedron'' even though
it does not bound a 3-ball.  This tessellation was already described
in detail from the point of view of combinatorial topology in the 19th
century \cite{KleinIkosa}.  In 1926, Brahana and Coble
\cite{Brahana1926}, also arrived at the same tessellation of a sphere
with five crosscaps as a map of 12 countries with five sides, and
studied the symmetry group in detail (see also recent work by Weber
\cite{WeberDodec2005} for additional historical background).  Such
tessellations were generalized by Stasheff for use in his study of the
homotopy theory of H-spaces
\cite{StasheffTAMS1963,StasheffHSpaces,Stasheff2004}, and, in
particular, the analogous tiles in higher dimensions are called
associahedra.  These have played a prominent role, e.g., in the work
of Devadoss \cite{Devadoss1999}.  Our discovery of the relation
between the five-crosscap dodecahedral tessellation and the 5-point
cross-ratios, as well as the apparent relation between the
higher-dimensional analogs and the $N$-point cross-ratios, should thus
be of further interest.

\section{Closure of the 5-Point Cross-Ratio Set in $\RP{5}$}
\label{sec:b5closure}

  We now present a detailed treatment of the pentagonal  tessellation
for $M$.  In the homogeneous coordinates
of $\RP{5}$, we will write the parameterization (\ref{ust5.eq}) as
\begin{eqnarray}
p(s,t) & = & \left[1,s,\frac{t}{s},\frac{1-s}{1 - t}, (1 - t),
        \frac{s - t}{s(1-t)}\right] \nonumber\\
 & = & \left[s(1 - t), s^2 (1 - t), t(1 - t), s(1 - s), s(1 -
  t)^2, s - t\right] \ .
\label{b5hom.eq}
\end{eqnarray}

On the triangular connected component $0<t<s<1$, as $(s,t)\rightarrow
(0,0)$ or $(s,t)\rightarrow (1,1)$, the images do not converge to
a point.  To extend the parameterization to the boundary of the
domain, we will replace the parameters $(s,t)$ as follows.

First, let
\begin{equation}
\!\!(x,y) = \left\{\!\!\begin{array}{ll}
                 (2 u - u v,\, u v), & \!\mbox{for $0 < u \leq
                 \frac{1}{2},\,\,0<v<1$ }\\
                (1 - v + u v, \,  -1 + 2 u + v -u v), &
             \!\mbox{for $ \frac{1}{2} \leq u < 1,\,0<v<1$.}
          \end{array} \right.
\label{xyuv.eq}
\end{equation}
The formula (\ref{xyuv.eq}) defines a 1-1 map between the open square
$(0,1)\times (0,1)$ in the $uv$-plane and the open triangle $0<y<x<1$
in the $xy$-plane, as shown in Figure \ref{uvmap.fig}.

\begin{figure}[htb]
\parbox[t]{5.0in}{\centering
\mbox{\psfig{width=2.2in, figure=figures/squvmapst.eps}}
\hspace{0.1in}\raisebox{1in}{$\Longrightarrow$}\hspace{0.1in}
\mbox{\psfig{width=2.2in, figure=figures/triuvmapst.eps}}
  \caption[]{The map from the square $0<u,v<1$  in the $uv$-plane
 to the triangular region $0<y<x<1$ in the $xy$-plane.}
  \label{uvmap.fig}}
\end{figure}

Next, in Table \ref{b5-12reg.tbl}, we present twelve formulas that give
1-1 maps between the open
triangular domain $0<y<x<1$ and each of the twelve connected
components shown in Figure \ref{12st.fig}.

\begin{table}[t!]
\begin{centering}
\begin{tabular}{|r|p{4.0in}|} \hline
\raisebox{-1.0ex}{n} &
The map from the region $0<y<x<1$ to the connected component
 \fbox{$n$} in Figure \ref{12st.fig}. \\ \hline
 \raisebox{-5pt}{\rule{0in}{15pt}}
${1}$  & $s=x$, $t=y$ \\

 ${2}$ & $s = (x-y)/(1-y),\, t =  y/(y -1)$  \\
 ${3}$ & $ s  =  1/(x y),\, t  =  x/y $  \\
 ${4}$ & $  s  =  y,\, t  =  x $  \\
 ${5}$  & $  s  =  y/(y-1),\, t  =  (x-y)/(1-y) $ \\
 ${6}$ &  $  s  =  x/(x-1),\, t  =  (x-y)/(x -1) $ \\
 ${7}$ &  $ s  =  (x-y)/(x-1),\, t  =  x/(x-1) $  \\
 ${8}$ &  $  s  =  x/y,\, t  =  1/(x y)  $  \\
 ${9}$ & $ s  =  (1-y)/(x-y),\, t  =  y/(y-x) $  \\
${10}$ &  $ s  =  1/(1-y),\, t  =  (1-x)/(1-y) $  \\
${11}$ & $ s  =  (1-x)/(1-y),\, t  =  1/(1-y)  $  \\
${12}$ & $  s  =  y/(y-x),\, t  =  (1-y)/(x-y) $  \\
\hline
\end{tabular}
\caption[]{Transformations from the triangular region $0<y<x<1$ to the
 12 connected components in Figure \ref{12st.fig}.}
\rule{5.375in}{0.01in}
\label{b5-12reg.tbl}
\end{centering}
\end{table}

Composing  (\ref{b5hom.eq}), the entries in Table \ref{b5-12reg.tbl},
and (\ref{xyuv.eq}), we get a parameterization for each of the 12
connected components of $\mathcal{C}$ on the common domain
$(0,1)\times (0,1)$ in the
$uv$-plane.  We will denote these parameterizations by
$f_{1},\ldots,f_{12}$, respectively.

It can be verified that each of the $f_{1},\ldots,f_{12}$ extends as a 1-1
parameterization to the closed square $[0,1]\times [0,1]$.  Each of
the twelve images is a smooth, closed, pentagonal  surface patch
whose vertices correspond to
\[ (u,v) =
({\textstyle\frac{1}{2}},0),\,(1,0),\,(1,1),\, (0,1),\,(0,0) \ . \]

Examining the pentagons one by one, we find they are joined together
to form the closed surface represented by Figure \ref{b5-12reg.fig}.

For future reference, we list below the homogeneous coordinates of the
15 vertices:
\begin{equation}
\begin{array}{rclp{0.5in}rcl}
v_{1} & = & [1, 1,\, 0,\, 0,\, 1,\, 1 ]&&
  v_{9} & = & [0,\, 0,\, 1,\, 0,\, 0,\, -1 ]\\
v_{2} & = & [1,  1,\, 1,\, 0,\, 0,\, 1 ]&&
  v_{10} & = & [0,\, -1,\, 0,\, 1,\, 0,\, 0 ]\\
v_{3} & = & [1, \, 1,\, 1,\, 1,\, 0,\, 0]&&
  v_{11} & = & [0,\, 0,\, 0,\, 1,\, 0,\, 0]\\
v_{4} & = & [1,\, 0,\, 1,\, 1,\, 1,\, 0 ]&&
  v_{12} & = & [0,\, 0,\, 0,\, 0,\, 0,\, 1]\\
v_{5} & = & [ 1, \,0,\, 0,\, 1,\, 1,\, 1 ]&&
    v_{13} & = & [0,\, 0,\, 1,\, 0,\, 0,\, 0 ]\\
v_{6} & = & [0,\, 0,\, -1,\, 0,\, 1,\, 0 ]&&
    v_{14} & = & [0,\, 0,\, 0,\, 0,\, 1,\, 0]\\
v_{7} & = & [0,\, 0,\, 0,\, -1,\, 0,\, 1]&&
  v_{15} & = & [0,\, 1,\, 0,\, 0,\, 0,\, 0] \ .\\
  v_{8} & = & [0, 1,\, 0,\, 0,\, -1,\, 0 ] && &&
\end{array}
\label{origcoords.eq}
\end{equation}

\section{Symmetries and the Double-Covering Lift}
\label{sec:symmetries}

One can see using combinatorial arguments that the tessellation of $M$
shown in Figure \ref{b5-12reg.fig} has many symmetries.  We will
present the group of symmetries as follows.

One of the symmetries, when restricted to face \fbox{$1$}, is a
rotation that transforms the vertices 1,2,3,4,5 to 2,3,4,5,1.  This
symmetry also transforms vertex 12 to vertex 15.  Using the
coordinates of these vertices from  (\ref{origcoords.eq}), we
construct the matrix
\[X_{5} = B \cdot A^{-1}\]
with $A$  the $6\times 6$ matrix
given by $A=[v_1 \, v_2 \, v_3 \, v_4 \,  v_5 \, v_{12}]$, where
$v_{1}$, etc., are written as column vectors whose components are
specified by (\ref{origcoords.eq}).
Similarly, $B=[ v_2 \, v_3 \, v_4 \, v_5 \,  v_1 \, v_{15}]$,
yielding
\begin{equation}
X_5 = \left[\begin{array}{cccccc}
1 &  0 &  0 &  0 &  0 &  0\\
0 &  0 &  1 &  0 &  0 &  0\\
0 &  0 &  0 &  1 &  0 &  0\\
0 &  0 &  0 &  0 &  1 &  0\\
0 &  0 &  0 &  0 &  0 &  1\\
0 &  1 &  0 &  0 &  0 &  0
\end{array} \right] \ ,
\label{permutationsym.eq}
\end{equation}

Another symmetry is the reflection along the edge joining $v_3$ and
$v_4$, which transforms vertices 1,2,3,4,5,12 to 12,7,3,4,9,1.  As
above, one constructs
\begin{equation}
X_2= \left[\begin{array}{cccccc}
 1 & 0 &  0 & 0  & 0 &  -1 \\
 2 & 0 &  -1 & 0  & -1 &  -1 \\
  0 & 0 &  0 & 1 & 0 &  0 \\
  0 & 0 & 1 & 0  & 0 &  0 \\
 2 & -1 &  0 & -1  & 0 & - 1 \\
  0 & 0 &  0 & 0  & 0 &  -1
\end{array} \right]\ .
\label{reflect2sym.eq}
\end{equation}

The following observations are essential: Viewing $X_5$ and $X_2$ as
elements of $\PGL{6}$, one can verify that the zero-locus of
(\ref{b5projzi.eq}) is invariant under the corresponding
transformations of $\RP{5}$.  $X_5$ and $X_2$ generate a group of
order 120, which is isomorphic to the group of automorphisms of $M$
mentioned above.  In other words, we have embedded the automorphism
group of $M$ in $\PGL{6}$.

In fact, $X_5$ and $X_2$ generate a subgroup $G$ of order 120 in
$\GL{6}$.  We let
\begin{equation}
Q = \frac{1}{70} \sum_{g\in G} (g)^{t} \cdot g \ .
\label{invargrpmetric.eq}
\end{equation}
More explicitly,
\begin{equation}
Q =
 \left[\begin{array}{cccccc}
 20 &  -6 & -6 & -6 & -6 & -6 \\
 -6 &   4 &  1 &  2 &  2 &  1 \\
 -6 &   1 &  4 &  1 &  2 &  2 \\
 -6 &   2 &  1 &  4 &  1 &  2 \\
 -6 &   2 &  2 &  1 &  4 &  1 \\
 -6 &   1 &  2 &  2 &  1 &  4
\end{array} \right] \ .
\label{qtransposeq.eq}
\end{equation}
As is well-known, $Q$ defines a $G$-invariant quadratic form on
$\mathbb{R}^{6}$.  Then the algebraic subvariety in $\mathbb{R}^{6}$ defined by
\begin{eqnarray}
  z_{0}^{\ 2} - z_0 z_1 - z_3 z_4 & = & 0 \nonumber\\
  z_{0}^{\ 2} - z_0 z_2 - z_4 z_5 & = & 0 \nonumber\\
  z_{0}^{\ 2} - z_0 z_3 - z_5 z_1 & = & 0 \label{b5projall.eq} \\
  z_{0}^{\ 2} - z_0 z_4 - z_1 z_2 & = & 0 \nonumber \\
  z_{0}^{\ 2} - z_0 z_5 - z_2 z_3  & = & 0 \nonumber\\
  \left[z_{0}\, z_{1}\, z_{2}\, z_{3}\, z_{4}\, z_{5}\right]\cdot Q \cdot
    \left[z_{0}\, z_{1}\,  z_{2}\, z_{3}\, z_{4}\, z_{5}\right]^{t} & = & 1 \nonumber
\end{eqnarray}
is $G$-invariant and we denote it by $\widetilde{M}$.

