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6 results for “Lattice polygons”
Rational polygons with exactly one interior lattice point
<p>This record contains</p> <ul> <li>maximal \(k\)-rational polygons with exactly one interior lattice point for \(1 \leq k \leq 10\),</li> <li>all \(k\)-rational polygons with exactly one interior lattice point for \(1 \leq k \leq 4\),</li> <li>all Ehrhart quasipolynomials of these polygons for \(1 \leq k \leq 5\).</li> <li>all \(k\)-rational LDP polygons (=almost \(k\)-hollow LDP polygons) for \(1 \leq k \leq 5\).</li> </ul> <p>The maximal polygons are stored as text files with one polygon per line. Each line contains the vertices of the polygon multiplied by \(k\) (so that they are integral). The other polygons as well as the Ehrhart quasipolynomials are stored using the <a href="https://www.hdfgroup.org/solutions/hdf5/">HDF5</a> file format. All polygons and Ehrhart quasipolynomials have been obtained using <a href="https://github.com/justus-springer/RationalPolygons.jl">RationalPolygons.jl</a>.</p> <h3>Structure of the HDF5 files for polygons</h3> <p>In the HDF5 files "all_k<k>.h5", the polygons are split into different datasets according to their normalized area and number of vertices. We use "a" to denote the normalized area (twice the euclidian area) and "n" to denote the number of vertices. For example, the \(3\)-rational polygons with normalized area \(30\) having \(5\) vertices are located in "k3_all.h5" under the dataset "/a30/n5". Each dataset of polygons is one-dimensional with one entry per polygon. A polygon is stored using a compound datatype with \(2 \cdot n\) fields of integers. These integers are the vertices of the polygon multiplied by \(k\), stored in column major layout. For example, a triangle with vertices \((x_1,y_1), (x_2,y_2)\) and \((x_3,y_3)\) is stored as the tuple \((k\cdot x_1,k\cdot y_1,k \cdot x_2,k\cdot y_2,k \cdot x_3,k \cdot y_3)\). Moreover, each file contains the special two-dimensional dataset "numbers_of_polygons" that stores the numbers of polygons for a given normalized volume and number of vertices.</p> <h3>Structure of the HDF5 files for Ehrhart quasipolynomials</h3> <p>In the HDF5 files "ehrhart_k<k>.h5", the Ehrhart quasipolynomials are split into different datasets according to the normalized area of the associated polygon. We use "a" for the normalized area (twice the euclidian area). For example, the Ehrhart quasipolynomials of all \(3\)-rational polygons with normalized area \(30\) are located in "ehrhart_k3.h5" under the dataset "/a30". Each Ehrhart quasipolynomial is stored as a \(3 \times k\) integral matrix. Hence each dataset is three-dimensional, with the first dimension to enumerate the quasipolynomials and the latter two dimensions to store each quasipolynomial itself. For the way Ehrhart quasipolynomials are encoded as \(3 \times k\) integral matrices, we refer to the <a href="https://justus-springer.github.io/RationalPolygons.jl/dev/polygons/#Ehrhart-Theory">relevant documentation of RationalPolygons.jl</a>.</p>
Rational polygons with no interior lattice points
<p>This record contains</p> <ul> <li>maximal \(k\)-rational polygons with no interior lattice points for \(1 \leq k \leq 21\),</li> <li>all \(k\)-rational polygons with no interior lattice points, except those that can be realized in the strip \( \mathbb{R} \times [0,1] \) for \(1 \leq k \leq 6\),</li> <li>all Ehrhart quasipolynomials of these polygons for \(1 \leq k \leq 6\).</li> </ul> <p>The maximal polygons are stored as text files with one polygon per line. Each line contains the vertices of the polygon multiplied by \(k\) (so that they are integral). For \(1 \leq k \leq 6\), the files that contain all polygons as well as the Ehrhart quasipolynomials are stored using the <a href="https://www.hdfgroup.org/solutions/hdf5/">HDF5</a> file format. All polygons and Ehrhart quasipolynomials have been obtained using <a href="https://github.com/justus-springer/RationalPolygons.jl">RationalPolygons.jl</a>.</p> <h3>Structure of the HDF5 files for polygons</h3> <p>In the HDF5 files "all_k<k>.h5", the polygons are split into different datasets according to their normalized area and number of vertices. We use "a" to denote the normalized area (twice the euclidian area) and "n" to denote the number of vertices. For example, the \(3\)-rational polygons with normalized area \(30\) having \(5\) vertices are located in "all_k3.h5" under the dataset "/a30/n5". Each dataset of polygons is one-dimensional with one entry per polygon. A polygon is stored using a compound datatype with \(2 \cdot n\) fields of integers. These integers are the vertices of the polygon multiplied by \(k\), stored in column major layout. For example, a triangle with vertices \((x_1,y_1), (x_2,y_2)\) and \((x_3,y_3)\) is stored as the tuple \((k\cdot x_1,k\cdot y_1,k \cdot x_2,k\cdot y_2,k \cdot x_3,k \cdot y_3)\). Moreover, each file contains the special two-dimensional dataset "numbers_of_polygons" that stores the numbers of polygons for a given normalized volume and number of vertices.