improve README

.gitignore

@@ -0,0 +1,6 @@
+*.aux
+*.log
+*.out
+*.pdf
+*.synctex.*
+auto/

README.md

@@ -1,58 +1,82 @@
 ---
 license: cc-by-sa-4.0
-pretty_name: Weight Systems Defining Five-Dimensional Reflexive and Non-Reflexive Polyhedra
+pretty_name: Weight Systems Defining Five-Dimensional IP Lattice Polytopes
 configs:
-- config_name: non-reflexive
-  data_files:
-  - split: full
-    path: non-reflexive/*.parquet
-- config_name: reflexive
-  data_files:
-  - split: full
-    path: reflexive/*.parquet
+  - config_name: non-reflexive
+    data_files:
+      - split: full
+        path: non-reflexive/*.parquet
+  - config_name: reflexive
+    data_files:
+      - split: full
+        path: reflexive/*.parquet
 tags:
-- physics
-- math
+  - physics
+  - math
 ---
 
-# Dataset of Weight Systems Defining Five-Dimensional Reflexive and Non-Reflexive Polyhedra
+# Weight Systems Defining Five-Dimensional IP Lattice Polytopes
 
 This dataset contains all weight systems defining five-dimensional reflexive and
-non-reflexive polyhedra, instrumental in the study of Calabi-Yau fourfolds in mathematics
-and theoretical physics. The data was compiled by Friedrich Schöller and Harald Skarke in
-[arXiv:1808.02422](https://arxiv.org/abs/1808.02422). More information is available at the
-[Calabi-Yau data website](http://hep.itp.tuwien.ac.at/~kreuzer/CY/). The dataset can be
-explored using the [search frontend](http://rgc.itp.tuwien.ac.at/fourfolds/).
+non-reflexive IP lattice polytopes, instrumental in the study of Calabi-Yau fourfolds in
+mathematics and theoretical physics. The data was compiled by Harald Skarke and Friedrich
+Schöller in [arXiv:1808.02422](https://arxiv.org/abs/1808.02422). More information is
+available at the [Calabi-Yau data website](http://hep.itp.tuwien.ac.at/~kreuzer/CY/). The
+dataset can be explored using the [search
+frontend](http://rgc.itp.tuwien.ac.at/fourfolds/). See below for a short mathematical
+exposition on the construction of polytopes.
+
+Please cite the paper when referencing this dataset:
+
+```
+@article{Scholler:2018apc,
+    author = {Schöller, Friedrich and Skarke, Harald},
+    title = "{All Weight Systems for Calabi-Yau Fourfolds from Reflexive Polyhedra}",
+    eprint = "1808.02422",
+    archivePrefix = "arXiv",
+    primaryClass = "hep-th",
+    doi = "10.1007/s00220-019-03331-9",
+    journal = "Commun. Math. Phys.",
+    volume = "372",
+    number = "2",
+    pages = "657--678",
+    year = "2019"
+}
+```
 
 ## Dataset Details
 
-The dataset consists of two subsets: weight systems defining reflexive polyhedra and
-weight systems defining non-reflexive polyhedra. Each subset is split into 4000 files in
-Parquet format. Rows within each file are sorted lexicographically by weights.
+The dataset consists of two subsets: weight systems defining reflexive (and therefore IP)
+polytopes and weight systems defining non-reflexive IP polytopes. Each subset is split
+into 4000 files in Parquet format. Rows within each file are sorted lexicographically by
+weights.
 
-Each row in the dataset represents a polyhedron and contains the six weights defining it,
+Each row in the dataset represents a polytope and contains the six weights defining it,
 along with the vertex count, facet count, and lattice point count. The reflexive dataset
 also includes the Hodge numbers \\( h^{1,1} \\), \\( h^{1,2} \\), and \\( h^{1,3} \\) of
-the corresponding Calabi-Yau manifold, and the lattice point count of the dual polyhedron.
+the corresponding Calabi-Yau manifold, and the lattice point count of the dual polytope.
 
 For any Calabi-Yau fourfold, the Euler characteristic \\( \chi \\) and the Hodge number
 \\( h^{2,2} \\) can be derived as follows:
+
 $$ \chi = 48 + 6 (h^{1,1} − h^{1,2} + h^{1,3}) $$
+
 $$ h^{2,2} = 44 + 4 h^{1,1} − 2 h^{1,2} + 4 h^{1,3} $$
 
-This dataset is licensed under the [CC BY-SA 4.0 license](http://creativecommons.org/licenses/by-sa/4.0/).
+This dataset is licensed under the
+[CC BY-SA 4.0 license](http://creativecommons.org/licenses/by-sa/4.0/).
 