Comparing to (\ref{b5projzi.eq}), we see that
$\widetilde{M}$ is  the double covering of $M$ lifted from
$\RP{5}$ to $\mathbb{R}^{6}$ and is therefore topologically the
orientable surface of {\it genus four\/}.  Notice that $\widetilde{M}$ is also
invariant under the action of $-I_{6}$, where $I_{6}$ denotes the
$6\times 6$ identity matrix.  Hence,  $\widetilde{M}$ is invariant under
the group $\widetilde{G}$  generated by $X_{5}$, $X_{2}$, and $-I_{6}$, which has 240
elements in $\GL{6}$.

Let $P$ be a $6 \times 6$ matrix satisfying
\begin{equation}
P^{t} P = Q \ .
\label{PtPeqQ.eq}
\end{equation}
Then $P\widetilde{M} \subset \Sphere{5}$, where
 $\Sphere{5} = \{(x_{0},\ldots,x_{5})\in \mathbb{R}^{6} \, :\,
 x_{0}^{\ 2} + \cdots +  x_{5}^{\ 2} = 1\}$ is the unit sphere in
 $\mathbb{R}^{6}$, and it is invariant under the subgroup of order 240
 in $\OO{6}$ generated by $P X_{5} P^{-1}$,  $P X_{2} P^{-1}$,
and  $-I_{6}$.

With the parameterization $f_{1}(u,v),\ldots,f_{12}(u,v)$ for $M$ from
Section \ref{sec:b5closure}, we can now easily write down the
following parameterization for $\widetilde{M}$:
\begin{equation}
\tilde{f}_{i}^{\pm} = \frac{\pm f_{i}}{\sqrt{f^{t}_{i} \cdot Q \cdot
    f_{i}}} \ \ \ \ i=1,\ldots,12 \ .
\label{r6lift.eq}
\end{equation}
Each $\tilde{f}_{i}^{\pm}$ maps $[0,1]\times [0,1]$ in the $uv$-plane
to a pentagonal surface patch.  This yields a tessellation of
 $\widetilde{M}$ with 24 pentagonal faces.  As mentioned at the end of Section
\ref{sec:crossratios}, such a tessellation for the genus-four
surface has long been known.  Here we have performed this
tessellation symmetrically in $\mathbb{R}^{6}$.

The coordinates of the vertices appearing in the tessellation of
$\widetilde{M}$ can be computed from (\ref{r6lift.eq}) at the points
$(u,v)\,=\, (\frac{1}{2},0),\, (1,0),\,(1,1),\,(0,1),\,(0,0)$.  They
are in fact the same as those presented in (\ref{origcoords.eq}),
together with their negatives, viewed now as coordinates in
$\mathbb{R}^{6}$.

We now identify the 24 faces in  $\widetilde{M}$ with ordered sets of
vertices in $\mathbb{R}^{6}$:  the oriented faces are labeled in
terms of the indices of the vertices in (\ref{origcoords.eq}), where a
minus sign indicates the negative mirror
vertex and conjugate  faces are denoted with bars:
\begin{equation}
\begin{array}{rlrl}
\mathrm{face\ } 1: & (1, 2, 3, 4, 5)&
\mathrm{face\ } \bar{1}: & (-1, -5, -4, -3, -2)\\
\mathrm{face\ } 2:& (1,5,- 9, - 13, 6) &
\mathrm{face\ } \bar{2}:& (-1, -6, +13, +9, -5)\\
\mathrm{face\ } 3: & ( 8, 15, 11, -7, 3)&
\mathrm{face\ } \bar{3}: & ( -8, -3, +7, -11, -15)\\
\mathrm{face\ } 4: & (-12, 9, 4, 3, -7) &
\mathrm{face\ } \bar{4}: & (12, 7, -3, -4, -9)\\
\mathrm{face\ } 5: & (12, -9, 5, 10, 11)&
\mathrm{face\ } \bar{5}: & (-12,-11, -10, -5, 9)\\
\mathrm{face\ } 6: & (-8, -15, 10, 5, 4)&
\mathrm{face\ } \bar{6}: & (+8, -4, -5, -10, +15)\\
\mathrm{face\ } 7: &  (-8, 4, 9, 13, 14)&
\mathrm{face\ } \bar{7}: &  (+8,-14, -13, -9, -4)\\
\mathrm{face\ } 8: & (8, 3, 2, -6, -14)&
\mathrm{face\ } \bar{8}: & (-8, +14, +6, -2, -3)\\
\mathrm{face\ } 9:  & (1, 6, 14, 15, -10)&
\mathrm{face\ } \bar{9}:  & (-1, +10, -15, -14, -6)\\
\mathrm{face\ } 10: &  (1, -10, -11, 7, 2)&
\mathrm{face\ } \overline{10}: &  (-1, -2, -7, +11, +10)\\
\mathrm{face\ } 11: & (12, 13, -6, 2, 7)&
\mathrm{face\ } \overline{11}: & (-12, -7, -2, +6,-13)\\
\mathrm{face\ } 12: & (12, 11, 15, 14, 13 ) &
\mathrm{face\ } \overline{12}: & (-12, -13, -14, -15, -11)\ .\\
\end{array}
\label{24facelabels.eq}
\end{equation}
The correspondence between the 12 projective faces and the 12 $(s,t)$
regions of Figure \ref{12st.fig} is shown in Figure  \ref{b5-12reg.fig};
the correspondence between the 24 faces in the double cover
$\widetilde{M}$ and the 12 regions is shown in
Figure \ref{b5-12orient.fig}.

\begin{figure}[!htb]
\centering
  \psfig{width=2.5in, figure=figures/b5-12orientA.eps}
\hspace{0.1in}
  \psfig{width=2.5in, figure=figures/b5-12orientB.eps}
  \caption[]{The $24$ face identifications of the double covering.}
 \label{b5-12orient.fig}
\end{figure}

\section{Review of the Pochhammer Contour for B${}_{4}$}\label{sec:B4Poch}

  In this section, we  review Pochhammer's construction
\cite{pochhammer1890,jordan1887,whittakerwatson} of the contour
integral leading to the formula (\ref{pochratio.eq}) for $B_{4}$.
This will lead us to the contour integral representation for $B_{5}$
to be presented in Section \ref{sec:B5function}.

Let
\begin{equation}
\beta(z;\,\alpha_{1},\alpha_{2}) = z^{\alpha_{1}-1}\,(1-z)^{\alpha_{2}-1} \ .
\label{b4integrand.eq}
\end{equation}
where $(\alpha_{1},\alpha_{2})$ is a pair of arbitrary complex numbers.
 Considered as a function of $z$,
$\beta$ defines a family of holomorphic
functions on a proper Riemann covering sheaf $S$ over
$\mathbb{C} \setminus\{0,1\}$. Let $C$ be a closed and oriented curve in $S$ that
is the lift of the closed and oriented curve in $\mathbb{C}
\setminus\{0,1\}$ shown in Figure \ref{pochsketch.fig},  where we label
the line segments by their relative phases in the lift.  This is known as the
 Pochhammer contour.

\begin{figure}[!htb]
\centering
  \mbox{\psfig{width=8.5cm, figure=figures/pochsketchalpha.eps}}
  \caption[]{The Pochhammer contour $C$ for the Euler Beta function.}
  \label{pochsketch.fig}
\end{figure}

Next, define
\[\epsilon(\alpha_{1},\alpha_{2})=\int_C\,\beta(z;\,\alpha_{1},\alpha_{2})\,dz\, \ .\]
Clearly $\epsilon(\alpha_{1},\alpha_{2})$ is a holomorphic function of
$(\alpha_{1},\alpha_{2})$ and is
invariant under continuous deformations of $C$. Therefore,  letting $r
\rightarrow 0^+$ in Figure \ref{pochsketch.fig}, one easily sees that, if
${\rm Re}\,\alpha_{1} >0$ and ${\rm Re}\,\alpha_{2} >0$, then
\[\epsilon(\alpha_{1},\alpha_{2})=(1-e^{2\pi i \alpha_{1}})(1-e^{2\pi
  i \alpha_{2}})\int_0^1\,x^{\alpha_{1}-1}(1-x)^{\alpha_{2}-1}\,dx \ ,
\]
which yields the formula  (\ref{pochratio.eq}).

If $\alpha_{1}$ (or $\alpha_{2}$, resp.) is an integer $\ge 1$, then the holomorphic
1-form $\beta(z;\,\alpha_{1},\alpha_{2})\,dz\,$ on $S$ descends to a holomorphic 1-form
on a proper Riemann covering sheaf over $\mathbb{C} \setminus \{0\}$
(or $\mathbb{C} \setminus \{1\}$, resp.). Since the curve in Figure
\ref{pochsketch.fig} is contractible in $\mathbb{C} \setminus \{0\}$
(or $\mathbb{C} \setminus \{1\}$, resp.), $\epsilon(\alpha_{1},\alpha_{2})=0$.

\begin {figure}[htb!]
\centering
  \mbox{\psfig{width=8.5cm, figure=figures/pochzeroes.eps}}
  \caption[]{The deformation of the Pochhammer contour to a null
  contour when the conditions $\alpha_{1}+\alpha_{2} = 0,-1,-2,\ldots$ remove
  the branch point at infinity (open circle).}
\label {pochdeform.fig}
\end {figure}

Notice that, by letting  $w=1/z$, we have
\[z^{\alpha_{1}-1}\,(1-z)^{\alpha_{2}-1}\,dz\, =
  -(w-1)^{\alpha_{2}-1}w^{-\alpha_{1}-\alpha_{2}}\,dw\, \ . \]
This shows that, if $(\alpha_{1}+\alpha_{2})$ is a non-positive integer
($\alpha_{1}+\alpha_{2} = 0,\,-1,\,-2,\ldots$),
the holomorphic 1-form $\beta(z;\,\alpha_{1},\alpha_{2})\,dz\,$ on $S$
descends to a holomorphic 1-form on a proper Riemann covering sheaf
over $\CP{1} \setminus\{0,1\}$, where as usual we identify $\CP{1}$
with $\mathbb{C}\cup\{\infty\}$.  Since the curve in Figure
\ref{pochsketch.fig} is contractible in $\CP{1} \setminus\{0,1\}$
(see Figure \ref{pochdeform.fig}), it therefore follows that
$\epsilon(\alpha_{1},\alpha_{2})=0$ also in this case.   Figure \ref{pochdeform.fig}
shows how the contractibility of the contour can be made explicit.

From these observations and (\ref{pochratio.eq}), one concludes in
particular that the poles of $B(\alpha_{1},\alpha_{2})$ can only occur
at points where either $\alpha_{1}$ or $\alpha_{2}$ is a non-positive
integer. Furthermore, $B(\alpha_{1},\alpha_{2})=0$ if neither
$\alpha_{1}$ nor $\alpha_{2}$ is a non-positive integer, but
$(\alpha_{1}+\alpha_{2})$ is a non-positive integer.  These properties
of course also follow directly from (\ref{B4gamma.eq}); in fact these
are precisely all the poles and zeroes of $B(\alpha_{1},\alpha_{2})$.

