279 lines
9.9 KiB
TeX
279 lines
9.9 KiB
TeX
% !TEX root = thesis.tex
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\chapter{A short dictionary of dimension-based GIS terms}
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\label{ch:dictionary}
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\begin{multicols}{2}
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\begin{description}
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\footnotesize
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\item[ambient space]
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the space in which objects are embedded.
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\item[area]
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1.\ measure of the 2D extent of an object;
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2.\ 2D combinatorial element;
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{\color{gray} 3.\ 2D geometric object}.
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\item[ball]
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1.\ ($\rightarrow$\ same as 3-\emph{ball}) topological definition of the space within a 2-sphere (\ie\ a sphere), often defined as the space within a certain distance (the radius) from a point in $\mathbb{R}^3$ (the centre);
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2.\ ($\rightarrow$\ often $n$-\emph{ball}) topological definition of the space within an $(n-1)$-sphere, often defined as the space within a certain distance (the radius) from a point in $\mathbb{R}^n$ (the centre);
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3.\ geometric definition of a perfectly round filled 3D object.
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\item[body]
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$n$-dimensional combinatorial element in an $n$D context.
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\item[box]
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1.\ cuboid;
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2.\ orthotope of any dimension.
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\item[cavity]
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3D hole.
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\item[cell]
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1.\ ($\rightarrow$\ sometimes $n$-\emph{cell}) $n$D combinatorial element;
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{\color{gray} 2.\ 3D combinatorial element}.
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\item[cell complex]
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topological space formed by a set of cells glued along common faces, all faces of a cell in the complex should be in the complex.
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\item[circle]
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1.\ topological definition of the space at a certain distance (the radius) from a point in $\mathbb{R}^2$ (the centre);
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2.\ geometric definition of a perfectly round hollow 2D object.
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\item[closed]
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1.\ topological definition of an object that includes its boundary;
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2.\ geometric definition of an object in $n$D, usually defined using boundary representation, that encloses an $n$D subspace.
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\item[congruent]
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having the same shape and size.
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\item[cube]
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1.\ ($\rightarrow$\ same as $3$-\emph{cube}) 3D polyhedron with 6 square facets;
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2.\ ($\rightarrow$\ usually $n$-\emph{cube}) $n$-orthotope with identical $(n-1)$-cube facets, akin to a square in 2D and a cube in 3D.
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\item[cuboid]
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1.\ box-shaped 3D polyhedron with 6 rectangular facets, its opposite facets are congruent and parallel;
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{\color{gray} 2.\ parallelotope}.
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\item[curve]
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1.\ 1D geometric object, often with a non-linear geometry;
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2.\ 1D combinatorial element;
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3.\ 1-manifold.
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\item[cut-line]
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degenerate part of a 2D shape forming a curve and protruding inwards from its boundary.
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\item[disk]
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topological definition of the space within a circle, often defined as the space within a certain distance (the radius) from a point in $\mathbb{R}^2$ (the centre).
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\item[edge]
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1D combinatorial element.
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\item[face]
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1.\ 2D combinatorial element;
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2.\ ($\rightarrow$\ sometimes \emph{$n$-face of $X$}) $n$D combinatorial element on the boundary of $X$;
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{\color{gray} 3.\ ($\rightarrow$\ usually \emph{face of $X$}) $(n-1)$D combinatorial element on the boundary of an $n$D combinatorial element $X$}.
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\item[facet]
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1.\ $(n-1)$D combinatorial element in an $n$D context;
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2.\ ($\rightarrow$\ often \emph{facet of $X$}) $(n-1)$D combinatorial element on the boundary of an $n$D combinatorial element $X$;
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{\color{gray} 3.\ 2D combinatorial element}.
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\item[flat]
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unbounded geometric object with linear geometry of any dimension, such as a point, line or plane.
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\item[hole]
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1.\ 2D void region;
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2.\ $n$D void region.