</p> <h3>Structure of the HDF5 files for Ehrhart quasipolynomials</h3> <p>In the HDF5 files "ehrhart_k<k>.h5", the Ehrhart quasipolynomials are split into different datasets according to the normalized area of the associated polygon. We use "a" for the normalized area (twice the euclidian area). For example, the Ehrhart quasipolynomials of all \(3\)-rational polygons with normalized area \(30\) are located in "ehrhart_k3.h5" under the dataset "/a30". Each Ehrhart quasipolynomial is stored as a \(3 \times k\) integral matrix. Hence each dataset is three-dimensional, with the first dimension to enumerate the quasipolynomials and the latter two dimensions to store each quasipolynomial itself. For the way Ehrhart quasipolynomials are encoded as \(3 \times k\) integral matrices, we refer to the <a href="https://justus-springer.github.io/RationalPolygons.jl/dev/polygons/#Ehrhart-Theory">relevant documentation of RationalPolygons.jl</a>.</p>
Half-integral polygons with few interior lattice points
<p>This record contains</p> <ul> <li>maximal half-integral polygons with up to 40 interior lattice points,</li> <li>all half-integral polygons with up to 12 interior lattice points,</li> <li>all Ehrhart quasipolynomials of half-integral polygons with up to 16 interior lattice points,</li> <li>all half-integral polygons with up to 15 collinear interior lattice points.</li> </ul> <p>The maximal polygons are stored as text files with one polygon per line. Each line contains the vertices of the polygon multiplied by two (so that they are integral). The other polygons as well as the Ehrhart quasipolynomials are stored using the <a href="https://www.hdfgroup.org/solutions/hdf5/">HDF5</a> file format. All polygons and Ehrhart quasipolynomials have been obtained using <a href="https://github.com/justus-springer/RationalPolygons.jl">RationalPolygons.jl</a>.</p> <h3>Structure of the HDF5 files for polygons</h3> <p>In the HDF5 files "all_i<i>.h5", the polygons are split into different datasets according to their normalized area and number of vertices. We use "a" to denote the normalized area (twice the euclidian area) and "n" to denote the number of vertices. For example, the half-integral polygons with \(8\) interior lattice points having normalized area \(60\) with \(5\) vertices are located in "all_i8.h5" under the dataset "/a60/n5". Each dataset of polygons is one-dimensional with one entry per polygon. A polygon is stored using a compound datatype with \(2 \cdot n\) fields of integers. These integers are the vertices of the polygon multiplied by two, stored in column major layout. For example, a triangle with vertices \((x_1,y_1), (x_2,y_2)\) and \((x_3,y_3)\) is stored as the tuple \((2\cdot x_1,2\cdot y_1,2 \cdot x_2,2\cdot y_2,2 \cdot x_3,2 \cdot y_3)\). Moreover, each file contains the special two-dimensional dataset "numbers_of_polygons" that stores the numbers of polygons for a given normalized volume and number of vertices.</p> <h3>Structure of the HDF5 files for Ehrhart quasipolynomials</h3> <p>In the HDF5 files "ehrhart_k<k>.h5", the Ehrhart quasipolynomials are split into different datasets according to the normalized area of the associated polygon. We use "a" for the normalized area (twice the euclidian area). For example, the Ehrhart quasipolynomials of all half-integral polygons \(8\) interior lattice points having normalized area \(60\) are located in "ehrhart_i8.h5" under the dataset "/a60". Each Ehrhart quasipolynomial is stored as a \(3 \times 2\) integral matrix. Hence each dataset is three-dimensional, with the first dimension to enumerate the quasipolynomials and the latter two dimensions to store each quasipolynomial itself. For the way Ehrhart quasipolynomials are encoded as \(3 \times 2\) integral matrices, we refer to the <a href="https://justus-springer.github.io/RationalPolygons.jl/dev/polygons/#Ehrhart-Theory">relevant documentation of RationalPolygons.jl</a>.</p>
3-rational polygons with few interior lattice points
<p>This record contains</p> <ul> <li>maximal 3-rational polygons with up to 5 interior lattice points,</li> <li>all 3-rational polygons with up to 4 interior lattice points,</li> <li>all Ehrhart quasipolynomials of 3-rational polygons with up to 5 interior lattice points.</li> </ul> <p>The maximal polygons are stored as text files with one polygon per line. Each line contains the vertices of the polygon multiplied by two (so that they are integral). The other polygons as well as the Ehrhart quasipolynomials are stored using the <a href="https://www.hdfgroup.org/solutions/hdf5/">HDF5</a> file format. All polygons and Ehrhart quasipolynomials have been obtained using <a href="https://github.com/justus-springer/RationalPolygons.jl">RationalPolygons.jl</a>.