\section {Contour Representation of the Function B${}_{5}$}\label{sec:B5function}

We now view $M$ as the real two-dimensional surface in $M^{c}$, as
defined at the end of Section \ref{sec:crossratios}.  The manifold
$M^{c}$ can be visualized by the complexification of Figure
\ref{b5-12reg.fig};  with the edges of the pentagon taken off,
$M^{c}$ is now parameterized by two complex parameters that we denote
as $(z_{1},\,z_{2})$, replacing $(s,t)$ in (\ref{b5hom.eq}).  Following
the procedure in Section \ref{sec:B4Poch}, let
\begin{eqnarray}
\lefteqn{\beta_5(z_1,z_2;\,\alpha_1,\alpha_2,\alpha_3,\alpha_4,\alpha_5)=}\nonumber\\
& = &
z_1^{\alpha_1-\alpha_2-\alpha_5}z_2^{\alpha_2-1}(1-z_1)^{\alpha_3-1}
  (z_1-z_2)^{\alpha_5-1}  (1-z_2)^{\alpha_4-\alpha_3-\alpha_5} \ .
\label{b5integrand.eq}
\end{eqnarray}
Then  (\ref{triB5.eq}) can be viewed as the integral of the (locally)
holomorphic 2-form $\beta_5\,dz_1\wedge dz_2$ (with branched
singularities) on $M^{c}$ over the domain \fbox{$1$} on $M$.

The function $\beta_5$ can be viewed as a
5-complex-parameter family of locally holomorphic functions on $M^{c}$
with branched singularities at the edges of the pentagons evident in the
complexification of Figure \ref{b5-12reg.fig}.

 Let $S$ be the Riemann covering sheaf of $\beta_5$ over $M^{c} \setminus
\{{\rm branch \ lines}\}$. We will construct an orientable closed
surface in $S$ that is the lift of a closed surface in $M^{c}$ obtained
by wrapping properly around the five (complex) edges of the pentagonal
domain \fbox{$1$}.  This will then lead to the formula (\ref{B5ratio.eq}).

A function such as $\beta_5$ is only defined
on $M^{c}$, away from singularities, up to a  factor $e^{2\pi i\gamma}$,
i.e., by a {\it phase} $\gamma$ which is an integer linear combination of
$\alpha_1,\cdots,\alpha_5$.  To lift a surface wrapping around the branch lines to
the covering sheaf $S$, on which $\beta_5$ is a holomorphic function,
we first need to understand how the phase of $\beta_5$ changes on a
piece of surface as it makes a simple fold back around one branch line
(see Figure \ref{foldingbranch.fig}).
\begin {figure}[bh!]
\centering
  \mbox{\psfig{width=5cm, figure=figures/foldingbranch.eps}}
\caption [] {A surface sheet folds back around a codimension two branch line.}
\label {foldingbranch.fig}
\end {figure}

It is obvious that if a surface folds back around the branch line
$z_2=0$, $z_1=1$, or $z_1-z_2=0$, then the phase of $\beta_5$
changes by $\pm \alpha_2$, $\pm \alpha_3$, or $\pm \alpha_5$, respectively, where the
sign $+$ or $-$ depends on the folding direction, i.e., whether
the direction is {\it counterclockwise} or {\it clockwise}.

\begin {figure}[ht!]
\centering
  \mbox{\psfig{width=5cm, figure=figures/b5cp2blowup.eps}}
\hspace{.5in}
  \mbox{\psfig{width=5cm, figure=figures/b5reg1blowup.eps}}
\caption [] {The pentagonal domains of the $B_{5}$ integrand, with detailed
pentagonal branch structure shown for region 1.}
\label{cp2blowup.fig}
\end {figure}

\begin {figure}[ht!]
\centering
  \mbox{\psfig{width=4.cm, figure=figures/blowupA.a.eps}}
  \raisebox{1in}{\hspace{.4in}$\mathbf\Longrightarrow$  \hspace{.4in}}
  \mbox{\psfig{width=5cm, figure=figures/blowupA.b.eps}}
\caption [] {The coordinates around the branch line at A.}
\label {blowupA.fig}
\end {figure}

To see how the the phase of $\beta_5$ changes on a surface folding
around the branch line $A$ (see Figure  \ref{cp2blowup.fig}),
let  $(w_1,w_2)$ be the coordinates around
$A$ chosen so that $A$ is given by $w_1=0$, and away from $A$,
$w_1=z_1$, $w_2=z_{2}/z_{1}$  (cf. \cite{GriffithsHarris1978} and
Figure \ref{blowupA.fig}).
 We can then write $\beta_5$ as
\begin{eqnarray*}
\lefteqn{\beta_5(z_1,z_2;\,\alpha_1,\alpha_2,\alpha_3,\alpha_4,\alpha_5)=}\\
& = &
w_1^{\alpha_1-2}w_2^{\alpha_2-1}(1-w_1)^{\alpha_3-1}(1-w_2)^{\alpha_5-1}
(1-w_1w_2)^{\alpha_4-\alpha_3-\alpha_5} \ .
\end{eqnarray*}
It is now easy to see that as a surface
folds back around A, the
phase of $\beta_5$ changes by $\pm \alpha_{1}$.  Similarly, one can show that
as a surface folds back around B, the phase of $\beta_5$ changes by
$\pm \alpha_{4}$.

We now construct an immersed surface in $M$ in three steps as follows:
\begin{itemize}
\item[Step 1.]  Consider a set of 32 copies of the pentagonal sheets
stacked over the region \fbox{1} in Figure \ref{cp2blowup.fig}, with a small
neighborhood of the five corners taken off for now.  From what we have
shown above, it is appropriate to label the edges of each pentagonal
sheet at $A, \ z_2=0, \ z_1=1, \ B, \ z_1-z_2=0$ by $\alpha_1, \ \alpha_2, \
\alpha_3, \ \alpha_4, \ \alpha_5$, respectively (see Figure \ref{pentagonblowup.fig}).

\begin {figure}
\centering
\psfig{width=5cm, figure=figures/pentagonblowup.eps}
\hspace{.8cm}
\psfig{width=5cm, figure=figures/pentagonstacks.eps}
\caption [] {The pentagonal sheets in  region \fbox{1}.}
\label {pentagonblowup.fig}
\end{figure}

We attach to each of these pentagonal sheets  a {\it phase label}
\[ p_1 \alpha_1+p_2 \alpha_2+p_3 \alpha_3+p_4
\alpha_4+p_5 \alpha_5 \ ,
\]
where $p_j=0$ or $1$ for $j=1,\dots,5$.  Each pentagonal sheet is
given an orientation which is same as or opposite to the original
natural orientation on region \fbox{1} according to whether $\sum p_j$ is
even or odd.

\item[Step 2.]
 Two pentagonal sheets in Step 1 are joined along the
edge $\alpha_j$ by folding around the corresponding branch line in the proper
direction (see Figure \ref{foldingbranch.fig})
if and only if their phase numbers  differ by
$\alpha_j$. It is easy to see that we end up with an immersed oriented
surface in $M \setminus \{{\rm branch \  lines}\}$ that can be lifted
to $S$. However, this surface has  {\it 40 holes\/}
caused by the small neighborhoods that we removed around the corners
where the branch lines intersect.  In Figure \ref{b5commutator.fig},
we show a single instance of one of these holes.

\begin {figure}
\centering
\psfig{width=6cm, figure=figures/b5Commutator.eps}
\psfig{width=6cm, figure=figures/b5CommutatorTorus.eps}\\
\hspace{1.5in} (a) \hfill (b) \hspace{1.5in}
\caption [] {(a) The hole at a single corner.
(b) A hole-filling disk in $\Sphere{3}$ enclosing the intersection
point of the two (complex) branch lines.}
\label {b5commutator.fig}
\end {figure}

\item[Step 3.]  To show that one can fill in these holes, we observe
 that a small 3-sphere around an intersection of two branch lines,
 with the branch lines taken off, is homotopic to a torus and hence
 its fundamental group is isomorphic to the Abelian group $\mathbb Z
 \times \mathbb Z$. It is easy to see that the boundary of a small
 hole at this intersection, which lies in the surrounding 3-sphere,
 represents the element $(1,0)+(0,1)+(-1,0)+(0,-1)=(0,0)$ and
 therefore is contractible (a filled-in disk as illustrated in Figure
 \ref{b5commutator.fig}(b)).

{\it Remark\/}: If one tries to construct such a surface directly in
$\mathbb{C}^{2}$, then there would be a hole at a corner where
three branch lines intersect, and for such holes the argument above fails.

\end{itemize}

We have therefore obtained an oriented closed surface $F$ immersed in
$S$, whose Euler characteristic can be easily seen to be
$\chi(F)=40-80+32=-8$.  Hence the surface $F$ is of genus 5, i.e., a
{\it sphere with five handles\/}.

Now, let
\[
\epsilon(\alpha_1,\alpha_2,\alpha_3,\alpha_4,\alpha_5)=\int_F\,\beta_5(z_1,z_2;\,
  \alpha_1,\alpha_2,\alpha_3,\alpha_4,\alpha_5)\,dz_1\wedge dz_2 \ .
\]
Clearly $\epsilon(\alpha_1,\alpha_2,\alpha_3,\alpha_4,\alpha_5)$ is a holomorphic function of
$(\alpha_1,\alpha_2,\alpha_3,\alpha_4,\alpha_5)$ and is invariant
under continuous deformations of $F$.  As in the case of the  Pochhammer contour
described in  Section \ref{sec:B4Poch}, we can  take the limit and
 calculate for suitably restricted $(\alpha_1,\alpha_2,\alpha_3,\alpha_4,\alpha_5)$ that
\begin{eqnarray*}
\lefteqn{\epsilon(\alpha_1,\alpha_2,\alpha_3,\alpha_4,\alpha_5)=}\\
 && = (1-e^{2\pi i\alpha_1})(1-e^{2\pi i\alpha_2})(1-e^{2\pi
i\alpha_3})(1-e^{2\pi i\alpha_4})(1-e^{2\pi i\alpha_5}) \\
&& \!\!  \cdot  \!\! \iint\limits_{0<x_{2}<x_{1}<1}\,
 \!\!\! x_1^{\alpha_1-\alpha_2-\alpha_5}x_2^{\alpha_2-1}(1-x_1)^{\alpha_3-1}
  (x_1-x_2)^{\alpha_5-1}(1-x_2)^{\alpha_4-\alpha_3-\alpha_5}\,dx_1\,dx_2\, \ .
\end{eqnarray*}
This proves (\ref{B5ratio.eq}). $\Box$

Following the analogy to the $B_{4}$ Pochhammer analysis to determine
further constraints on the poles and zeroes of $B_{5}$ is an
interesting challenge for future work.

 \section{Visualizations of Connected Components and  Pochhammer Contours}
 \label{sec:Vis}

  The analysis of the $B_{5}$ function in the previous sections has
been based entirely on algebraic manipulations and line drawings
sketching the essential features of the geometry.  This section is
motivated by the observation that, since there are  algebraic
constructions for  every geometric concept, we can go one step further
and show precise images of each construction, helping the reader to
develop a quantitative as well as a qualitative understanding of the
framework we have developed.  We establish the basic context with
some examples based on the Euler Beta function, and then proceed to
show some of the remarkable manifolds that occur in the $B_{5}$ analysis.