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\item[hyperball]
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higher-dimensional ball.
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\item[hypercell]
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1.\ higher-dimensional combinatorial element;
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2.\ 4D combinatorial element.
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\item[hypercube]
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1.\ higher-dimensional cube;
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2.\ 4-cube.
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\item[hyperplane]
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1.\ $(n-1)$D linear subspace in an $n$D context;
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{\color{gray} 2.\ plane in a higher-dimensional context.}
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\item[hyperrectangle]
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higher-dimensional orthotope.
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\item[hypersphere]
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higher-dimensional sphere.
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\item[hypersurface]
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$(n-1)$D subspace in an $n$D context, often curved;
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\item[interval]
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topological definition of the space between two points in $\mathbb{R}$ (the endpoints).
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\item[Lebesgue measure]
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measure of the $n$D extent of an object, akin to length in 1D, area in 2D and volume in 3D.
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\item[length]
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measure of the 1D extent of an object.
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\item[line segment]
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1D geometric object.
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\item[manifold]
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($\rightarrow$\ sometimes $n$-\emph{manifold}) topological space that resembles $\mathbb{R}^n$ at every point.
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\item[manifold with boundary]
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($\rightarrow$\ sometimes $n$-\emph{manifold with boundary}) topological space that resembles $\mathbb{R}^n$ at every point in its interior.
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\item[node]
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0D combinatorial element.
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\item[open]
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1.\ topological definition of an object that does not include its boundary;
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2.\ geometric definition of an object in $n$D, usually defined using boundary representation, that does not enclose any $n$D subspace.
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\item[orthotope]
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($\rightarrow$\ sometimes $n$-\emph{orthotope}) $n$-polytope with congruent parallel facets forming right angles to each other, akin to a rectangle in 2D and a cuboid in 3D.
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\item[parallelepiped]
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3D polyhedron bounded by 6 parallelogram-shaped 2D faces.
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\item[parallelogram]
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polygon bounded by 4 parallel and have the same shape.
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\item[parallelotope]
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a polytope whose opposite facets are parallel and have the same shape, akin to a parallelogram in 2D and a parallelepiped in 3D.
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\item[peak]
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1.\ $(n-3)$D combinatorial element in an $n$D context;
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2.\ ($\rightarrow$\ \emph{peak of $X$}) $(n-3)$D combinatorial element on the boundary of an $n$D combinatorial element $X$;
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\item[plane]
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2D unbounded linear subspace.
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\item[point]
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1.\ 0D geometric element;
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2.\ topological definition of the space at one location.
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\item[polychoron]
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4D geometric object with linear geometry.
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\item[polygon]
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2D geometric object with linear geometry, possibly with holes.
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\item[polygonal curve]
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curve formed by a sequence of line segments joined at their endpoints.
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\item[polyhedron]
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1.\ 3D geometric object with linear geometry, possibly with holes;
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{\color{gray} 2.\ an $n$D geometric object with linear geometry}.
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\item[polyline]
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curve formed by a sequence of line segments joined at their endpoints.
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\item[polyteron]
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5D geometric object with linear geometry.
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\item[polytope]
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$n$D geometric object with linear geometry.
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\item[puncture]
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0D (point) hole, often formed from a degenerate polyline or polygon.
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\item[ridge]
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1.\ $(n-2)$D combinatorial element in an $n$D context;
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2.\ ($\rightarrow$\ \emph{ridge of $X$}) $(n-2)$D combinatorial element on the boundary of an $n$D combinatorial element $X$.
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\item[line string]
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curve formed by joining a sequence of vertices by straight line segments, represented by these vertices.
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\item[linear ring]
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2D geometric object with linear geometry represented as a sequence of vertices.
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\item[ring]
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2D combinatorial element represented by its 1D boundary.