 ### Data Fields
 
-- `weight0 to weight5:` Weights of the weight system defining the polyhedron.
-- `vertex_count:` Vertex count of the polyhedron.
-- `facet_count:` Facet count of the polyhedron.
-- `point_count:` Lattice point count of the polyhedron.
-- `dual_point_count:` Lattice point count of the dual polyhedron (only for reflexive
-  polyhedra).
-- `h11:` Hodge number \\( h^{1,1} \\) (only for reflexive polyhedra).
-- `h12:` Hodge number \\( h^{1,2} \\) (only for reflexive polyhedra).
-- `h13:` Hodge number \\( h^{1,3} \\) (only for reflexive polyhedra).
+- `weight0` to `weight5`: Weights of the weight system defining the polytope.
+- `vertex_count`: Vertex count of the polytope.
+- `facet_count`: Facet count of the polytope.
+- `point_count`: Lattice point count of the polytope.
+- `dual_point_count`: Lattice point count of the dual polytope (only for reflexive
+  polytopes).
+- `h11`: Hodge number \\( h^{1,1} \\) (only for reflexive polytopes).
+- `h12`: Hodge number \\( h^{1,2} \\) (only for reflexive polytopes).
+- `h13`: Hodge number \\( h^{1,3} \\) (only for reflexive polytopes).
 
 ## Usage
 
@@ -70,8 +94,8 @@ for row in dataset.take(5):
 ```
 
 When cloning the Git repository with Git Large File Storage (LFS), data files are stored
-in the Git LFS storage directory, as well as in the working tree. To avoid occupying
-double the disk space, use a filesystem that supports copy-on-write and run the following
+both in the Git LFS storage directory and in the working tree. To avoid occupying double
+the disk space, use a filesystem that supports copy-on-write, and run the following
 commands to clone the repository:
 