\subsection{${\mathbf B_4}$ Connected Components Embedded in a
  Veronese Surface}

The Euler Beta function itself can be analyzed using cross-ratio
coordinates.  We begin with the two cross-ratio  variables, $x_1$ and $x_2$, obeying
the apparently uninteresting constraint
\[x_{1} = 1 - x_{2} \ . \]
However, when we put this into homogeneous coordinates
$\{x_{0},x_{1},x_{2}\}$, the constraint becomes $x_{0} = x_{1}
+x_{2}$, and we can solve these equations independently in the three component
regions, written as three intervals in inhomogeneous coordinates as  $A = [0,1]$, $B =
[1,\infty]$, and $C = [-\infty,0]$. Noting that the
space we are now dealing with is {\it not\/} $\,\mathbb{C}$ or $\CP{1}$,
but the real part of the $\CP{2}$ {\it cross-ratio-space\/},
we can parameterize each interval in homogeneous $\RP{2}$
coordinates as follows:
\begin{eqnarray}
A(t):&& \left[ 1,\, t,\, (1-t) \right] \nonumber\\
B(t):&& \left[ (1-t), \, 1,\, -t  \right] \\
C(t):&& \left[-t,\, (1-t),\, -1 \right] \nonumber \ .
 \end{eqnarray}
We see that region $A$ solves $1= x_1+x_2$ with $x_{1} = t$,
$B$ solves $1= x_1+x_2$ with $x_{1} = 1/(1-t)$ when all is multiplied
by $(1-t)$, and
$C$ solves $1= x_1+x_2$ with $x_{1} = (t-1)/t$ when all is multiplied
by $t$.  The interpolating functions in the three regions
obviously correspond to the three $B_4$ component regions introduced
initially in Figure (\ref{b4-3conn.fig}),
and they interpolate between the  points represented
in the Riemann-sphere depiction of Figure \ref{rsphere1.fig}:

\begin{figure}[!htb]
\centering \mbox{\psfig{width=8.5cm, figure=figures/rsphere1.eps}}
  \caption[]{The full base space of the $B_4$ branched covering, showing
the three regions corresponding to the three intervals $(0,1)$,
$(1,\infty)$, and $(-\infty,0)$ on the real projective  line.}
  \label{rsphere1.fig}
\end{figure}

\[
\begin{array}{rcrclclcl}
&&&& \RP{2} \mathrm{\ homog} &&
             \CP{1}  (z_{0},z_{1}) &&
            \mathbb{C}  \mathrm{\ inhomog\ } (z_{1}/z_{0})\\
p_{1} & = & A(0) & = & [1,0,1] & \approx &  z = [1,0]   & \approx &   x=0 \\
p_{2} & = & B(0) & = & [1,1,0] & \approx  & z = [1,1]  & \approx &   x=1 \\
p_{3} & = & C(0) & = & [0,1,-1] &\approx  & z = [0,1]& \approx &  x=\pm \infty \ .
\end{array}
\]

But there is a small problem: if we follow the coordinate
interpolations carefully, they only work projectively; the actual
interpolations close on one another only if we include the negative,
projectively-equivalent points $\bar{p}_{i} = - p_{i}$, for a total
of 6 points and six linear paths, rather than three.  Thus, the
projective coordinates for the three components  $A,\,B,\,C$
can be plotted either as a connected hexagon in $\mathbb{R}^{3}$, the
embedding space of the homogeneous $\RP{2}$ coordinates, or as a more
visually consistent projection onto a constant-radius $\Sphere{2}$,
the double-cover of $\RP{2}$, as shown in Figure \ref{b4-double-rp2.fig}(a).

\begin{figure}[!phtb]
\centerline{\psfig{width=6.5cm, figure=figures/b4-hexagon-s2.eps}}
\centerline{ (a) }
\centerline{   \psfig{width=6.5cm, figure=figures/b4-double-rp2.eps}
\psfig{width=7.0cm, figure=figures/b4-double-whitneyrp2.eps}}
\centerline{\hspace{1.in}  (b)\hfill  (c)\hspace{1.in}}
  \caption[]{(a) The double cover of $B_4$ branched covering, showing
the linear interpolations among the points
$p_{1} = (1,0,1), \bar{p}_{1} = (-1,0,-1),
p_{2}  =  (1,1,0), \bar{p}_{2}  =  (-1,-1,0),
p_{3}  =  (0,1,-1), \bar{p}_{3}  =  (0,-1,1)$
 and their projections to $\Sphere{2}$.  (b) The Veronese map, projected
 to the Steiner Roman Surface, $(w_1,w_2,w_3)$. (c) Projection onto the
crosscap, $(w_1,w_2,w_6)$.}
  \label{b4-double-rp2.fig}
\end{figure}

To actually achieve the desired end result of a visualization of
the $B_{4}$ cross-ratio coordinates in a logical embedding, we must
find a quadratic map that removes the distinction between
the positive and negative versions of the same projective coordinates
and maps $\Sphere{2}$ explicitly to $\RP{2}$.  This is achieved
classically by the Veronese surface (see, e.g., the traditional
embedding of $\RP{2}$ given in the appendix
of Hilbert and Cohn-Vossen, \cite{HilbertCV}):
\begin{equation}
\begin{array}{ccc}
w_1  = \sqrt{2} x_{0} x_{1} &
w_2  = \sqrt{2} x_{0} x_{2} &
w_3  = \sqrt{2} x_{1} x_{2} \\
w_4  = ( x_{0})^{2} &
w_5  = ( x_{1})^{2} &
w_6  = ( x_{2})^{2}
\end{array}
\label{b4vero.eq}
\end{equation}
where the spherical constraint $x_{0}^{\ 2} + x_{1}^{\ 2} + x_{2}^{\ 2} =1$
implies the standard Veronese surface constraint  $\sum (w_{i})^{2} =
\left(\sum(x_{k})^{2}\right)^{2} = 1$.
In  Figures \ref{b4-double-rp2.fig}(b) and (c), we see the exact
paths of the three component integrals of the Euler Beta function
as they are embedded in alternate projections of $\RP{2}$ (the
real part of $\CP{2}$) to 3D.
This is equivalent mathematically, and yet a significantly
contrasting viewpoint, to the conventional $\CP{1}$ alternative indicated in
Figure \ref{rsphere1.fig}.

\subsection{Visualizing the ${\mathbf B_4}$ Pochhammer Contour}

  We now illustrate explicitly the geometry of the Pochhammer contour for the
Euler Beta function.  Starting from (\ref{b4integrand.eq}), we choose
a pair of small relatively prime rational exponents $(\alpha_1,\,\alpha_2)$, and project
the 4D plot of $w=\beta(z;\alpha_1,\alpha_2)$ to 3D, with the horizontal plane
parameterized by $x={\rm Re}(z)$, $y={\rm Im}(z)$, and the vertical axis given
by ${\rm Re}(w)$.  Figure \ref{pochviews.fig}(a) shows a small region
of the branched Riemann cover of the complex plane punctured at $z=0$
and $z=1$,  and   Figure \ref{pochviews.fig}(b) shows the
precise path in this branched cover of the Pochhammer ``commutator''
contour sketched in Figure \ref{pochsketch.fig}, but now as an actual embedding in
$\mathbb{C}^{2}$ (technically $\mathbb{R}^{4}$ projected to $\mathbb{R}^{3}$).
Figure \ref{pochviews.fig}(c) combines the two views to show the
Pochhammer contour in its geometric context on the Riemann surface.

\begin{figure}[!htbp]
{\centering \vspace*{-1in}
  \mbox{\psfig{width=7.0cm, figure=figures/poch4mesh.eps}
\psfig{width=7.0cm, figure=figures/poch-v1.eps}}\\
 \mbox{(a) \hspace{3.0in} (b)}  \\
\mbox{\psfig{width=9.0cm, figure=figures/poch4-merge.eps}}\\
\hfill (c) \hfill }
  \caption[]{(a) A representative Riemann surface derived from the
integrand of the Euler Beta function with suitable rational values of
$(\alpha_{1},\,\alpha_{2})$.  (b) The actual geometry of the Pochhammer contour traced on
the representative Riemann surface.  The path on the Riemann surface
is a ``commutator,'' encircling each branch point once in each
direction; comparing to Figure \ref{pochsketch.fig}, the four end
loops shrink to points as $r \rightarrow 0^{+}$. (c) Combined plot. }
  \label{pochviews.fig}
\end{figure}

\subsection{${\mathbf B_5}$: Components of the 5-Point Cross-Ratios}

The four single lines joined by infinitesimal loops
shown in Figures \ref{pochsketch.fig} and
\ref{pochviews.fig} represent the four distinct {\it phases\/}
of the $B_{4}$ Pochhammer integration path.
For $B_{5}$, the analog of one of these lines is
a pentagonal surface, and the set of four lines representing integration domain
of $B_{4}$ with distinct phases is replaced in $B_{5}$ by 32
pentagons with distinct phases.  Just as the three lines in
Figure \ref{rsphere1.fig} or Figure
 \ref{b4-double-rp2.fig} describe the three {\it components\/}
of $B_{4}$ that followed from solving the cross-ratio constraint, the
twelve components of $B_{5}$ can be studied using parametric solutions
of its own cross-ratio constraint system:  re-indexing for convenience using
 $(z_{1},z_{2},z_{3},z_{4},x_{5})\equiv
(u_{12},u_{13},u_{23},u_{24},u_{34})$,
the $B_{5}$ cross-ratio system becomes:
\begin{eqnarray}
z_{1} &=& 1 - z_{3}z_{4} \nonumber \\
z_{2} &=& 1 - z_{4}z_{5} \nonumber\\
z_{3} &=& 1 - z_{5}z_{1} \label{b5constrnts.eq} \\
z_{4} &=& 1 - z_{1}z_{2} \nonumber\\
z_{5} &=& 1 - z_{2}z_{3}  \nonumber \ .
\end{eqnarray}

  Any pentagon can be represented algebraically by picking
two of the five $z_{i}$'s as independent, and plotting any of the
dependent variables found by solving the constraints in formula
(\ref{b5constrnts.eq}) on the third axis.  The typical result, shown in Figure
\ref{pentplotmesh.fig}, is an algebraic 2-manifold embedded in
$\mathbb{R}^{5}$ showing the integration
region over the variety given by (\ref{b5constrnts.eq}).
 Projected from a horizontal direction, the pentagon of Figure
\ref{pentplotmesh.fig}(a) becomes a square region, whereas when
projected from the vertical direction, it becomes a triangular region,
 \begin{figure}[!ht]
\centering \mbox{\psfig{width=2.5in, figure=figures/pentplot1.eps}}
\mbox{\psfig{width=2.5in, figure=figures/pentplot4.eps}}\\
\centerline{\hspace{1.5in}(a) \hfill (b) \hspace{1.5in}}
\centerline{\psfig{width=3.5in, figure=figures/pentplot3.eps}}
\vspace{0.1in}
\centerline{ (c) }
  \caption[]{(a) Plotting the $B_5$ variables $(x,(1-x)/(1-xy),y)$,
showing how the ``blown-up'' pentagonal manifold arises naturally
in the cross-ratio manifold. (b) Completely regular version of the
pentagonal, normalized to $\Sphere{5}$ and projected,
corresponding to the variables of  Figure \ref{uvmap.fig}.
(c) Side view of (b).}
  \label{pentplotmesh.fig}
  \end{figure}
corresponding to formula (\ref{triB5.eq}).

