remove 2d polytope construction notes

README.md

@@ -229,33 +229,9 @@ There are exactly three IP weight systems that define two-dimensional polytopes
 |     (1, 1, 2) |                                5 |                                  9 |
 |     (1, 2, 3) |                                7 |                                  7 |
 
-We will now construct these polytopes from their corresponding weight system. Fixing the
-first two vertices of the polytopes
-
-$$
-  \mathbf{v}_0 = (1, 0) \quad \text{and} \quad
-  \mathbf{v}_1 = (0, 1) \;,
-$$
-
-one can obtain the position of the third vertex by solving the weight system equation from
-before:
-
-$$
-  \mathbf{v}_2 = - \frac{q_0 \mathbf{v}_0 + q_1 \mathbf{v}_1}{q_2} \;.
-$$
-
-The resulting polytopes and their duals are depicted below. Lattice points are indicated
-by dots.
+The polytopes and their duals are depicted below. Lattice points are indicated by dots.
 <img src="pictures/ws-2d.png" style="display: block; margin-left: auto; margin-right: auto; width:520px;">
 
-One may notice that a simpler description could be obtained by fixing \\(\mathbf{v}_2 =
-(1, 0)\\) instead of \\(\mathbf{v}_0\\), which would avoid fractional vertex coordinates.
-However, this approach would not illustrate the construction of the lattice. This is
-because, in this scenario, the lattice points would invariably align with points having
-integer coordinates. In practice, coordinates are often chosen so that lattice points
-correspond to those with integer coordinates. In higher dimensions, this is not trivial,
-as a weight with a value of one is not always present in a weight system.
-
 ### General Dimension
 
 In higher dimensions, the situation becomes more complex. Not all IP polytopes are

pictures/ws-2d.tex

@@ -21,7 +21,7 @@
 
       \path [draw, fill opacity=\opacity, fill=fillColor, line width=\polytopeLineWidth] (1, 0) --(0, 1) --(-1, -1) --cycle;
 
-      \draw[step=1, dotted, line width=\gridLineWidth] (-10, -10) grid (10, 10);
+      % \draw[step=1, dotted, line width=\gridLineWidth] (-10, -10) grid (10, 10);
       \path [draw, line width=\gridLineWidth] (-10, 0) --(10, 0) (0, -10) --(0, 10);
 
       \foreach \i in {-2,...,2}
@@ -37,7 +37,7 @@
       \begin{scope}
         \path [draw, fill opacity=\opacity, fill=fillColor, line width=\polytopeLineWidth] (-1, 2) --(2, -1) --(-1, -1) --cycle;
 
-        \draw[step=1, dotted, line width=\gridLineWidth] (-10, -10) grid (10, 10);
+        % \draw[step=1, dotted, line width=\gridLineWidth] (-10, -10) grid (10, 10);
         \path [draw, line width=\gridLineWidth] (-10, 0) --(10, 0) (0, -10) --(0, 10);
 
         \foreach \i in {-2,...,2}
@@ -59,7 +59,7 @@
       \begin{scope}[scale=1.4142] % sqrt(2)
         \path [draw, fill opacity=\opacity, fill=fillColor, line width=\polytopeLineWidth] (1, 0) --(0, 1) --(-1/2, -1/2) --cycle;
 
-        \draw[step=1, dotted, line width=\gridLineWidth] (-10, -10) grid (10, 10);
+        % \draw[step=1, dotted, line width=\gridLineWidth] (-10, -10) grid (10, 10);
         \path [draw, line width=\gridLineWidth] (-10, 0) --(10, 0) (0, -10) --(0, 10);
 
         \foreach \i in {-2,...,2}
@@ -76,7 +76,7 @@
       \begin{scope}[scale=0.707]
         \path [draw, fill opacity=\opacity, fill=fillColor, line width=\polytopeLineWidth] (-1, 3) --(3, -1) --(-1, -1) --cycle;
 
-        \draw[step=1, dotted, line width=\gridLineWidth] (-10, -10) grid (10, 10);
+        % \draw[step=1, dotted, line width=\gridLineWidth] (-10, -10) grid (10, 10);
         \path [draw, line width=\gridLineWidth] (-10, 0) --(10, 0) (0, -10) --(0, 10);
 
         \foreach \i in {-2,...,2}
@@ -98,7 +98,7 @@
       \begin{scope}[scale=1.732] % sqrt(3)
         \path [draw, fill opacity=\opacity, fill=fillColor, line width=\polytopeLineWidth] (1, 0) --(0, 1) --(-1/3, -2/3) --cycle;
 
-        \draw[step=1, dotted, line width=\gridLineWidth] (-10, -10) grid (10, 10);
+        % \draw[step=1, dotted, line width=\gridLineWidth] (-10, -10) grid (10, 10);
         \path [draw, line width=\gridLineWidth] (-10, 0) --(10, 0) (0, -10) --(0, 10);
 
         \foreach \i in {-2,...,2}
@@ -116,7 +116,7 @@
         \begin{scope}[xshift=-2cm]
           \path [draw, fill opacity=\opacity, fill=fillColor, line width=\polytopeLineWidth] (5, -1) --(-1, 2) --(-1, -1) --cycle;
 
-          \draw[step=1, dotted, line width=\gridLineWidth] (-10, -10) grid (10, 10);
+          % \draw[step=1, dotted, line width=\gridLineWidth] (-10, -10) grid (10, 10);
           \path [draw, line width=\gridLineWidth] (-10, 0) --(10, 0) (0, -10) --(0, 10);
 
           \foreach \i in {-1,...,3}