 ```bash
@@ -95,22 +119,160 @@ git lfs fetch
 git lfs dedup
 ```
 
-## Citation
+## Construction of Polytopes
 
-Please cite the following research paper when referencing this dataset:
+This is an introduction to the mathematics involved in the construction of polytopes
+relevant to this dataset. For more details and precise definitions, consult the paper
+[arXiv:1808.02422](https://arxiv.org/abs/1808.02422) and references therein.
 
-```
-@article{Scholler:2018apc,
-    author = {Sch\"oller, Friedrich and Skarke, Harald},
-    title = "{All Weight Systems for Calabi-Yau Fourfolds from Reflexive Polyhedra}",
-    eprint = "1808.02422",
-    archivePrefix = "arXiv",
-    primaryClass = "hep-th",
-    doi = "10.1007/s00220-019-03331-9",
-    journal = "Commun. Math. Phys.",
-    volume = "372",
-    number = "2",
-    pages = "657--678",
-    year = "2019"
-}
-```
+### Polytopes
+
+A polytope is the convex hull of a finite set of points in \\(n\\)-dimensional Euclidean
+space, \\(\mathbb{R}^n\\). This means it is the smallest convex shape that contains all
+these points. The minimal collection of points that define a particular polytope are its
+vertices. Familiar examples of polytopes include triangles and rectangles in two
+dimensions, and cubes and octahedra in three dimensions.
+
+A polytope is considered an *IP polytope* (interior point polytope) if the origin of
+\\(\mathbb{R}^n\\) is in the interior of the polytope, not on its boundary or outside it.
+
+For any IP polytope \\(\nabla\\), its dual polytope \\(\nabla^*\\) is defined as the set
+of points \\(\mathbf{y}\\) satisfying
+
+$$
+  \mathbf{x} \cdot \mathbf{y}
+  \ge -1 \quad \text{for all } \mathbf{x} \in \nabla \;.
+$$
+
+This relationship is symmetric: the dual of the dual of a polytope is the polytope itself,
+i.e., \\( \nabla^{**} = \nabla \\).
+
+### Weight Systems
+
+Weight systems provide a means to describe simple polytopes known as *simplexes*. More
+broadly, *combined weight systems*, which are collections of individual weight systems,
+can describe any polytope. A combined weight system is a matrix consisting of real
+numbers. The construction process is outlined as follows:
+
+Consider a polytope in \\(\mathbb{R}^n\\) with vertex count \\(k\\), where \\(k\\) is
+bigger than \\(n\\). It is possible to position \\(n\\) of these vertices at arbitrary
+(linearly independent) locations through a linear transformation. The placement of the
+remaining \\(k - n\\) vertices is then determined. Their positions are the defining
+properties of a polytope. To specify these positions independently of the applied linear
+transformation, one can use the following system of equations. If \\(\mathbf{v}_0,
+\mathbf{v}_1, \dots \mathbf{v}_{k-1}\\) are the vertices of the polytope, these relations
+fix \\(k - n\\) vertices in terms of the other \\(n\\):
+
+$$
+  \sum_{i=0}^{k-1} q_i^{(j)} \mathbf{v}_i
+  = 0 \quad \text{for } 0 \le j \le k - n - 1 \;,
+$$
+
+where \\(q_i^{(j)}\\) is the matrix of real numbers, the combined weight system. In cases
+where \\(k = n + 1\\), \\(j\\) is limited to the value zero, reducing the matrix to a
+single weight system \\(q_i\\). In this scenario, the polytope is a simplex, and the
+equation simplifies to:
+
+$$ \sum_{i=0}^n q_i \mathbf{v}_i = 0 \;. $$
+
+It is important to note that scaling all weights in a weight system by a common factor
+results in an equivalent weight system that defines the same polytope.
+
+For this dataset, the focus is on a specific construction of lattice polytopes described
+in subsequent sections.
+
+### Lattice Polytopes
+
+A lattice polytope is a polytope with vertices at the points of a regular grid, or