To create an image of the twelve $B_{5}$ components, we now use the
constraints (\ref{b5constrnts.eq}) and proceed through the same
arguments that we used for $B_{4}$: We solve the constraints in
a family of homogeneous $\RP{5}$ coordinates based on choosing
non-singular parameterizations of
$(z_{0},z_{1},z_{2},z_{3},z_{4},z_{5})$, and find that the natural
connectivity actually gives us initially the 24-pentagon double cover
analogous to the six $B_{4}$ curves shown in Figure
\ref{b4-double-rp2.fig}(a).  Figure \ref{b5surfconnect.fig} is the
schematic analog of Figure \ref{rsphere1.fig} for $B_{4}$, showing
 the topological diagram of the
surface, which we can verify is non-orientable with 15 vertices, 30
edges, and 12 pentagonal faces, giving the advertised Euler
characteristic $\chi = -3$,  a sphere with five crosscaps.  However,
we can also see traced on this surface the family of complex lines
that form the symmetrized base of the branched cover enabled by the
blow-ups: there are 10 separate interlocking triangles, each denoting
the (circular) real
line of a $\CP{1}$ corresponding precisely to the $B_4$ diagram of
Figure \ref{rsphere1.fig} or  \ref{b4-double-rp2.fig}(b);
treating these schematically as  filled-in
triangles, we get the image in Figure \ref{pentTris.fig}, where the
boundaries of the 10 triangles taken five at a time bound the 12
pentagons.  The corresponding 12-pentagon figure can be thought
of as shown in Figure \ref{b5surfconnect.fig}, where the boundaries of the connected
components (corresponding to $-\alpha_{1}-\alpha_{2}$ for $B_{4}$)
are linear combinations of the $B_{5}$ $\alpha_{i}$'s.
In particular, the face labeled ``12'' corresponds to a $B_5$ function with
a set of exponents that is distinct from the values
$(\alpha_1,\alpha_2,\alpha_3,\alpha_4,\alpha_5)$  used in
(\ref{triB5.eq}), although they are closely related.
One can show with suitable variable changes that the exponents
corresponding to the primed branch lines (analogous to the exponent at
infinity for $B_{4}$) are
\begin {eqnarray}
{\alpha'}_1 & = & 1 + \alpha_{1} - \alpha_{2} - \alpha_{5} \nonumber \\
{\alpha'}_2 & = & 1 + \alpha_{2} - \alpha_{3} - \alpha_{1}  \nonumber \\
{\alpha'}_3 & = & 1 + \alpha_{3} - \alpha_{4} - \alpha_{2} \label {b5expon.eq}\\
{\alpha'}_4 & = & 1 + \alpha_{4} - \alpha_{5} - \alpha_{3}  \nonumber \\
{\alpha'}_5 & = & 1 + \alpha_{5} - \alpha_{1} - \alpha_{4}  \nonumber \ .
\end{eqnarray}

\begin {figure}[tp!]
\centering
  \mbox{\psfig{width=4.5in, figure=figures/5crosscapbar.eps}}
\caption [] {The diagram of how the 12 pentagonal pieces of surface
  join together to form a closed surface in $\RP{5}$.  Circles mark
  the 15 vertices, and squares mark the images of the corresponding
  regions in Figure \ref{12st.fig}.  Triangles label the exponent
of each of the branch lines delineating the connected components.
(See (\ref{b5expon.eq}).)}
\label {b5surfconnect.fig}
\end {figure}

\begin {figure}[tb]
\parbox[t]{5.00in}{\centering
  \psfig{file=figures/pentTris.eps,height=4.00in}
\caption[]{These 10 filled triangles represent the single
cover of the 10 $\CP{1}$
branch surfaces of the $B_{5}$ integrand Riemann manifold,
 The straight edges taken 5 at a time bound the 12 pentagons.}
\label{pentTris.fig}}
\end{figure}

\begin {figure}[tbp]
\parbox[t]{5.00in}{\centering
  \psfig{file=figures/10hexagons.eps,height=4.00in}
\caption[]{An explicit graph of the double covering of the
cross-ratio variable space, showing each vertex
and each of the 10 hexagons corresponding individually
to a double-covered $\CP{1}$ branch line, and also to
the hexagon in Figure \ref{b4-double-rp2.fig}. It is easy to
see how Figure \ref{pentTris.fig} emerges as the single-cover after
reducing each hexagon to a triangle.}
\label{pentHex.fig}}
\end{figure}

The solutions of the constraints are continuous only in the double
cover, so we will work first in an unnormalized $\RP{5}$ to produce
the analogs of the six end points and six straight interpolating edges that we
showed in Figure  \ref{b4-double-rp2.fig}(a) for $B_{4}$.
This equivalent set of vertices is the set of 10 hexagons representing
the double cover of the $\CP{1}$ branch lines denoted by
$(\alpha_{1},\ldots,{\alpha'}_{1},\ldots)$, as shown in Figure \ref{pentHex.fig}.
These  have the following
vertex assignments in the double cover:
\[
\begin{array}{rlrl}
\mbox{\rm line\ } \alpha_{1}: & (1, 5, 10,-1,-5,-10) &
\mbox{\rm line\ }  {\alpha'}_{1}: &(10,15,11,-10,-15,-11)\\
\mbox{\rm line\ } \alpha_{2}: & (2,1,6,-2,-1,-6)  &
\mbox{\rm line\ }  {\alpha'}_{2}: & (6,13,14,-6,-13,-14)\\
\mbox{\rm line\ } \alpha_{3}: &  (3,2,7,-3,-2,-7)&
\mbox{\rm line\ }  {\alpha'}_{3}: & (7,11,12,-7,-11,-12)\\
\mbox{\rm line\ } \alpha_{4}: & (4,3,8,-4,-3,-8)  &
\mbox{\rm line\ }  {\alpha'}_{4}: &  (8,14,15,-8,-14,-15)\\
\mbox{\rm line\ } \alpha_{5}: &  (5,4,9,-5,-4,-9)&
\mbox{\rm line\ }  {\alpha'}_{5}: &(9,12,13,-9,-12,-13) \ .
\end{array}
\]
Each region is bounded by a linear interpolation connecting the
(doubled) vertex set, as noted earlier.

 Figure \ref{b5cp1doubles.fig} shows the actual doubled geometry,
both as straight lines in $\mathbb{R}^{6}$, and as
curves in the sphere $\Sphere{5}$ (projected to 3D).

\begin {figure}[tp]
\centering
\psfig{width=2.5in, figure=figures/b5triindexDC.eps}
\psfig{width=2.5in, figure=figures/b5triS1symDC.eps}\\
\hspace{1.25in} (a) \hfill (b) \hspace{1.25in}
\caption [] {(a) Projection from homogeneous $\mathbb{R}^{6}$
coordinates of all the  $2\times 30 = 60$
straight edges giving edges of pentagons (or
real lines of the $\CP{1}$ branches) in the double cover, the analogs
of the six {\it end points\/} in Figure  \ref{b4-double-rp2.fig}(a).
(b) Normalization to $\Sphere{5}$, analogous to the same six
points on $\Sphere{2}$ in  \ref{b4-double-rp2.fig}(a).}
\label{b5cp1doubles.fig}
\end{figure}

\paragraph{Fully Symmetric Vertex Choice.}
  If all we were interested in was the descriptive topology of the
$B_5$ five-crosscap base manifold, any set of vertices with the proper
connectivity would be sufficient.  However, our dual purpose is to
understand not only the topology, but also any unique geometric
features or symmetries that might characterize this manifold, leading
to embeddings whose graphical depictions might be especially
informative.

  We have therefore pursued the search for special
embeddings one step further, and computed an orthonormalized set
of vertices along with the corresponding surface embedding that allows all
pentagons to be expressed as rigid transformations of one another
derived from the operations of the discrete symmetry operators of
Section \ref{sec:symmetries}.

 The next step is to use the matrix $P$ defined by (\ref{PtPeqQ.eq}).
Such a $P$ can be found from standard linear algebra methods; we will
not write $P$ explicitly because its entries are not rational numbers
and are very lengthy.

Let
\begin{eqnarray*}
  F^{\pm}(s,t)& =& P \cdot \tilde{f}^{\pm}(s,t) \\
     & = & \pm \frac{P\cdot f_{i}}{\sqrt{f^{t}_{i} \cdot Q \cdot f_{i}}}
\end{eqnarray*}
We then get the transformation $P\cdot \widetilde{M}$ of the surface
$\widetilde{M}$.  Notice that  $P\cdot \widetilde{M}$ is in
$\Sphere{5}$ and is invariant under
\[  \gamma = P \cdot g \cdot P^{-1} \]
for $g\in\widetilde{G}$;  the $\gamma$'s are now the orthogonal matrices
forming a subgroup of order 240 in $\OO{6}$.

We pick the following 24 elements from  this group:
\begin{scriptsize}
\begin{equation}
\begin{array}{cc}
\gamma^{\pm}_1 = \pm {\mbox{Identity}} &
 \gamma^{\pm}_2 = \pm P\cdot  \left[ \begin{array}{cccccc}
 1&0&0&-1&0&0 \\
 2&0&-1&-1&0&-1\\
 0&0&0&-1&0&0\\
 2&-1&0&-1&-1&0\\
  0&0&0&0&0&1\\
 0&1&0&0&0&0
  \end{array} \right] \cdot P^{-1}\\[0.50in]
\gamma^{\pm}_3 = \pm P\cdot   \left[ \begin{array}{cccccc}
  -1&0&0&1&0&1 \\
  0&0&0&0&0&1 \\
  2&-1&-1&0&-1&0 \\
  0&0&0&1&0&0 \\
  -2&0&1&1&0&1 \\
  -2&1&0&1&0&1
  \end{array} \right] \cdot P^{-1} &
 \gamma^{\pm}_4 = \pm P\cdot   \left[\begin{array}{cccccc}
1& 0& 0& 0& 0& -1 \\
 2& -1& 0& -1& 0& -1 \\
 0& 0& 1& 0& 0& 0 \\
 0& 0& 0& 1& 0& 0 \\
 2& 0& -1& 0& -1& -1 \\
 0& 0& 0& 0& 0& -1
 \end{array} \right] \cdot P^{-1}\\[0.50in]
\gamma^{\pm}_5 = \pm P\cdot    \left[\begin{array}{cccccc}
-1& 1& 0& 1& 0& 0 \\
 -2& 1& 0& 1& 0& 1 \\
 -2& 1& 0& 1& 1& 0 \\
 0& 0& 0& 1& 0& 0 \\
 2& 0& -1& 0& -1& -1 \\
 0& 1& 0& 0& 0& 0
 \end{array} \right] \cdot P^{-1}&
 \gamma^{\pm}_6 = \pm P\cdot   \left[\begin{array}{cccccc}
1& -1& 0& 0& 0& 0 \\
 0& -1& 0& 0& 0& 0 \\
 2& -1& -1& 0& -1& 0 \\
 0& 0& 0& 1& 0& 0 \\
 0& 0& 0& 0& 1& 0 \\
 2& -1& 0& -1& 0& -1
 \end{array} \right] \cdot P^{-1}\\[0.50in]
\gamma^{\pm}_7 = \pm P\cdot    \left[\begin{array}{cccccc}
-1& 0& 1& 0& 0& 1 \\
 -2& 0& 1& 1& 0& 1 \\
 0& 0& 1& 0& 0& 0 \\
 2& -1& 0& -1& -1& 0 \\
 0& 0& 0& 0& 0& 1 \\
 -2& 0& 1& 0& 1& 1
\end{array} \right] \cdot P^{-1}&
 \gamma^{\pm}_8 = \pm P\cdot   \left[\begin{array}{cccccc}
1& 0& 0& 0& -1& 0 \\
 0& 1& 0& 0& 0& 0 \\
 0& 0& 1& 0& 0& 0 \\
 2& -1& 0& -1& -1& 0 \\
 0& 0& 0& 0& -1& 0 \\
 2& 0& -1& 0& -1& -1
 \end{array} \right] \cdot P^{-1}\\[0.50in]
\gamma^{\pm}_{9} = \pm P\cdot   \left[ \begin{array}{cccccc}
-1& 1& 0& 0& 1& 0 \\
 0& 0& 0& 0& 1& 0 \\
 -2& 1& 0& 1& 1& 0 \\
 -2& 1& 1& 0& 1& 0 \\
 0& 1& 0& 0& 0& 0 \\
 2& 0& -1& -1& 0& -1
\end{array} \right] \cdot P^{-1}&
 \gamma^{\pm}_{10} = \pm P\cdot   \left[\begin{array}{cccccc}
1& 0& -1& 0& 0& 0 \\
 0& 0& 0& 0& 0& 1 \\
 2& -1& -1& 0& -1& 0 \\
 0& 0& -1& 0& 0& 0 \\
 2& 0& -1& -1& 0& -1 \\
 0& 0& 0& 0& 1& 0
 \end{array} \right] \cdot P^{-1}\\[0.50in]
\gamma^{\pm}_{11} = \pm P\cdot   \left[\begin{array}{cccccc}
-1& 0& 1& 0& 1& 0 \\
 2& -1& 0& -1& 0& -1 \\
 0& 0& 1& 0& 0& 0 \\
 -2& 1& 1& 0& 1& 0 \\
 -2& 0& 1& 0& 1& 1 \\
 0& 0& 0& 0& 1& 0
\end{array} \right] \cdot P^{-1}&
 \gamma^{\pm}_{12} = \pm P\cdot   \left[\begin{array}{cccccc}
3& -1& -1& -1& -1& -1 \\
 2& 0& -1& 0& -1& -1 \\
 2& -1& -1& 0& -1& 0 \\
 2& -1& 0& -1& -1& 0 \\
 2& -1& 0& -1& 0& -1 \\
 2& 0& -1& -1& 0& -1
 \end{array} \right] \cdot P^{-1}
\end{array}
\label{12xformmats.eq}
\end{equation}
\end{scriptsize}
Then, the entire 5-crosscap surface or its genus-5 double cover can be
constructed piece by piece starting from a single pentagon $F_{1}$ and
then transforming by $\gamma^{\pm}_{i}$.

