Wolfram Research

Function Repository Resource:

SimplexBoundary

Source Notebook

Find the topological boundary of a simplex or simplicial complex

Contributed by: Richard Hennigan (Wolfram Research)

ResourceFunction["SimplexBoundary"][simplex]

finds the topological boundary of simplex.

ResourceFunction["SimplexBoundary"][{simplex1,simplex2,}]

finds the topological boundary of the simplicial complex containing simplex1,simplex2,.

ResourceFunction["SimplexBoundary"][mesh]

finds the topological boundary of the MeshRegion mesh.

ResourceFunction["SimplexBoundary"][complex,k]

finds the topological boundary of the k-skeleton of complex.

Details and Options

A simplex can be considered any of the following:
Point[v] point
Line[{v1,v2}]line segment
Triangle[{v1,v2,v3}] or Polygon[{v1,v2,v3}]filled triangle
Tetrahedron[{v1,v2,v3,v4}] filled tetrahedron
Simplex[{v1,v2,,vn}] an (n-1)-dimensional simplex
A simplicial complex is either a list of simplices or a MeshRegion where all cells are valid simplices.
The boundary preserves orientations so that it can be used for simplicial homology computations.

Examples

Basic Examples (4) 

Retrieve the ResourceFunction:

In[1]:=
ResourceFunction["SimplexBoundary"]
Out[1]=

Get the boundary of a Simplex:

In[2]:=
ResourceFunction["SimplexBoundary"][Simplex[{1, 2, 3}]]
Out[2]=

Get the boundary of a simplicial complex, represented as a list of simplices:

In[3]:=
ResourceFunction[
 "SimplexBoundary"][{Simplex[{1, 2, 3}], Simplex[{1, 3, 4}]}]
Out[3]=

Get the boundary of a MeshRegion:

In[4]:=
mesh = MeshRegion[{{0, 0}, {1, 0}, {0, 1}, {1, 1}, {2, 1}, {2, 0}}, Polygon[{{1, 2, 3}, {4, 5, 6}}]]
Out[4]=
In[5]:=
ResourceFunction["SimplexBoundary"][mesh]
Out[5]=

Scope (2) 

By default, the boundary will be computed from simplices that match the dimension of the simplicial complex:

In[6]:=
ResourceFunction[
 "SimplexBoundary"][{Point[1], Line[{2, 3}], Triangle[{4, 5, 6}]}]
Out[6]=

A different dimension k can be specified, which will find the boundary of the k-skeleton of the complex:

In[7]:=
ResourceFunction[
 "SimplexBoundary"][{Point[1], Line[{2, 3}], Triangle[{4, 5, 6}]}, 1]
Out[7]=

Generalizations and Extensions (1) 

Some other primitives can represent a simplex as well:

In[8]:=
ResourceFunction["SimplexBoundary"][
 Triangle[{{0, 0}, {1, 0}, {0, 1}}]]
Out[8]=
In[9]:=
Graphics[%]
Out[9]=

Applications (2) 

Find holes in a 3D model:

In[10]:=
mesh = ResourceData["Stanford Bunny", "MeshRegion"]
Out[10]=
In[11]:=
boundary = ResourceFunction["SimplexBoundary"][mesh]
Out[11]=

Highlight the boundary edges:

In[12]:=
Show[HighlightMesh[mesh, Style[boundary, Red]], ViewPoint -> {-2.25, -0.5, -2.5}, ViewVertical -> {-1, 0, 0}]
Out[12]=

Properties and Relations (3) 

Orientation is preserved:

In[13]:=
pts = {{0, 0, 0}, {1, 0, 0}, {0, 1, 0}, {0, 0, 1}};
boundary = ResourceFunction["SimplexBoundary"][Tetrahedron[{1, 2, 3, 4}]];
insideOutBoundary = ResourceFunction["SimplexBoundary"][Tetrahedron[{2, 1, 3, 4}]];
{Graphics3D[{FaceForm[Red, Blue], GraphicsComplex[pts, boundary]}], Graphics3D[{FaceForm[Red, Blue], GraphicsComplex[pts, insideOutBoundary]}]}
Out[13]=
In[14]:=
pts = {{0, 0}, {1, 0}, {0, 1}};
boundary = ResourceFunction["SimplexBoundary"][Triangle[{1, 2, 3}]];
insideOutBoundary = ResourceFunction["SimplexBoundary"][Triangle[{2, 1, 3}]];
{Graphics[GraphicsComplex[pts, Arrow @@@ boundary]], Graphics[GraphicsComplex[pts, Arrow @@@ insideOutBoundary]]}
Out[14]=

The boundary of a boundary is always empty:

In[15]:=
NestList[ResourceFunction["SimplexBoundary"],
 MeshRegion[{{0, 0}, {1, 0}, {0, 1}, {1, 1}, {2, 1}, {2, 0}}, Polygon[{{1, 2, 3}, {4, 5, 6}}]], 2]
Out[15]=
In[16]:=
Graphics /@ NestList[ResourceFunction["SimplexBoundary"], Triangle[], 2]
Out[16]=
In[17]:=
Graphics /@ NestList[ResourceFunction[
  "SimplexBoundary"], {Simplex[{{0, 0}, {1, 0}, {0, 1}}], Simplex[{{1, 1}, {1, 0}, {0, 1}}]}, 2]
Out[17]=

A graph can be considered a one-dimensional simplicial complex:

In[18]:=
(* Evaluate this cell to get the example input *) CloudGet["https://www.wolframcloud.com/obj/7c3732eb-2f65-4916-9195-aa279a2784fc"]
Out[18]=
In[19]:=
HighlightGraph[g, boundary]
Out[19]=