+lattice. Using linear transformations, any lattice polytope can be transformed so that its
+vertices have integer coordinates, hence they are also referred to as integral
+polytopes.
+
+The dual of a lattice with points \\(L\\) is the lattice consisting of all points
+\\(\mathbf{y}\\) that satisfy
+
+$$
+  \mathbf{x} \cdot \mathbf{y} \in \mathbb{Z} \quad \text{for all } \mathbf{x} \in L \;.
+$$
+
+*Reflexive polytopes* are a specific type of lattice polytope characterized by having a
+dual that is also a lattice polytope, with vertices situated on the dual lattice. These
+polytopes play a central role in the context of this dataset.
+
+The weights of a lattice polytope are always rational. This characteristic enables the
+rescaling of a weight system so that its weights become integers without any common
+divisor. This rescaling has been performed in this dataset.
+
+Typically, the dual of a lattice polytope defined by a weight system is not a lattice
+polytope. However, our interest lies in a different construction than simply considering
+polytopes defined by (combined) weight systems, as described above. In this construction,
+they are just the starting point. We start with the polytope \\(\nabla\\), arising from a
+weight system as previously described. Then, we define the polytope \\(\Delta\\) as the
+convex hull of the intersection of \\(\nabla^*\\) with the points of the dual lattice. In
+the context of this dataset, the polytope \\(\Delta\\) is referred to as ‘the polytope’.
+Correspondingly, \\(\Delta^{\!*}\\) is referred to as ‘the dual polytope’. The lattice of
+\\(\Delta\\) is taken to be the coarsest lattice possible, such that \\(\nabla\\) is a
+lattice polytope, i.e., the lattice generated by the vertices of \\(\nabla\\). This
+construction is exemplified in the following sections.
+
+A weight system is considered an IP weight system if the corresponding \\(\Delta\\) is an
+IP polytope; that is, the origin is within its interior. Since only IP polytopes have
+corresponding dual polytopes, this condition is essential for the polytope \\(\Delta\\) to
+be classified as reflexive.
+
+### Two Dimensions
+
+In two dimensions, all IP weight systems define reflexive polytopes and every vertex of
+\\(\nabla^*\\) lies on the dual lattice, making \\(\Delta\\) and \\(\nabla^*\\) identical.
+There are exactly three IP weight systems that define two-dimensional polytopes
+(polygons). Each polytope is reflexive and has three vertices and three facets (edges):
+
+| weight system | number of points of \\(\nabla\\) | number of points of \\(\nabla^*\\) |
+|--------------:|---------------------------------:|-----------------------------------:|
+|     (1, 1, 1) |                                4 |                                 10 |
+|     (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.
+<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 general case in higher dimensions, where
+this is not possible since there is not always a weight equal to 1.
+
+### General Dimension
+
+In higher dimensions, the situation becomes more complex. Not all IP polytopes are
+reflexive, and generally, \\(\Delta \neq \nabla^*\\).
+
+This example shows the construction of the three-dimensional polytope \\(\Delta\\) with
+weight system (2, 3, 4, 5) and its dual \\(\Delta^{\!*}\\). Lattice points lying on the
+polytopes are indicated by dots. \\(\Delta\\) has 7 vertices and 13 lattice points,
+\\(\Delta^{\!*}\\) also has 7 vertices, but 16 lattice points.
+<img src="pictures/ws-3d-2-3-4-5.png" style="display: block; margin-left: auto; margin-right: auto; width:450px;">
+
+The counts of reflexive single-weight-system polytopes by dimension \\(n\\) are:
+
+| \\(n\\) | reflexive single-weight-system polytopes |
+|--------:|-----------------------------------------:|
+|       2 |                                        3 |
+|       3 |                                       95 |
+|       4 |                                  184,026 |
+|       5 |           (this dataset) 185,269,499,015 |

pictures/Makefile

@@ -0,0 +1,21 @@
+SOURCES = $(wildcard *.tex)
+PNGS = $(SOURCES:.tex=.png)
+PDFS = $(SOURCES:.tex=.pdf)
+
+png: $(PNGS)
+pdf: $(PDFS)
+
+%.pdf: %.tex
+	pdflatex $<
+	rm $(<:.tex=.aux) $(<:.tex=.log)
+
+%.png: %.pdf
+	convert -density 600 $< -flatten $@
+
+.PHONY: clean
+clean:
+	rm -rf auto *.aux *.log *.synctex.gz
+
+.PHONY: cleanall
+cleanall: clean
+	rm -rf *.png *.pdf

pictures/ws-2d.tex

@@ -0,0 +1,132 @@
+\documentclass[tikz,11pt]{standalone}
+
+\usepackage{tikz}
+
+\pgfmathsetmacro{\gridLineWidth}{0.6pt}
+\pgfmathsetmacro{\polytopeLineWidth}{1.2pt}
+\pgfmathsetmacro{\dotSize}{0.5mm}
+\definecolor{fillColor}{rgb}{0.8,0.8,0.8}
+\pgfmathsetmacro{\opacity}{0.7}
+
+\begin{document}
+\begin{tikzpicture}
+  % (1, 1, 1)
+  \begin{scope}[yshift=5.2cm, scale=0.7]
+    \node at (-5, 0) {$\mathbf{q} = (1, 1, 1)$};
+    
+    \begin{scope}
+      \node at (0, -3.3) {$\nabla = \Delta^{\!*}$};
+
+      \clip (0,0) circle (2.7);
+
+      \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);
+      \path [draw, line width=\gridLineWidth] (-10, 0) --(10, 0) (0, -10) --(0, 10);
+
+      \foreach \i in {-2,...,2}
+      \foreach \j in {-2,...,2}{
+        \node at (\i, \j) [circle, fill, inner sep=\dotSize] {};
+      };
+    \end{scope}
+
+    \begin{scope}[xshift=6.5cm]
+      \node at (0, -3.3) {$\nabla^* = \Delta$};
+
+      \clip (0,0) circle (2.7);
+      \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);
+        \path [draw, line width=\gridLineWidth] (-10, 0) --(10, 0) (0, -10) --(0, 10);
+
+        \foreach \i in {-2,...,2}
+        \foreach \j in {-2,...,2}{
+          \node at (\i, \j) [circle, fill, inner sep=\dotSize] {};
+        };
+      \end{scope}
+    \end{scope}
+  \end{scope}
+
+  % (1, 1, 2)
+  \begin{scope}[yshift=0cm, scale=0.7]
+    \node at (-5, 0) {$\mathbf{q} = (1, 1, 2)$};
+
+    \begin{scope}
+      \node at (0, -3.3) {$\nabla = \Delta^{\!*}$};
+
+      \clip (0,0) circle (2.7);
+      \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);
+        \path [draw, line width=\gridLineWidth] (-10, 0) --(10, 0) (0, -10) --(0, 10);
+
+        \foreach \i in {-2,...,2}
+        \foreach \j in {-2,...,2}{
+          \node at (\i / 2 - \j / 2, \i / 2 + \j / 2) [circle, fill, inner sep=\dotSize] {};
+        };
+      \end{scope}
+    \end{scope}
+
+    \begin{scope}[xshift=6.5cm]
+      \node at (0, -3.3) {$\nabla^* = \Delta$};
+
+      \clip (0,0) circle (2.7);
+      \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);
+        \path [draw, line width=\gridLineWidth] (-10, 0) --(10, 0) (0, -10) --(0, 10);
+
+        \foreach \i in {-2,...,2}
+        \foreach \j in {-2,...,2}{
+          \node at (\i - \j, \i + \j) [circle, fill, inner sep=\dotSize] {};
+        };
+      \end{scope}
+    \end{scope}
+  \end{scope}
+
+  % (1, 2, 3)
+  \begin{scope}[yshift=-5.2cm, scale=0.7]
+    \node at (-5, 0) {$\mathbf{q} = (1, 2, 3)$};
+
+    \begin{scope}
+      \node at (0, -3.3) {$\nabla = \Delta^{\!*}$};
+
+      \clip (0,0) circle (2.7);
+      \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);
+        \path [draw, line width=\gridLineWidth] (-10, 0) --(10, 0) (0, -10) --(0, 10);
+
+        \foreach \i in {-2,...,2}
+        \foreach \j in {-4,...,4}{