In Figure \ref{b5rp5flat.fig}, we plot a pair of projections of the 24
surface patches $\gamma^{\pm}_{i} \cdot F_{1}(s,t)$ from $\Sphere{5}$
in $\mathbb{R}^{6}$ to $\mathbb{R}^{3}$. These are global solutions of the
5-point cross-ratio constraints with diametrically opposite copies of
each of the 12 pentagons forming the genus 5 double cover.

\begin {figure}[htb!]
\centering
\mbox{\psfig{width=2.5in, figure=figures/b5-DCpents1.eps}} \hspace{0.1in}
\mbox{\psfig{width=2.5in, figure=figures/b5-DCpents236.eps}}
\caption [] {Projections from $\RP{5}$ to $\Sphere{5}$ of the double cover
of the 12 blown-up pentagons forming a 5-crosscap dodecahedron. The
first projection emphasizes the smooth nature of the overall surface,
while the second projection emphasizes the pentagonal structure.}
\label {b5rp5flat.fig}
\end {figure}

\paragraph{$\mathbf B_{5}$ Veronese Map.}
These vertices and the polygonal faces of the dodecahedron inscribed
on the $B_{5}$'s ``real'' integration manifold, a sphere with five
crosscaps, can be compactly embedded for visualization purposes using
a quadratic form that is a  straightforward generalization of
the Veronese surface parameterization.  Given the homogeneous $\RP{5}$
variables $r = (r_0,\,r_1,\,r_2,\,r_3,\,r_4,\,r_5)$
above, we can construct an $\mathbb{R}^{21}$ embedding as
\begin{eqnarray*}
\begin{array}{ccccccccccc}
w_{1} & = & c r_0 r_1 &&
w_{2} & = & c r_0 r_2&&
w_{3} & = & c r_0 r_3 \\
w_{4} & = & c r_0 r_4 &&
w_{5} & = & c r_0 r_5 &&
w_{6} & = & c r_1 r_2 \\
w_{7} & = & c r_1 r_3 &&
w_{8} & = & c r_1 r_4 &&
w_{9} & = & c  r_1 r_5 \\
w_{10} & = & c r_2 r_3 &&
w_{11} & = & c r_2 r_4 &&
w_{12} & = & c r_2 r_5 \\
w_{13} & = & c r_3 r_4 &&
w_{14} & = & c r_3 r_5&&
w_{15} & = & c r_4 r_5 \\
w_{16} & = & (r_{0})^{2} &&
w_{17} & = & (r_{1})^{2} &&
w_{18} & = & (r_{2})^{2} \\
w_{19} & = & (r_{3})^{2} &&
w_{20} & = & (r_{4})^{2} &&
w_{21} & = & (r_{5})^{2}  \ .
\end{array}
\end{eqnarray*}
This map is constructed to lie on the sphere $ \sum_{i} (w_{i})^{2} = 1$
when $c= \sqrt{2}$ and the homogeneous coordinates are normalized
to obey $\sum_{k} (r_{k})^{2}=1$.  Note that the analog of the Steiner
Roman Surface immersion projecting $\RP{2}$  into $\mathbb{R}^{3}$ is achieved
by selecting the variables $(w_{1},\,\ldots,\,w_{15})$ mapping
 $\RP{5}$  into $\mathbb{R}^{15}$.

An alternative, but less symmetric,  $\mathbb{R}^{18}$ embedding is
\begin{eqnarray*}
\begin{array}{ccccccccccc}
w_{1} & = & 2 r_0 r_1 &&
w_{2} & = & \sqrt{2} r_0 r_2&&
w_{3} & = & \sqrt{2} r_0 r_3 \\
w_{4} & = & \sqrt{2} r_0 r_4 &&
w_{5} & = & \sqrt{2} r_0 r_5 &&
w_{6} & = & \sqrt{2} r_1 r_2 \\
w_{7} & = & \sqrt{2} r_1 r_3 &&
w_{8} & = & \sqrt{2} r_1 r_4 &&
w_{9} & = & \sqrt{2}  r_1 r_5 \\
w_{10} & = & 2 r_2 r_3 &&
w_{11} & = & \sqrt{2} r_2 r_4 &&
w_{12} & = & \sqrt{2} r_2 r_5 \\
w_{13} & = & \sqrt{2} r_3 r_4 &&
w_{14} & = & \sqrt{2} r_3 r_5&&
w_{15} & = & 2 r_4 r_5 \\
w_{16} & = & (r_{0})^{2} - (r_{1})^{2} &&
w_{17} & = & (r_{2})^{2} - (r_{3})^{2} &&
w_{18} & = & (r_{4})^{2} - (r_{5})^{2} \ ,
\end{array}
\end{eqnarray*}
which also lies on the sphere $ \sum_{i} (w_{i})^{2} = 1$
when $\sum_{k} (r_{k})^{2}=1$.

Projections of the (non-double-covered) 5-crosscap surface can at last
be drawn using  these quadratic maps, and typical results are
shown in Figure \ref{b5vero21.fig}.

\begin {figure}[htbp!]
\centerline{\psfig{width=3.25in, figure=figures/b5-vero21-125.eps}}
\centerline{\psfig{width=3.25in,figure=figures/b5-NDVero21-width2.eps}}
\caption [] {Projections of the 5-crosscap surface embedded in
  $\mathbb{R}^{21}$ using  the quadratic map.
Above: the (1,2,5) projection color coded by pentagon.
Below: the (1,2,16) projection with shaded surface and grid.
These are roughly the analogs of the projections of the circle
embedded in the projective plane in Figure \ref{b4-double-rp2.fig}.}
\label {b5vero21.fig}
\end {figure}

\subsection{The $\mathbf B_5$ Pochhammer Contour}
 \label{sec:VisPoch}

Within the domain of a {\it single pentagon\/},
we can now finally begin
to piece together a picture of the global topology of the $B_5$
Pochhammer contour.  This manifold can be drawn explicitly in various
ways by joining together the sets of commutators that eventually
return to the same phase, forming the closed surface; Figure
\ref{B5CommAB.fig} illustrates a single commutator element.    Figure
\ref{b5pochform3.fig} shows the schematic diagram of the full set of
commutators as they return cyclically to the home phase; this diagram
can be unfolded in various ways to show the overall structure, as
illustrated in Figures \ref{b5pochform1.fig} and \ref{b5pochform2.fig}.

\begin {figure}[htp]
\centering
  \mbox{\psfig{width=8.5cm, figure=figures/b5Commutator-w-lines.eps}}
\caption [] {The $B_{5}$ ``commutator'' $aba^{-1}b^{-1}$ bounding
a patch that covers one-fifth of each of four different phases,
which for the example branch lines with exponents $\alpha_{1}$ and
 $\alpha_{2}$, are labeled as $\{0,\,1,\,12,\,2\}$.}
\label {B5CommAB.fig}
\end {figure}

\begin {figure}[p]
\centering
  \mbox{\psfig{width=3in, figure=figures/b5pochcommdiag.eps}}
\caption [] {The commutator diagram of the $B_5$ Pochhammer surface. }
\label {b5pochform3.fig}
\end {figure}

\begin {figure}[p]
\centering

 \mbox{\psfig{width=3.5in, figure=figures/b5poch5ccstar.eps}}
\caption [] {The analog of the Pochhammer surface for $B_5$
is a 32-fold cover of a single pentagon, joining each of the
$2^{5}$ possible combinations of the 5 phases
$\{\alpha_{1},\ldots,\alpha_{5}\}$.
This surface can be constructed from (32/4)*5=40 individual
commutator patches.}
\label {b5pochform1.fig}
\end {figure}
\begin {figure}[p]
\centering

 \mbox{\psfig{width=2.5in, figure=figures/b5pochcommface.eps}}\\
\caption [] {A global picture of the $B_5$ Pochhammer surface
as an unrolled, thickened torus with four punctures,
showing more clearly the origin of its genus 5 structure.}
\label {b5pochform2.fig}
\end {figure}

\begin {figure}[p]
\centering
\centerline{
  \mbox{\psfig{width=3in,height=4in, figure=figures/5commut.eps}}
  \mbox{\psfig{width=3in,height=4in, figure=figures/5commutFill.eps}}
}
\centerline{\hspace{0.75in} (a) \hfill (b) \hspace{1.5in}}
\caption [] {(a)
The block of four commutator strips
surrounding one of the five corners of a single $B_5$ Pochhammer
pentagon.
 (b) Filling in the remainder of the surface at one
corner of the pentagon, giving  four ``handkerchief'' surface areas
filling the corresponding branched cover threading its
way around the branch points in the Riemann surface.
}
\label {multikerchief.fig}
\end {figure}

Finally, the explicit algebraic form of the Pochhammer can be
embedded directly in the Riemann manifold of
$\beta_{5}(x,y;\alpha_{1},\ldots,\alpha_{5})$, following the fashion
of Figure  \ref{pochviews.fig} to yield the surfaces shown in
Figure \ref{multikerchief.fig}.  This image shows one-fifth
of the Pochhammer contour covering a set of eight of the  32
total surfaces; sewing together all the corresponding copies
yields the entire surface.

\section{Remarks on the General Case }\label{sec:BNfcn}

 The affine variety defined by the $N$-point cross-ratio constraints
(\ref{uijconstr.eq}) is of dimension $(N-3)$ and has a natural
decomposition into $(N-1)!/2$ smooth components delineated by the
varieties $u_{ij}=0$.  The $N$-point function $B_{N}$ is initially
defined as an integral of an $(N-3)$-form over a {\it single one\/}
of these components.  Each of the $(N-1)!/2$ components is an {\it
$N(N-3)/2$-polytope\/} --- its $(i,j)$-th face is on the projective
{\it hyperplane\/} given by $u_{ij}=0$; these are not in general
regular polytopes, but reflect the existence of various  poles that
correspond to multiparticle combinations in elementary spinless
string theory.  Table \ref{components.tbl} summarizes for low $N$
the number of cross-ratio variables appearing in the standard
constraints, which is also the number of faces of the polytope
defining a single component, along with the total number of
components.  These polytopes have an exact and previously
unsuspected correspondence with the Stasheff associahedra
\cite{StasheffHSpaces,StasheffTAMS1963,Stasheff2004}, in all dimensions.  Each of the
$B_{6}$ components, for example, is a nonahedron, as pictured in
Figure \ref{nonahedron.fig};  this structure is described in detail
by Devadoss \cite{Devadoss1999}, who also gives, for example, a
tessellation of the moduli space $\overline{\mathcal{M}}^{6}_{0}(\mathbb{R})$
tiled by 60 nonahedral associahedra.  Our work seems to indicate
that the moduli spaces $\overline{\mathcal{M}}^{N}_{0}(\mathbb{R})$ studied
by Devadoss can also be viewed as the space of $N$-point
cross-ratios with the tessellations we have described in this paper.

