====================== Then we write down the equations of a variety which are computed from the pair by a three-step procedure: skeleton, unprojection to basic equations, more calculation to recover the full ideal. Most of this paper is devoted to explaining the origin of these pictures and working out how to interpret them and calculate with them. This variety will be the canonical cover of a flip. So we finish by adding a flipping $\C^*$ action. The construction is then complete. For the time being one can take the relationship between the pictures and the equations for granted, treating the data of the example as a whole, although anyone versed in Newton polygons, toric geometry or finite quotient singularities will see what we are doing. The name of the example, $X$, refers to the numerical classification of varieties in Section~\ref{sec!numeric}. The subscript is the codimension of the flip. \subsection*{$\II_9(3,2,3)$} \[ \begin{array}{ccccccc} 3 & \quad & 1 &\quad& 0 & & -1 \\ \cline{1-3} \cline{5-7} \begin{array}{c} 2 \\ 3 \\ 2 \\ 3 \end{array} && \begin{array}{c} 2 \\ 4 \\ 3 \\ \end{array} && \begin{array}{c} 2 \\ 3 \\ 2 \\ 3 \end{array} && \begin{array}{c} 2 \\ 4 \\ 3 \\ \end{array} \\ \cline{1-3} \cline{5-7} 0 && -2 && 2 & \quad & 1 \end{array} \] We imagine this picture to be of two vertical rectangles with a few marked points on their boundaries tagged by integers. The left-hand rectangle is called the {\em elephant} and the right-hand is the {\em section}. There are some hidden extra labels called {\em annotations}: $\al$ and $\be$ at the top left and top right corner of the elephant, $\la$ and $\mu$ at the bottom left and bottom right hand corner of the section. We give the annotation $0$ to all side points in the picture while the four remaining corners get annotations calculated from the tags shown together with $\al$, $\be$, $\la$, $\mu$ using a continued fraction routine. In this case, for instance, the top left corner annotation of the section is $19\la+18\mu$. See Section~\ref{lrpdefs} for details of these definitions and calculations. Together with all its labels, this pair of long rectangles is intended to represent equations in $13$ variables, $x_0,\dots,x_5$, $y_0,\dots,y_4$, $u$, $t$. The $x$ variables correspond to the marked points up the left-hand side of the rectangles (which between them are thus displaying two tags for each variable, only differing at the corners). The $y$ variables correspond to those on the two right-hand sides. For each $x$ or $y$ variable there is a {\em tag equation}. The tag equations at the corners are the most important. At $x_5$, the tag equation is \[ x_4y_4=x_5^3u^\al+t^{19\la+18\mu}. \] The $x_4y_4$ monomial is the product of the variables adjacent to $x_5$ in the rectangles. The $x_5^3u^\al$ monomial is read from the elephant: the power of $x_5$ appearing is the tag at $x_5$ while the power of $u$ appearing is the annotation at $x_5$. The final term in $t$ is calculated from the section: there is a factor $x_5^0$ corresponding to the tag, while the power of $t$ appearing is again the annotation on the section at $x_5$. You notice that this equation is similar to that of a terminal singularity before taking a finite quotient, and indeed it will be the equation of a singularity on $X^-$ before the flip. Basic equations for non-corner variables are harder to compute. For instance, at $x_1$ the equation is in fact \[ x_0x_2=x_1^3+u^{5\al+12\be}t^\mu. \] The final term cannot be read directly from the picture but must be calculated, one way or another, using syzygies. In this case, our method of Gorenstein projection and pentagrams makes the calculation tolerable. Eventually, at least in cases whose equations exhibit some kind of regularity, the result is the ideal of equations of a Gorenstein $4$-fold in a $13$-dimensional affine space, a codimension~9 variety as expected. We denote this $4$-fold by $\cA$. To make a flip we add a $\C^*$ action. We do this by taking the variables as eigencoordinates and giving them weights in such a way that all the equations are homogeneous with respect to these weights. It is enough to choose weights for just four variables since the homogeneity of the equations determines all other weights. We define the weight of $u$ to be $0$, that of $t$ to be $1$, $x_0$ to be $-a$ and $y_0$ to be $-b$ for positive integers $a$ and $b$. These are the choices of name that we will often make, but beware that $a$ and $b$ will be either determined by other variables already chosen, or else satisfy some divisibility conditions. The homogeneity of the basic equation at $x_0$, \[ x_1y_0=u^?+x_0^2t^\la \] implies that the weight of $x_1$ is $a$ since $u^?$ has weight $0$ whatever power it is. Notice too that $a$ is exactly $\la/2$: there are lots of relations between all the labels and here we have been careless in noting them. Similarly we find that all other weights are strictly positive. A flip arises by taking the different $\C^*$ quotients of $\cA$. The base of the flip is the categorical quotient while the $X^+$ in the flipping diagram is the relative $\Proj$ of the negatively weighted variables. As will always be the case, this has two patches coming from the two negative variables $x_0$ and $y_0$ which cover an exceptional $\PP(a,b)=\PP^1$ having $x_0$, $y_0$ as homogeneous coordinates. For instance, the $x_0$ patch is the hypersurface \[ x_1y_0=u^?+t^\la \] (coming from the basic equation above) divided by a finite cyclic stabiliser of order $a$. This singularity is clearly isolated so is also terminal, as in \cite{ypg} \S6. The main result of this paper is to associate a pair of long rectangles to any flip and classify those pair of long rectangless which admit a flipping $\C^*$ action. ====================== Our classification is not a list of all flips of type~$A$, rather a classification of certain numerical data that can occur together with a collection of methods for constructing flips of each numerical type. These constructions should correspond to deformation families of flips in a straightforward way, although particular calculations appear prohibitively complicated. The numerical data already suggests interesting structures within these flips which we begin to study. In particular, we find that codimension is not necessarily a good measure of complexity for flips. More likely is the way that they can decompose into families of distinct projection types. ====================== the coordinate ring $R_0=R[\C^6]=\linebreak[2] k[x_0,y_0,A,B,L,M]$ to $R_1=R_0[x_1]$. Thus $R_1$ is the ring \[ R_1=k[x_0,y_0,A,B,L,M,x_1]/(x_1y_0-A^3B^5-x_0L), \] that is, This is clearly normal. This is a trivial case of a Kustin--Miller unprojection. Setting $L=M=0$ reduces the dimension by 2, so that $L,M$ is a regular sequence. Moreover, the section $V_{1+1}\cap(L=M=0)$ is a normal toric variety, so that $V_{1+1}$ is also normal. \paragraph{Pentagrams and Pfaffians} Step~3 can be mechanised in terms of the {\em penta\-gram} The tag equations (\ref{eq!codim2}) at $x_0$ and $y_1$ that form the input to Step~3 are equations for $x_1y_0$ and $x_0y_1$. The output of Step~3 consists of three new equations (\ref{eq!x_2}) for $y_0x_2$, $y_1x_1$ and $x_2x_0$. All five equations are the $4\x4$ Pfaffians of the skewsymmetric $5\x5$ matrix \begin{equation} \begin{pmatrix} x_1 & -AB^2&-L&x_2 \\ & x_0 & -A^2B^3&-y_0^2M \\ && y_0 & -x_1\\ &&& y_1 \end{pmatrix}. \end{equation} (By convention, here and in what follows, we only write the upperdiagonal terms $a_{ij}$ with $1\le i