DuganHammock/ ConvexPolytope

Computes convex polytopes in n-dimensions

Contributed by: Dugan Hammock

ConvexPolytope computes the polytope defined by the convex hull of a set of vertices in n-dimensional space.

Installation Instructions

To install this paclet in your Wolfram Language environment, evaluate this code:
PacletInstall["DuganHammock/ConvexPolytope"]

To load the code after installation, evaluate this code:
Needs["DuganHammock`ConvexPolytope`"]

Details

Methods are defined for taking the polytope duals and for taking hyper-planar slices of a polytope.

Examples

Basic Examples

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polytope = PolyhedronData["Dodecahedron", "VertexCoordinates"] // N // ConvexPolytope;
polytope = Join[polytope["Vertices"], Map[Mean, polytope["Facets"]]] // Map[Normalize] // ConvexPolytope;
polytope // Polygon // Graphics3D

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Hexagon as equatorial slice of a dodecahedron:

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polytope = PolyhedronData["Dodecahedron", "VertexCoordinates"] // N // ConvexPolytope;
polytopeSlice = polytope["Slice", {1, 1, 1}, 0.0];

Graphics3D[{
{Opacity[0.5], polytope // Polygon},
{Red, polytopeSlice // Polygon},
{Red, polytopeSlice // Tube[.05]},
}, Boxed -> False]

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4D hypercube (tesseract):

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$Manipulate[ skewMatrix4D = {{1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 0}, {3, 1, 1, 1}}; hypercubeVertices = Tuples[{-1, 1}, {4}] // N // Map[Normalize]; hypercubeVertices = hypercubeVertices . RotationMatrix[\[Theta], {{1, 1, 1, 1}, {0, 0, 0, 1}}]\[Transpose]; hypercubeVertices = hypercubeVertices . skewMatrix4D; hypercubePolytope = hypercubeVertices // N // ConvexPolytope; hypercubeSlicePolytope = hypercubePolytope["Slice", {0, 0, 0, 1}, sliceValue]; Graphics3D[{ Specularity[1, 20], {Gray, hypercubePolytope // Tube[.03]}, If[Positive[hypercubeSlicePolytope["Dimension"]], { {Red, hypercubeSlicePolytope // Polygon}, {Black, hypercubeSlicePolytope // Tube[0.05]}, {Black, hypercubeSlicePolytope // Sphere[0.1]} } ] } , PlotRange -> 3 , Boxed -> False ] , {{sliceValue, 0.}, -1, 1} , {{\[Theta], 1}, 0., Pi/2.} , {skewMatrix4D, None} , {hypercubeVertices, None} , {hypercubePolytope, None} , {hypercubeSlicePolytope, None} , TrackedSymbols :> {sliceValue, \[Theta]} ]$