\begin{table}[t!]\centerline{
\begin{tabular}{|r|c|c|} \hline
 \raisebox{-1.5ex}{\rule{0in}{4ex}}$N$ & $\frac{N(N-3)}{2}\, = \,\RP{}$ dim. & %
     $\frac{(N-1)!}{2}$ = no.\  components \\ \hline
 4 &  2 &  3 \\
 5 &  5 &  12\\
 6 &  9 &  60\\
 7 &  14 &  360\\
8 &  20 &  2520\\
 9 &  27 &  20160\\
 10 &  35 &  181440\\
\hline
\end{tabular}}
\caption[]{The first column is the value of $N$ for a given function
  $B_{N}$.  The second column gives the dimension of the projective
  space that implements the blow-up in cross-ratio coordinates; this
  is the same as the number of faces of the polytope defined by the
  natural $u_{ij}=0$ boundaries of the integration region for one
  component  using the cross-ratio coordinates; these are not
  necessarily regular polytopes. The third column is the total number
  of components, i.e., the number of polytopes that fit together to
  give the analogs of the 5-crosscap dodecahedron for $B_{5}$.\\
  \rule{5.375in}{.01in} }
\label{components.tbl}
\end{table}

\begin {figure}[ht]
\centering
\centerline{
  \mbox{\psfig{width=3in, figure=figures/nonahedron.eps}}
}
\caption [] {This nonahedron is the elementary connected component of
the 6-point cross-ratios forming the basis for the analysis of
$B_{6}$; just as 12 pentagons tessellate the 5-crosscap surface,
60 of these nonahedra tessellate the analogous 3-manifold.
}
\label {nonahedron.fig}
\end {figure}

We conjecture that the real integral form of $B_{N}$ can
always be expressed alternatively by a Pochhammer-like contour
integral in the corresponding smooth complex algebraic variety $M^{c}$ of
dimension $(N-3)$ in $\CP{N(N-3)/2}$. The contour is a real
$(N-3)$-dimensional submanifold and is obtained by wrapping copies of
the $N(N-3)/2$-polytope integral domain properly around the branch
hyperplanes where its faces are located.  Notice that it is fairly
easy to see, by the description above and (\ref{BNintegral.eq}), that
the branch hyperplane at each face -- say, face $(i,j)$ -- is of
complex codimension 1 in $M^{c}$ and, when folded around it, the {\it
phase\/} of the integrand in the lift to the Riemann covering sheaf
changes by $\pm \alpha_{ij}$.  By this mechanism, (\ref{B5ratio.eq})
should generalize in an obvious way.

\section*{Acknowledgments}
This research was supported in part by NSF grant numbers CCR-0204112
and IIS-0430730.  AJH is grateful to Tullio Regge for his early
encouragement and interest in this problem.   Special thanks are due
to Charles Livingston, Philip Chi-Wing Fu, and Sidharth Thakur for
advice, insights, and assistance with graphics tools.  We also thank
James Stasheff for introducing us to the literature on associahedra.

\begin{small}
\bibliographystyle{plain}

redpajama_set_name: RedPajamaArXiv


Includes English translation and discussion of Book of tkhines for a pregnant woman, a 17th century Yiddish and Hebrew manuscript in the Bodleian library.
Commentary. Introducing tkhines -- Profit and prayer: women and a unique pan-European printing industry -- The Seyder tkhines: a book for a messianic age -- Seyder tkhines: definitions -- Side by side by seyder -- A rare manuscript lost and found: Book of tkhines for a pregnant woman -- Other Yiddish prayers and religious songs and their authors -- Seyder tkhines in the twenty-first century: the messianic inheritance -- Prayers in translation. The Seyder tkhines -- Daily tkhines -- Sabbath tkhines -- Nide tkhines -- Fast-day tkhines -- Four Graveside tkhines -- New Year tkhines and Day of Atonement tkhines -- Book of tkhines for a pregnant woman -- Single prayers and songs in tkhine language -- Sabbath prayer / Khane Katz -- How good (Ma toyvu) / Khane Katz -- Simkhes toyre song / Rivkke Tiktiner -- A tkhine to be said before kedushe / Beyle Hurvits -- A song of the Ten commandments / Sheyndele, wife of Gershon ben Shmuel -- Song (lid) / Yankev ben Elyohu -- Father, king / Toybe Pan -- Daily tkhine (an extract) / Rokhl, daughter of Mordkhe Soyfer.
Tehinnot -- Translations into English.
Judaism -- Prayers and devotions -- English.
Tehinnot -- History and criticism.
Jewish women -- Prayers and devotions -- History and criticism.
Book of tkhines for a pregnant woman.
The Seyder Tkhines , translated from its original Yiddish by noted tkhines scholar, Devra Kay, and centerpiece of this groundbreaking work, was a standard Yiddish prayer book for women. It first appeared in Amsterdam in 1648, and continued to be published for the next three generations, usually inside the Hebrew synagogue prayer book. A product of an age when mysticism pervaded mainstream Judaism, the Seyder Tkhines provided women with newly composed, alternative daily prayers that were more specific to their needs. Included in this volume is a unique Yiddish manuscript dating from the 17th century - a collection of prayers written specifically for a rich, pregnant woman, which Kay discovered among the rare books of the Bodleian Library in Oxford, England.
Now, for the first time, these prayers have been skillfully translated and brought to public view. In addition to her translations, Kay presents her own extensive commentary, providing a deeper understanding of the historic, religious, and cultural background of this period in Jewish history. This unparalleled book will have special appeal to those interested in the social, literary, and religious history of women, as well as the history of the Yiddish language and literature. The interest in these forgotten prayers and their significance to the lives of women has now been revived, and these tkhines are ready to be rediscovered by a modern readership.
According to Kay, ?Jewish women today who are seeking a precedent for women?s prayer need look no further? than the ?seyder thkines,? or Yiddish prayer books for women that were popular in the mid- to late 17th century. Now largely forgotten, these works of popular piety emerged in a time of remarkable messianic expectation and fervent holiness. Kay, who began researching these prayer books for her doctoral dissertation, writes that the prayer books flourished for approximately three generations and had no precedents in the history of Jewish prayer, since they were vernacular prayers written by and for women. Kay traces the development of the cottage industry of these prayer books, which proved to be financially profitable for printers and spiritually beneficial for women, who did not participate in many of the male-oriented rituals in the synagogue. Particularly interesting is a prayer book written specifically for an unnamed pregnant woman, who is encouraged to repent of her sins and be ever-ready for a possible departure from this world. One prayer explicitly beseeches God for an easy labor and a safe delivery. Kay does readers a great service by collecting these historic prayers into this volume and providing helpful commentary. (Sept.) Copyright 2004 Reed Business Information.
Kay's reclamation of this nearly lost tradition in Jewish writing and practice is a great accomplishment. The tkhines were prayers directed to, in the voices of, and most likely written by Jewish women, and the tradition seems to have begun as early as the 17th century. Kay's translations are flat and even awkward ("God Almighty You have punished we [sic] women" is a notable gaffe), but her brief commentaries, which discuss the significance of the tkhines in a newly literate and keenly Messianic moment in European Jewish culture, are intelligent and valuable. While much more remains to be said about these women's prayers, Kay's work is a fine introduction. Highly recommended. (c) Copyright 2010. Library Journals LLC, a wholly owned subsidiary of Media Source, Inc. No redistribution permitted.


US Supreme Court holds a warrant is normally required before ordering a blood draw | The Collins Law Firm, P.C.
On behalf of The Collins Law Firm, P.C. posted in Drunk Driving on Wednesday, April 24, 2013.
The Fourth Amendment of the U.S. Constitution provides that people should be free from unreasonable searches and seizures. The U.S. Supreme Court recently issued a decision regarding that Fourth Amendment protection.
The case before the high court - Missouri v. McNeely - involved a motorist who was pulled over by a police officer for speeding. When the law enforcement officer requested that he take a breathalyzer test, he refused. The officer then took the driver to a hospital, where he instructed that a blood draw take place. The driver did not consent to having his blood taken.
After a blood alcohol test, it was revealed that the driver's BAC was almost two times the legal limit. The Missouri Supreme Court refused to consider the evidence, however, because the police officer made no attempt to obtain a warrant before ordering the blood draw.
The U.S. Supreme Court, in an 8-1 decision, ruled that in most cases, a warrant is required before blood can be drawn in a suspected case of driving under the influence. Justice Sotomayor wrote that in many cases the police officer will have enough time to get a warrant, as email and cellphones help to expedite the process.
Although the justices noted that alcohol dissipates in the body over time, Justice Sotomayor stated that determining when there is not sufficient time to obtain a warrant will be on a "case-by-case" basis. In addition, officers who fail to obtain a warrant will have to explain why they did not do so in court.
When someone is facing charges of driving under the influence, obtaining counsel to establish a strong defense on his or her behalf is a wise step.
Source: NPR, "Supreme Court Backs Warrants For Blood Tests In DUI Cases," Nina Totenberg, April 17, 2013.


With independent art galleries, jewelry stores, clothing boutiques and gift stores lining Cannery Row, shoppers will find meaningful ideas for a variety of friends and family members. The pedestrian-friendly shopping district promises unique gifts for even the most hard-to-shop-for loved ones, along with plenty of fun and unexpected items along the way.
Head to Cannery Row for your retail fix, and make the most of the trip with this small-business shopping list.
• Steinbeck Jewelers Cannery Row's original jeweler dazzles with a breathtaking, eclectic collection of gems from around the world and upscale fashion jewelry.


text:
What is our Philosophy?
Learning a language helps children to develop a natural curiosity and confidence to explore, other countries, cultures and languages, accepting that, in a multi-lingual society it is a valuable skill to be able to communicate effectively with others in another language. It also deepens their understanding of the world and their place in it. By the time children leave The Winchcombe School, they will have a natural curiosity and confidence to explore, other countries, cultures and languages, accepting that, in a multi-lingual society it is a valuable skill to be able to communicate effectively with others in another language.
What is taught?
At The Winchcombe School, we teach French through the Primary French 'Niveau Bleu' Project. It introduces children to French culture through the use of songs, rhymes, significant landmarks and cultural traditions and celebrations, encouraging children to think about the similarities and differences between this and their own culture and why these exist. It also explores places in the UK and Europe, looking at where in the world France and the UK are and how they are linked.
How it's taught?
French is taught in key stage two once a week.
Long Term overview Mfl

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text:
Талія Мінт (;  26 листопада 1989, Київ) — українська порноакторка.

Вона почала свою кар'єру у 2015 році. Проживає в Іспанії.

Співпрацювала з Girlfriends Films, vivthomas.com, Harmony Films та nubiles.net.