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\item[subfacet]
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1.\ $(n-2)$D combinatorial element in an $n$D context;
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2.\ ($\rightarrow$\ \emph{subfacet of $X$}) $(n-2)$D combinatorial element on the boundary of an $n$D combinatorial element $X$.
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\item[shell]
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3D geometric object with linear geometry without 3D holes, usually defined as the volume enclosed by its 2D boundary, often represented as a set of 2D faces.
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\item[simplex]
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1.\ ($\rightarrow$\ sometimes \emph{$n$-simplex}) combinatorial element with $n+1$ vertices and $n+1$ facets;
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2.\ ($\rightarrow$\ sometimes \emph{$n$-simplex}) geometric object based on $n+1$ affinely independent vertices, akin to a point in 0D, a line segment in 1D, a triangle in 2D, or a tetrahedron in 3D.
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\item[simplicial complex]
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1.\ ($\rightarrow$\ sometimes \emph{abstract simplicial complex}) topological space formed by a set of simplices glued along common faces, all faces of a simplex in the complex should be in the complex;
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2.\ ($\rightarrow$\ sometimes \emph{geometric simplicial complex}) abstract simplicial complex where simplices are embedded into Euclidean space, their interiors should not intersect geometrically.
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\item[solid]
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1.\ 3D geometric object with linear geometry, possibly with 3D holes, often represented as a set of shells;
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2.\ object defined based on solid modelling, often as opposed one based on boundary representation;
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3.\ an object containing its interior (as opposed to \emph{hollow}).
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\item[spike]
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degenerate part of a 2D shape forming a curve, often protruding outwards from its boundary.
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\item[sphere]
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1.\ ($\rightarrow$\ same as $2$-\emph{sphere}) topological definition of the space at a certain distance (the radius) from a point in $\mathbb{R}^3$ (the centre);
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2.\ ($\rightarrow$\ often $n$-\emph{sphere}) topological definition of the space at a certain distance (the radius) from a point in $\mathbb{R}^{n+1}$ (the centre);
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3.\ geometric definition of a perfectly round hollow 3D object.
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\item[surface]
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1.\ 2-manifold;
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2.\ 2D combinatorial element.
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\item[tesseract]
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4-cube: polychoron with 8 cubical facets.
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\item[vertex]
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0D combinatorial element.
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\item[volume]
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1.\ measure of the 3D extent of an object;
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2.\ 3D combinatorial element.
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\item[wire]
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2D combinatorial element represented by its 1D boundary.
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\end{description}
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\end{multicols}
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% \begin{table}
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% \scriptsize
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% \begin{tabular}{ccccccc}
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% \toprule
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% \multicolumn{7}{c}{dimension} \\
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% $n$ & 0 & 1 & 2 & 3 & 4 \\
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% \midrule
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% \multicolumn{7}{c}{topological} \\
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% $n$-simplex & vertex & edge & face & volume \\
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% $n$-cell & vertex & edge & face & volume \\
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% $n$-sphere & pair of points & circle & sphere \\
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% $n$-ball & & interval & disk & ball \\
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% open $n$-ball & & open interval & open disk & open ball \\
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% $n$-manifold & discrete space & curve & surface \\
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% \midrule
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% \multicolumn{7}{c}{geometric} \\
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% $n$-simplex & point & line segment & triangle & tetrahedron & pentachoron \\
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% $n$-cube & point & square & cube & tesseract \\
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% $n$-orthotope & point & line segment & rectangle & cuboid/box \\
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% $n$-parallelotope & point & line segment & parallelogram & parallelepiped \\
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% prismatic $n$-polytope & & line segment & rectangle & prism & prismatic polychoron \\
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% $n$-polytope & point & line segment & polygon & polyhedron & polychoron \\
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% \midrule
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% \multicolumn{7}{c}{other} \\
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% $n$-th Lebesgue measure & length & area & volume \\
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% & & collinear & coplanar \\
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% \bottomrule
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% \end{tabular}
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% \end{table}
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