+          \node at (\i / 3 - \j / 3, \i * 2 / 3 + \j / 3) [circle, fill, inner sep=\dotSize] {};
+        };
+      \end{scope}
+    \end{scope}
+
+    \begin{scope}[xshift=6.5cm]
+      \node at (0, -3.3) {$\nabla^* = \Delta$};
+
+      \clip (0,0) circle (2.7);
+      \begin{scope}[scale=0.577]
+        \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);
+          \path [draw, line width=\gridLineWidth] (-10, 0) --(10, 0) (0, -10) --(0, 10);
+
+          \foreach \i in {-1,...,3}
+          \foreach \j in {-4,...,4}{
+            \node at (3 * \i + \j, \j) [circle, fill, inner sep=\dotSize] {};
+          };
+        \end{scope}
+      \end{scope}
+    \end{scope}
+  \end{scope}
+
+\end{tikzpicture}
+\end{document}

pictures/ws-3d-2-3-4-5.tex

@@ -0,0 +1,266 @@
+\documentclass[tikz,11pt]{standalone}
+
+\usepackage{tikz}
+\usepackage{tikz-3dplot}
+
+\usetikzlibrary{arrows.meta}
+
+\pgfmathsetmacro{\polytopeLineWidth}{1.2pt}
+\pgfmathsetmacro{\dotSize}{0.5mm}
+\definecolor{fillColor}{rgb}{0.8,0.8,0.8}
+\pgfmathsetmacro{\opacity}{0.7}
+\pgfmathsetmacro{\rotation}{72}
+\pgfmathsetmacro{\zRotation}{30}
+% \pgfmathsetmacro{\rotation}{100}
+% \pgfmathsetmacro{\zRotation}{30}
+
+\newcommand{\point}[1]{
+  \path (#1) node[fill, black, circle, inner sep=\dotSize] {};
+}
+
+\begin{document}
+
+\begin{tikzpicture}[line join=bevel, line width=\polytopeLineWidth, scale=1.1]
+  \begin{scope}
+    \path [draw, -Stealth] (1.7, 2.4) -- node [above] {dual} (3.7, 2.4);
+    \path [draw, -Stealth] (5, -0.5) -- node [right] {convex hull} (5, -1.5);
+
+    \node at (0, 0) {$\nabla$};
+
+    \tdplotsetmaincoords{\rotation}{-\zRotation}
+    \tdplotsetrotatedcoords{0}{90}{90}
+
+    % \begin{tikzpicture}[tdplot_main_coords]
+    %   \draw[thick,->] (0,0,0) -- (1,0,0) node[anchor=north east]{$x$};
+    %   \draw[thick,->] (0,0,0) -- (0,1,0) node[anchor=north west]{$y$};
+    %   \draw[thick,->] (0,0,0) -- (0,0,1) node[anchor=south]{$z$};
+    % \end{tikzpicture}
+
+    \begin{scope}[tdplot_rotated_coords, xshift=0.2cm, yshift=3.4cm]
+      % back faces
+      \path [draw, fill opacity=\opacity, fill=fillColor] (2.,0.,-1.)--(0.,0.,1.)--(-3.,-2.,-1.)--cycle;
+      \path [draw, fill opacity=\opacity, fill=fillColor] (0.,1.,0.)--(0.,0.,1.)--(2.,0.,-1.)--cycle;
+
+      % back points
+      \point{1,0,0};
+      \point{0.,0.,0.}; % inside
+
+      % front faces
+      \path [draw, fill opacity=\opacity, fill=fillColor] (0.,1.,0.)--(2.,0.,-1.)--(-3.,-2.,-1.)--cycle;
+      \path [draw, fill opacity=\opacity, fill=fillColor] (0.,0.,1.)--(0.,1.,0.)--(-3.,-2.,-1.)--cycle;
+
+      % front points
+      \point{2.,0.,-1.};
+      \point{0.,1.,0.};
+      \point{0.,0.,1.};
+      \point{-3.,-2.,-1.};
+    \end{scope}
+  \end{scope}
+
+  % \begin{scope}
+  %   \path [draw, -Stealth] (1.7, 2.4) --(3.7, 2.4);
+  %   \path [draw, -Stealth] (5, -0.5) --(5, -1.5);
+
+  %   \node at (0, 0) {$\nabla$};
+
+  %   \tdplotsetmaincoords{\rotation}{180 - \zRotation}
+  %   \tdplotsetrotatedcoords{0}{90}{90}
+
+  %   \begin{scope}[tdplot_rotated_coords, xshift=0.2cm, yshift=3.4cm]
+  %     % \begin{scope}[tdplot_rotated_coords]
+  %     %   \draw[thick,->] (0,0,0) -- (1,0,0) node[anchor=north east]{$x$};
+  %     %   \draw[thick,->] (0,0,0) -- (0,1,0) node[anchor=north west]{$y$};
+  %     %   \draw[thick,->] (0,0,0) -- (0,0,1) node[anchor=south]{$z$};
+  %     % \end{scope}
+
+  %     % back faces
+  %     \path [draw, fill opacity=\opacity, fill=fillColor] (0.,1.,0.)--(2.,0.,-1.)--(-3.,-2.,-1.)--cycle;
+  %     \path [draw, fill opacity=\opacity, fill=fillColor] (0.,0.,1.)--(0.,1.,0.)--(-3.,-2.,-1.)--cycle;
+
+  %     % back points
+  %     \point{2.,0.,-1.};
+  %     \point{0.,1.,0.};