Нагороди та номінації

Примітки

Посилання

Уродженці Києва
Бісексуальні жінки
Українські порноакторки

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text:
Nine Potential Studs You Forgot About for 2010
August 21, 2010 by Greg Brosh
When it comes to fantasy football snake drafts, you have one and only one true weapon to use against your opponents: the element of surprise. You know the guy going first is going to take Chris Johnson. You know the next guy will take Adrian Peterson. The next pick? Maurice Jones-Drew. By the time the first round is over, it's also very safe to assume that any or all of Drew Brees, DeAngelo Williams, Aaron Rodgers, Frank Gore, and Steven Jackson will be off the board. But what about after all the big names and studs are gone, and you are left with the task of filling out your roster with veterans who refuse to go away, or the young kids who aren't even guaranteed playing time. These, however, are the picks that count the most.
Going back to the theme of surprise, it is important to make at least a few bold picks in your draft. It helps liven up the predictable order of the fantasy football draft, and if you end up being right? Your friends will hate you and/or think you're a genius! So you can effectively use the element of surprise in your draft, here are nine guys you have almost surely forgotten about in the last year or so:
1. Clinton Portis, RB, Washington Redskins
After a nice 2008 in which he totaled 1,705 all-purpose yards with 9 scores, Portis seemed to fall off the face of the earth last year. In only eight games, Portis averaged an even 4.0 yards per carry on 124 carries; 494 yards total. He had just one touchdown, and his 61.8 yards per game was easily the worst of his career.
Still, Portis, who turns 29 in September, has some game left in him (or at least some stats). His end zone dry-spell was worrisome, but Portis wasn't the only one struggling. The entire Redskins team stumbled out of the gate to a 2-5 record heading into their Week Nine match-up with the Atlanta Falcons, which saw Portis limited to 4 carries as he sustained a season-ending concussion.
He may be a shell of the guy who had 1,500 or more rushing yards with 11 or more touchdowns in 3 of his first 4 seasons in the league, but he isn't done yet either. This year, the Redskins will have a new head coach in Mike Shanahan, who coached Portis in his first two years in the NFL with the Denver Broncos. Shanahan loves the ground game, and the numbers show it. From 1995 to 2008 in Denver, Shanahan's Broncos ran the ball 46.8% of the time (fourth most in the NFL), which led to Denver having 11 individual 1,000 yard rushers during that time span. Finally, it is also pertinent to know that during that same time span, Shanahan ran the ball 56.9% of the time while in the red zone, third highest in the NFL.
Even with quarterback Donovan McNabb figuring to be the focal point of the Redskins' offense and Shanahan being noncommittal on the topic of who will start in the backfield with Willie Parker and Larry Johnson in the mix, Portis has to be the feature back of this offense. He is the most talented of the bunch, and given Shanahan's tendencies plus an improved offensive line, he also looks primed to have a solid year. Given his current ADP (which has him going in the 8th round or later), Portis is a steal and should be considered a solid second or third option.
2. LaDainian Tomlinson, RB, New York Jets
As a rabid New England Patriots fan, Tomlinson is practically dead to me. As a running back whose workload has dipped from 315 carries in 2007 to 223 in 2009 and whose yards per carry during that same time has gone from 4.7 to 3.3, and his total yardage during that time has dropped off the cliff from 1,474 yards in 2007 to 730 yards last year; Tomlinson is probably dead to you too. Still, the lovable (even though he plays for that disgusting New York franchise) LT refuses to say die.
Most fantasy football players consider running backs to be over the hill once they reach 30, and prefer to tread cautiously when it comes to gambling on 30+ running backs being a large part of your fantasy offense. LT might have hit rock bottom last year in terms of yards per carry, or it might get worse. The fact is, we don't know.
However, that's not to say there isn't still upside for Tomlinson. Despite his horrid 3.3 yards per carry last year, LT still had 12 touchdowns on the ground, an increase from 2008. He might not be that superstar, workhorse running back that he was a few years ago, but guys who have such a great nose for the end zone don't just go away after one bad year. Remember 2006, when LT scored 31 touchdowns on his own? If anyone has the talent to come back from such a miserable season, it's LaDainian Tomlinson.
Tomlinson's new team, the New York Jets, ran the ball 607 times last year, and quite possibly have the best offensive line in the league. That said, they also lost the guy who had over half of their carries last year (331, with 1,400 yards and 14 scores) in Thomas Jones. Sure, Shonn Greene figures to get a big increase in carries, but he had 108 carries last year. He can't possibly take all of the carries that Jones had last year, and the departure of Leon Washington (who had 72 carries before breaking his leg) leaves even more carries up for grabs. It's probably safest to assume both Greene and Tomlinson to amass around 200 carries. Hell, given his ability to score, it's possible that Tomlinson could be the primary red zone back.
Per Tomlinson's current ADP, he is around the 40th running back taken off the board, well out of the top 100 players. But given the offensive line that will be blocking for him, the explosive offense that he finally won't be expected to carry that will be surrounding him, and the amount of carries he is sure to get, don't you think LT is worth a late round flier? Even at this point of his career, he has far more upside than the other running backs available at his position, which include Tim Hightower, Chester Taylor, and Tashard Choice.
3. John Carlson, TE, Seattle Seahawks
Don't sleep on John Carlson. Carlson may not be a fallen star like Portis or Tomlinson, but he does seem poised to have a big year for the Seahawks in 2010. Last season, despite poor quarterback play and terrible blocking from the offensive line that forced Carlson to block more often than he normally would, the tight end hauled in 7 touchdown passes on 574 yards. In the two years he has been with the team, Carlson has evolved into Seattle's most dependable pass catcher.
With blocking tight end Chris Baker signed in free agency and Russell Okung added in the draft, plus the tight end-heavy philosophy of Offensive Coordinator Jeremy Bates, Carlson figures to take on a huge role for the 2010 Seahawks. If he got 7 touchdowns last year while in a minimized role, imagine the damage he could do this year while seeing a ton more looks. The best part? His ADP currently sandwiches him in between Heath Miller and Dustin Keller, who have considerably less upside.
4. Cincinnati Bengals D/ST
Currently in fantasy football drafts, the Bengals average out as the 13th defensive unit taken off the board, behind the Giants, the Bears, and the Chargers, who all have had serious holes and/or inconsistent play from their defenses plague them at various times during the past few seasons.
The Bengals allowed the fourth least total yards per game last year at just over 301, behind only Baltimore, Green Bay, and New York (Jets). Their 18.2 points allowed per game was also good for 6th best in the league. They addressed their middle of the road pass-rush by adding the wildebeest otherwise known as Carlos Dunlap and the pass-rushing defensive tackle Geno Atkins in the draft. They also tallied a healthy 19 interceptions, ninth most in the league.
They might not be world-beaters, but Cincinnati has a very solid defensive unit, and at the very least can be used as a spot start to rack up points in their meetings with Cleveland (twice), Buffalo, and Tampa.
5. Laurence Maroney, RB, New England Patriots
Many New England fans may be quick to jump the gun on eulogizing Maroney's Patriots career, but the fact is that he can still be a valuable fantasy player. First, Maroney is still 25, which means he does have a little bit of time to figure things out and revitalize his game. Second, he may not be living up to his first-round expectations, but he is still, at the very least, a helpful contributor.
On only 194 carries last year (although it was a career high), Maroney made the most of what he got and turned it into 757 yards (3.90 yards/carry) and 9 touchdowns. Pretty good, huh? And the news gets better. As the Patriots look to achieve a better balance between the run the passing game, Maroney should see more carries come his way, and as they also move away from the spread, Maroney also has the potential to be used as a receiver out of the backfield.
Last year, Maroney had some issues with fumbling (coughing one up in the end zone in the game vs. Indy that the Pats eventually lost by one score particularly comes to mind) and was not always used on a regular basis, as he had 5 games in which he was given 10 carries or less and did not play in week 17. Confidence in him is low, which is why Maroney has yet to be handed the No. 1 running back job, but he figures to have a much bigger role in the 2010 Patriots nonetheless.
With his ADP sitting at 120, just behind LaDainian Tomlinson (and the 46th ranked running back), Maroney could provide great value with a late round pick. He appears to be a sure thing bye-week fill in, and could possibly be a second or third running back if he finally takes that next step.
6. Ricky Williams, RB, Miami Dolphins
The reasons to forget about Ricky are simple: (1) the Dolphins love to use Ronnie Brown despite having an infinitely better back in Williams, and (2) his workload is a roller coaster. When Ricky Williams gets 18 to 20, or even more carries, he is as big a stud as they come. That's when he will tally up huge yardage and score. A lot. The problem is, with a healthy Ronnie Brown, it will be hard for Ricky to really show his stuff.
Before Brown was injured in 2009 (Week 10), Ricky was given 20 carries once and 10 or more carries four times. When he had the starting job all to himself, Run Ricky Run was going wild over opposing defenses, ultimately finishing the year with 1,121 yards (4.7 yards/carry) and 13 total touchdowns.
Even though Williams is 33, those years he took off in the middle of his prime have certainly kept the wear-and-tear on his body at a minimum, which is why he can still be such a terror for opposing defenses. It's also important to consider that Ronnie Brown has played exactly one full season since entering the league, and isn't amazingly good when he is healthy.
Most fantasy experts agree that Ricky is a handcuff player for fantasy purposes, but his situation makes him the uber-handcuff. Even when Brown is playing, Ricky is capable of putting up big numbers (7 total touchdowns before Brown was lost for 2009 season), but if/when Brown goes down, Williams automatically becomes the feature back for Miami. If you own Ronnie Brown, you absolutely have to get Ricky. Even if you don't own Brown, Williams is still a good pick, as his ADP clumps him in around Brandon Jacobs and Ben Tate as the 32nd ranked running back.
7. Ahmad Bradshaw, RB, New York Giants
Remember 2008, when the New York Football Giants had both Brandon Jacobs and Derrick Ward rush for over 1,000 yards? The way this year's training camp and preseason has been going, it is becoming increasingly apparent that the Giants will have the same type of backfield timeshare with Jacobs and Ahmad Bradshaw for the 2010 season. It is certainly warranted—Jacobs rushed for only 835 and 5 touchdowns while playing in 15 games (most since 2006) and getting 224 rushing attempts (career high), and Bradshaw averaged just under 5 yards per carry as a backup, totaling 778 yards with 7 touchdowns.
With 21 catches for 207 yards, Bradshaw is also a receiving threat out of the backfield, which makes him a versatile tool for the Giants to move the chains with, and if given more opportunities this year, he should easily eclipse the 10 touchdown mark. With a solid offensive line blocking the way for him plus a good passing game to keep defenses honest, Bradshaw could have a very nice year. With his upside plus ADP (82, just ahead of Jerome Harrison, just behind Jacobs) taken into account, Bradshaw could wind up being one of the draft's biggest bargains. Consider him a very high quality backup or a solid flex play.
8. Alex Smith, QB, San Francisco 49ers
To this point, Smith, the former first overall pick, has been, to say the very least, a colossal bust. But in his defense, he has yet to work with the same Offensive Coordinator for more than one season, and for his whole career his surrounding cast mates have been mediocre, at best. After taking over in Week 7 for the 49ers last season, Smith finished with 2,350 passing yards, an 81.5 QB rating, and 18 touchdowns to 12 interceptions.
Despite having a career renaissance of sorts, Smith does have to worry about one thing, and that's getting too comfortable. His backups David Carr and Nate Davis are unlikely to provide much real competition for his job, which may not keep him at the top of his game.
Still, in 2010 Smith figures to have the best supporting cast of his career. The 49ers drafted blocking help this spring in the form of offensive linemen Mike Iupati and Anthony Davis, which should allow him more time in the pocket to find his targets. That segues us perfectly into the next point: the targets at his disposal.
After a bumpy transition into the NFL, Michael Crabtree hopes to make the jump into the elite class of NFL wide receivers, and Smith also has a solid receiver to target in Josh Morgan. Vernon Davis has established himself as one of the best tight ends in the league and figures to be a prominent safety valve for Smith, and Frank Gore was also used heavily as a pass-catcher last year.
Having more time to find good targets with some new-found familiarity in the coaching staff should lead Smith to the best season of his career this year. At the very least, considering his ADP around 140, Smith looks like a dirt cheap bye week or injury fill-in. Who knows, if he really surprises, he could become a regular starter.
9. Lance Moore, WR, New Orleans Saints
Moore suffered some serious injuries (torn labrum, dislocated shoulder) in 2009 that kept him from building on his breakout 2008 season, in which he caught 79 balls for 928 yards and 10 touchdowns. He only saw his way into a handful of games because of those injuries, but looked good against the Giants when he caught 6 passes for 78 yards and a touchdown.
The New Orleans Saints have a prolific passing offense, and in 2008, Moore became one of Drew Brees' favorite targets. For a New Orleans team that passed the ball 544 times last year, Brees will have plenty of love to spread around to his guys throughout the air, and Moore is a prime candidate to be one of the main beneficiaries of the Saints' pass-happy scheme.
His ADP of 178 makes him an absolute bargain given his immense upside as a sleeper, and given a full, healthy, season, Moore can easily crack the top 20 in fantasy wide receivers. Yet, because many have forgotten about Moore with his quiet 2009 and the abundance of productive New Orleans wide receivers, Moore is currently the 73rd wide receiver going off the board. Bottom line: Moore is a no-brainer late round pick.
Category - Fantasy Football News and Analysis - Nine Potential Studs You Forgot About for 2010