The boundary can give you information about degree of vertices:

In[20]:=
Sort[boundary] === Sort[Keys@
   Select[AssociationThread[VertexList[g], VertexDegree[g]], # == 1 &]]
Out[20]=

Possible Issues (5) 

Not all graphics primitives are valid simplices:

In[21]:=
arc = Circle[{0, 0}, 1, {Pi/6, 3 Pi/4}];
Graphics[arc]
Out[21]=
Out[21]=

They can often be converted to simplicial complexes using DiscretizeGraphics that are reasonable approximations:

In[22]:=
d = DiscretizeGraphics[arc]
Out[22]=
In[23]:=
Graphics[{arc, Style[MeshPrimitives[ResourceFunction["SimplexBoundary"][d], 0], Red, PointSize[Large]]}]
Out[23]=

Valid n-dimensional simplices must have n+1 vertices:

Out[23]=
In[24]:=
ResourceFunction["SimplexBoundary"][Polygon[{1, 2, 3}]]
Out[24]=

Mesh regions are not necessarily composed of simplices:

In[25]:=
mesh = DiscretizeGraphics[Rectangle[]]
Out[25]=
Out[25]=

TriangulateMesh can often be used to create a valid simplicial complex:

In[26]:=
ResourceFunction["SimplexBoundary"][
 TriangulateMesh[mesh, MaxCellMeasure -> Infinity]]
Out[26]=

Vertices must be unique:

Out[26]=

If orientations are not consistent within a simplicial complex, the boundary will not have consistent orientations, either:

In[27]:=
s = {Simplex[{{0, 0, 0}, {1, 0, 0}, {0, 1, 0}}], Simplex[{{1, 1, 0}, {1, 0, 0}, {0, 1, 0}}]};
Graphics3D[{FaceForm[Red, Blue], s}, BoxRatios -> {1, 1, .1}, Boxed -> False]
Out[27]=
In[28]:=
Graphics3D[Arrow @@@ ResourceFunction["SimplexBoundary"][s], BoxRatios -> {1, 1, .1}, Boxed -> False]
Out[28]=

Interactive Examples (1) 

One can visualize 4D shapes by rotating and projecting the boundary into 3D. Here is an example using a hexadecachoron:

In[29]:=
rotationFunction = Block[{\[Theta]1, \[Theta]2, \[Theta]3, \[Theta]4, \[Theta]5, \[Theta]6}, With[{rotateExpr = Simplify[
       N[(Composition @@ RotationTransform @@@ Transpose[{{\[Theta]1, \[Theta]2, \[Theta]3, \[Theta]4, \[Theta]5, \[Theta]6}, Subsets[IdentityMatrix[4], {2}]}])[{x, y, z, w}]], Reals]},
    Compile[{{r, _Real, 1}, {v, _Real, 1}},
     Block[{\[Theta]1, \[Theta]2, \[Theta]3, \[Theta]4, \[Theta]5, \[Theta]6, x, y, z, w},
      {\[Theta]1, \[Theta]2, \[Theta]3, \[Theta]4, \[Theta]5, \[Theta]6} = r;
      {x, y, z, w} = v;
      rotateExpr
      ],
     CompilationOptions -> {"ExpressionOptimization" -> True},
     RuntimeOptions -> "Speed",
     RuntimeAttributes -> {Listable},
     Parallelization -> True
     ]
    ]];

vertices = Flatten[Table[
    Chop[rotationFunction[r, Prepend[IdentityMatrix[4], {0, 0, 0, 0}]]], {r, Pi*IdentityMatrix[6]}], 1];
hexadecachoron = Table[Simplex[{1, 2, 3, 4, 5} + 5 n], {n, 0, 5}]
Out[29]=
In[30]:=
Manipulate[
 Graphics3D[{EdgeForm[Thick], Opacity[.25], GraphicsComplex[
    rotationFunction[{\[Theta]1, \[Theta]2, \[Theta]3, \[Theta]4, \[Theta]5, \[Theta]6}, vertices][[All, ;; 3]], ResourceFunction["SimplexBoundary"][hexadecachoron]]}, PlotRange -> 1, ViewAngle -> Pi/10, Boxed -> False],
 {{\[Theta]1, 1}, -2 Pi, 2 Pi},
 {{\[Theta]2, 1}, -2 Pi, 2 Pi},
 {{\[Theta]3, 1}, -2 Pi, 2 Pi},
 {{\[Theta]4, 1}, -2 Pi, 2 Pi},
 {{\[Theta]5, 1}, -2 Pi, 2 Pi},
 {{\[Theta]6, 1}, -2 Pi, 2 Pi}
 ]
Out[30]=

Neat Examples (2) 

The boundary of an annulus is two circles, while the boundary of a Moebius strip is a single circle:

In[31]:=
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GrayLevel[0.3], 
ImageScaled[{2, 0, 2}]}, {"Directional", 
GrayLevel[0.33], 
ImageScaled[{2, 2, 2}]}, {"Directional", 
GrayLevel[0.3], 
ImageScaled[{0, 2, 2}]}},
Method->{"ShrinkWrap" -> True},
ViewPoint->{1.3, -2.4, 2.},
ViewVertical->{0., 0., 1.}]\)]}
Out[31]=

Get text outlines:

In[32]:=
ResourceFunction["SimplexBoundary"][
 DiscretizeGraphics[Text["Hello world"], _Text]]
Out[32]=

Requirements

Wolfram Language 11.3 (March 2018) or above

Resource History

Related Resources

License Information