+  %     \point{0.,0.,1.};
+  %     \point{-3.,-2.,-1.};
+  %     \point{0.,0.,0.}; % inside
+
+  %     % front faces
+  %     \path [draw, fill opacity=\opacity, fill=fillColor] (2.,0.,-1.)--(0.,0.,1.)--(-3.,-2.,-1.)--cycle;
+  %     \path [draw, fill opacity=\opacity, fill=fillColor] (0.,1.,0.)--(0.,0.,1.)--(2.,0.,-1.)--cycle;
+
+  %     % front points
+  %     \point{1,0,0};
+  %   \end{scope}
+  % \end{scope}
+
+  \begin{scope}[xshift=5cm]
+    \node at (0, 0) {$\nabla^*$};
+
+    \tdplotsetmaincoords{\rotation}{180 - \zRotation}
+    \tdplotsetrotatedcoords{0}{90}{90}
+    \begin{scope}[tdplot_rotated_coords, yshift=1.8cm]
+      % back faces
+      \path [draw, fill opacity=0.7, fill=fillColor] (1.333,-1.,-1.)--(-1.,-1.,-1.)--(-1.,2.5,-1.)--cycle;
+      \path [draw, fill opacity=0.7, fill=fillColor] (-1.,-1.,-1.)--(0.4,-1.,1.8)--(-1.,2.5,-1.)--cycle;
+      \path [draw, fill opacity=0.7, fill=fillColor] (0.4,-1.,1.8)--(-1.,-1.,-1.)--(1.333,-1.,-1.)--cycle;
+
+      % back points
+      \point{-1.,-1.,-1.};
+      \point{0.,-1.,-1.};
+      \point{1.,-1.,-1.};
+      \point{-1.,0.,-1.};
+      \point{-1.,1.,-1.};
+      \point{-1.,2.,-1.};
+      \point{0.,-1.,1.};
+      \point{0.,0.,-1.}; % on face
+      \point{0.,-1.,0.}; % on face
+      \point{0.,0.,0.}; % inside
+
+      % front faces
+      \path [draw, fill opacity=0.7, fill=fillColor] (0.4,-1.,1.8)--(1.333,-1.,-1.)--(-1.,2.5,-1.)--cycle;
+
+      % front points
+      \point{0.,1.,-1.};
+      \point{0.,0.,1.};
+      \point{1.,-1.,0.};
+    \end{scope}
+  \end{scope}
+
+  \begin{scope}[xshift=5cm, yshift=-6.5cm]
+    \node at (0, 0) {$\Delta$};
+
+    \tdplotsetmaincoords{\rotation}{180 - \zRotation}
+    \tdplotsetrotatedcoords{0}{90}{90}
+    \begin{scope}[tdplot_rotated_coords, xshift=-0.2cm, yshift=1.8cm]
+      % back edges
+      \path [draw, densely dotted] (0.4,-1.,1.8)--(1.333,-1.,-1.);
+      \path [draw, densely dotted] (1.333,-1.,-1.)--(-1.,2.5,-1.);
+      \path [draw, densely dotted] (-1.,2.5,-1.)--(0.4,-1.,1.8);
+      \path [draw, densely dotted] (1.333,-1.,-1.)--(-1.,-1.,-1.);
+      \path [draw, densely dotted] (-1.,-1.,-1.)--(-1.,2.5,-1.);
+      \path [draw, densely dotted] (-1.,-1.,-1.)--(0.4,-1.,1.8);
+
+      % back faces
+      \path [draw, fill opacity=\opacity, fill=fillColor] (-1.,-1.,-1.)--(0.,-1.,1.)--(0.,0.,1.)--(-1.,2.,-1.)--cycle;
+      \path [draw, fill opacity=\opacity, fill=fillColor] (-1.,-1.,-1.)--(-1.,2.,-1.)--(0.,1.,-1.)--(1.,-1.,-1.)--cycle;
+      \path [draw, fill opacity=\opacity, fill=fillColor] (0.,-1.,1.)--(-1.,-1.,-1.)--(1.,-1.,-1.)--(1.,-1.,0.)--cycle;
+
+      % back points
+      \point{-1.,-1.,-1.};
+      \point{-1.,0.,-1.};
+      \point{-1.,1.,-1.};
+      \point{0.,-1.,-1.};
+      \point{0.,-1.,0.}; % on face
+      \point{0.,0.,-1.}; % on face
+      \point{0.,0.,0.}; % inside
+
+      % front faces
+      \path [draw, fill opacity=\opacity, fill=fillColor] (0.,0.,1.)--(0.,1.,-1.)--(-1.,2.,-1.)--cycle;
+      \path [draw, fill opacity=\opacity, fill=fillColor] (0.,1.,-1.)--(1.,-1.,0.)--(1.,-1.,-1.)--cycle;
+      \path [draw, fill opacity=\opacity, fill=fillColor] (1.,-1.,0.)--(0.,1.,-1.)--(0.,0.,1.)--cycle;
+      \path [draw, fill opacity=\opacity, fill=fillColor] (1.,-1.,0.)--(0.,0.,1.)--(0.,-1.,1.)--cycle;
+
+      % front points
+      \point{1.,-1.,-1.};
+      \point{1.,-1.,0.};
+      \point{0.,1.,-1.};
+      \point{0.,-1.,1.};
+      \point{0.,0.,1.};
+      \point{-1.,2.,-1.};
+    \end{scope}
+  \end{scope}
+
+  \begin{scope}[yshift=-6.5cm]
+    \path [draw, -Stealth] (3.7, 2.7) -- node [above] {dual} (1.7, 2.7);
+
+    \node at (0, 0) {$\Delta^{\!*}$};
+
+    \tdplotsetmaincoords{\rotation}{-\zRotation}