</p> <h3>Structure of the HDF5 files for polygons</h3> <p>In the HDF5 files "all_i<i>.h5", the polygons are split into different datasets according to their normalized area and number of vertices. We use "a" to denote the normalized area (twice the euclidian area) and "n" to denote the number of vertices. For example, the 3-rational polygons with \(8\) interior lattice points having normalized area \(60\) with \(5\) vertices are located in "all_i8.h5" under the dataset "/a60/n5". Each dataset of polygons is one-dimensional with one entry per polygon. A polygon is stored using a compound datatype with \(2 \cdot n\) fields of integers. These integers are the vertices of the polygon multiplied by three, stored in column major layout. For example, a triangle with vertices \((x_1,y_1), (x_2,y_2)\) and \((x_3,y_3)\) is stored as the tuple \((3\cdot x_1,3\cdot y_1,3 \cdot x_2,3\cdot y_2,3 \cdot x_3,3 \cdot y_3)\). Moreover, each file contains the special two-dimensional dataset "numbers_of_polygons" that stores the numbers of polygons for a given normalized volume and number of vertices.</p> <h3>Structure of the HDF5 files for Ehrhart quasipolynomials</h3> <p>In the HDF5 files "ehrhart_k<k>.h5", the Ehrhart quasipolynomials are split into different datasets according to the normalized area of the associated polygon. We use "a" for the normalized area (twice the euclidian area). For example, the Ehrhart quasipolynomials of all 3-rational polygons \(8\) interior lattice points having normalized area \(60\) are located in "ehrhart_i8.h5" under the dataset "/a60". Each Ehrhart quasipolynomial is stored as a \(3 \times 3\) integral matrix. Hence each dataset is three-dimensional, with the first dimension to enumerate the quasipolynomials and the latter two dimensions to store each quasipolynomial itself. For the way Ehrhart quasipolynomials are encoded as \(3 \times 3\) integral matrices, we refer to the <a href="https://justus-springer.github.io/RationalPolygons.jl/dev/polygons/#Ehrhart-Theory">relevant documentation of RationalPolygons.jl</a>.</p>
Polygons with small denominator containing a small number of lattice points
<p>The denominator of a rational polytope \(P\) is an integer \(r\) such that the dilated polytope \(rP\) has lattice point vertices. The size of a polytope is the number of lattice points it contains. This dataset contains polygons with denominator 2 and 3 with small size, classified using a growing algorithm as described in [HHK24].</p> <p>The data consists of files "denom_r_size_k_polygons.txt" which record the denominator \(r\) size \(k\) polygons \(P\). Each entry consists of the vertices and volume of \(rP\), the Ehrhart \(\delta\)-vector/\(h^*\)-vector of \(P\), and an ID number, which is unique among polygons of given size and denominator. Entries are ordered by their ID number. There are 50,564 entries in total.</p> <p><strong>Example entry:</strong></p> <p>ID=1<br>Vertices=[[ 1, 0 ], [ 0, 1 ], [ 3, 5 ]]<br>Volume=7<br>DeltaVec=[ 1, 0, 3, 7, 3, 0 ]</p> <p>If you make use of this data, please cite [HHK24] and the DOI for this data:</p> <p>doi:10.5281/zenodo.14230584</p> <p><strong>References:</strong></p> <p>[HHK24] Girtrude Hamm, Johannes Hofscheier, Alexander Kasprzyk, Classification and Ehrhart Theory of Denominator 2 Polygons. (preprint) arxiv:2411.19183</p>
Lattice polygons with at most 70 lattice points
<p>This record contains</p> <ul> <li>All lattice polygons with at most 70 lattice points,</li> <li>All interior lattice polygons with at most 112 lattice points,</li> </ul> <p>where a lattice polygon is called interior if it is the convex hull of interior lattice points of another lattice polygon. The algorithm to classify lattice polygons by lattice points is due to Koelman, see section 4.4 of his dissertation (reference below). His original classification was up to 42 lattice points, but his data was not publicily available. We have implemented his algorithm in <a href="https://github.com/justus-springer/RationalPolygons.jl">RationalPolygons.jl</a>, recreating and extending his classification.</p> <h3>Structure of the HDF5 files</h3> <p>The two archives "koelman70.tgz" and "koelman_interiors112.tgz" contain files l3.h5, l4.h5, etc... storing the polygons with given number of lattice points in the <a href="https://www.hdfgroup.org/solutions/hdf5/">HDF5</a> file format. In the HDF5 files, the polygons are split into different datasets according to their number of vertices, which we denote by "n". There is one dataset for every number of vertices. For example, the lattice polygons with 6 vertices having 50 lattice points are located in "l50.h5" under the dataset "n6". Each dataset of polygons is one-dimensional with one entry per polygon. A polygon is stored using a compound datatype with \(2 \cdot n\) fields of integers, which are the vertices of the polygon. For example, a triangle with vertices \((x_1,y_1), (x_2,y_2)\) and \((x_3,y_3)\) is stored as the tuple \((x_1,y_1,x_2,y_2,x_3,y_3)\). Moreover, each file contains the special one-dimensional dataset "numbers_of_polygons" that stores the numbers of polygons for given number of vertices.</p>
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