+    \tdplotsetrotatedcoords{0}{90}{90}
+    \begin{scope}[tdplot_rotated_coords, xshift=0.2cm, yshift=3.4cm]
+      % back edges
+      \path [draw, fill opacity=\opacity, fill=fillColor] (0.,0.,1.)--(-2.,-2.,-1.)--(2.,0.,-1.) --cycle;
+      \path [draw, fill opacity=\opacity, fill=fillColor] (-3.,-2.,-1.)--(-2.,-2.,-1.)--(0.,0.,1.)--(-2.,-1.,0.) --cycle;
+      \path [draw, fill opacity=\opacity, fill=fillColor] (0.,1.,0.)--(0.,0.,1.)--(2.,0.,-1.) --cycle;
+
+      % back faces
+      \path [draw, densely dotted] (2.,0.,-1.)--(-3.,-2.,-1.);
+      \path [draw, densely dotted] (-3.,-2.,-1.)--(0.,1.,0.);
+      \path [draw, densely dotted] (0.,0.,1.)--(-3.,-2.,-1.);
+
+      % back points
+      \point{-1.,-1.,0.};
+      \point{1.,0.,0.};
+      \point{0.,0.,0.}; % inside
+
+      % front faces
+      \path [draw, fill opacity=\opacity, fill=fillColor] (-1.,0.,-1.)--(-3.,-2.,-1.)--(-2.,-1.,0.)--(0.,1.,0.) --cycle;
+      \path [draw, fill opacity=\opacity, fill=fillColor] (-1.,0.,-1.)--(0.,1.,0.)--(2.,0.,-1.) --cycle;
+      \path [draw, fill opacity=\opacity, fill=fillColor] (-1.,0.,-1.)--(2.,0.,-1.)--(-2.,-2.,-1.)--(-3.,-2.,-1.) --cycle;
+      \path [draw, fill opacity=\opacity, fill=fillColor] (-2.,-1.,0.)--(0.,0.,1.)--(0.,1.,0.) --cycle;
+
+      % front points
+      \point{-3.,-2.,-1.};
+      \point{-2.,-2.,-1.};
+      \point{-2.,-1.,-1.};
+      \point{-2.,-1.,0.};
+      \point{-1.,0.,-1.};
+      \point{-1.,0.,0.};
+      \point{0.,-1.,-1.};
+      \point{0.,0.,-1.};
+      \point{0.,0.,1.};
+      \point{0.,1.,0.};
+      \point{1.,0.,-1.};
+      \point{2.,0.,-1.};
+      \point{-1.,-1.,-1.}; % on face
+    \end{scope}
+  \end{scope}
+
+  % \begin{scope}[yshift=-6.5cm]
+  %   \path [draw, -Stealth] (3.7, 2.7) --(1.7, 2.7);
+
+  %   \node at (0, 0) {$\Delta^{\!*}$};
+
+  %   \tdplotsetmaincoords{\rotation}{180 - \zRotation}
+  %   \tdplotsetrotatedcoords{0}{90}{90}
+  %   \begin{scope}[tdplot_rotated_coords, xshift=0.2cm, yshift=3.4cm]
+  %     % back faces
+  %     \path [draw, fill opacity=\opacity, fill=fillColor] (-1.,0.,-1.)--(-3.,-2.,-1.)--(-2.,-1.,0.)--(0.,1.,0.) --cycle;
+  %     \path [draw, fill opacity=\opacity, fill=fillColor] (-1.,0.,-1.)--(0.,1.,0.)--(2.,0.,-1.) --cycle;
+  %     \path [draw, fill opacity=\opacity, fill=fillColor] (-1.,0.,-1.)--(2.,0.,-1.)--(-2.,-2.,-1.)--(-3.,-2.,-1.) --cycle;
+  %     \path [draw, fill opacity=\opacity, fill=fillColor] (-2.,-1.,0.)--(0.,0.,1.)--(0.,1.,0.) --cycle;
+
+  %     % back points
+  %     \point{-3.,-2.,-1.};
+  %     \point{-2.,-2.,-1.};
+  %     \point{-2.,-1.,-1.};
+  %     \point{-2.,-1.,0.};
+  %     \point{-1.,0.,-1.};
+  %     \point{-1.,0.,0.};
+  %     \point{0.,-1.,-1.};
+  %     \point{0.,0.,-1.};
+  %     \point{0.,0.,1.};
+  %     \point{0.,1.,0.};
+  %     \point{1.,0.,-1.};
+  %     \point{2.,0.,-1.};
+  %     \point{-1.,-1.,-1.}; % on face
+  %     \point{0.,0.,0.}; % inside
+
+  %     % front edges
+  %     \path [draw, fill opacity=\opacity, fill=fillColor] (0.,0.,1.)--(-2.,-2.,-1.)--(2.,0.,-1.) --cycle;
+  %     \path [draw, fill opacity=\opacity, fill=fillColor] (-3.,-2.,-1.)--(-2.,-2.,-1.)--(0.,0.,1.)--(-2.,-1.,0.) --cycle;
+  %     \path [draw, fill opacity=\opacity, fill=fillColor] (0.,1.,0.)--(0.,0.,1.)--(2.,0.,-1.) --cycle;
+
+  %     % front faces
+  %     \path [draw, densely dotted] (2.,0.,-1.)--(-3.,-2.,-1.);
+  %     \path [draw, densely dotted] (-3.,-2.,-1.)--(0.,1.,0.);
+  %     \path [draw, densely dotted] (0.,0.,1.)--(-3.,-2.,-1.);
+
+  %     % front points
+  %     \point{-1.,-1.,0.};
+  %     \point{1.,0.,0.};
+  %   \end{scope}
+  % \end{scope}
+\end{tikzpicture}
+
+\end{document}