Tetrahedral Graph

Author

Eric W. Weisstein
November 27, 2023

This notebook downloaded from http://mathworld.wolfram.com/notebooks/GraphTheory/TetrahedralGraph.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/TetrahedralGraph.html.

©2023 Wolfram Research, Inc. except for portions noted otherwise

Platonic graph embeddings

Names

GraphData["TetrahedralGraph","StandardName"]

GraphData["TetrahedralGraph","StandardNames"]

Primary

GraphData["TetrahedralGraph"]//StyleGraphs[#,ImageSize->Small]&

Construction from Read and Wilson

All

GraphData["TetrahedralGraph","Graph","All"]//StyleGraphs

Embeddings

GraphDataEmbeddings["TetrahedralGraph"]

Degenerate

GraphData["TetrahedralGraph","Graph","Degenerate"]//StyleGraphs

IntegerCoordinates

GraphData["TetrahedralGraph","Graph","IntegerCoordinates"]//StyleGraphs

Integral

IntegralDrawing/@GraphData["TetrahedralGraph","Graph","Integral"]//StyleGraphs

LCF

GraphData["TetrahedralGraph","Graph","LCF"]//StyleGraphs

MinimalCrosssing

GraphData["TetrahedralGraph","Graph","MinimalCrossing"]//StyleGraphs

MinimalIntegral

IntegralDrawing/@GraphData["TetrahedralGraph","Graph","MinimalIntegral"]//StyleGraphs

MinimalPlanarIntegral

IntegralDrawing/@GraphData["TetrahedralGraph","Graph","MinimalPlanarIntegral"]//StyleGraphs

Construction

ShowLabeledGraph[g=Graph[List/@{{1,2},{2,3},{1,3},{3,4},{1,4},{2,4}},List/@{{-8,0},{8,0},{0,6},{0,15}}]]

FullSimplify[Norm[Subtract@@Vertices[g][[#]]]]&/@Edges[g]

RecognizeGraph[g]

IntegralDrawing[g]

ToCommonEdges[g,"TetrahedralGraph"]

Perspective

GraphData["TetrahedralGraph","Graph","Perspective"]//StyleGraphs

Planar

GraphData["TetrahedralGraph","Graph","Planar"]//StyleGraphs

3-D

GraphData["TetrahedralGraph","Graph","3D"]//StyleGraphs

Projections

Manipulate[PolyhedronProjectionGraph[PolyhedronData["Tetrahedron","Polyhedron"],{a,b,c}],{a,-1,1},{b,-1,1},{c,-1,1}]

{0, 0, 1}

PolyhedronProjectionGraph[PolyhedronData["Tetrahedron","Polyhedron"],{0,0,1},VertexLabels->"Name"]

{0, 1, 0}

PolyhedronProjectionGraph[PolyhedronData["Tetrahedron","Polyhedron"],{0,1,0},VertexLabels->"Name"]

{1, 0, 0}

PolyhedronProjectionGraph[PolyhedronData["Tetrahedron","Polyhedron"],{1,.00001,0},VertexLabels->"Name"]

PolyhedronProjectionGraph[PolyhedronData["Tetrahedron","Polyhedron"],{1,z,0},VertexLabels->"Name"]

Lookup[Options[%],VertexCoordinates]//FullSimplify[#,z>0]&

Limit[%,z->0]

SetProperty[PolyhedronProjectionGraph[PolyhedronData["Tetrahedron","Polyhedron"],{1,.00001,0},VertexLabels->"Name"],VertexCoordinates->{{0,-((Sqrt[(3/2)])/2)},{(1/2),(1/(2 (Sqrt[6])))},{-(1/2),(1/(2 (Sqrt[6])))},{0,(1/(2 (Sqrt[6])))}}]

{1, 1, -1}

PolyhedronProjectionGraph[PolyhedronData["Tetrahedron","Polyhedron"],{1,1,-1},VertexLabels->"Name"]

{(Sqrt[2]),0,1}

PolyhedronProjectionGraph[PolyhedronData["Tetrahedron","Polyhedron"],{(Sqrt[2]),0,1},VertexLabels->"Name"]

Stereoscopic projections

PolyhedronData["Tetrahedron","Orientations"]

Quiet@GraphPlot[SkeletonGraph[PolyhedronData[{"Tetrahedron","C2"},"Faces"]//Normal,#],Method->None]&/@Range[.2,1.5,.1]

GraphPlot[SkeletonGraph[PolyhedronData[{"Tetrahedron","C3"},"Faces"]//Normal,#],Method->None]&/@Range[.2,2,.1]

Embeddings

Minimal Integral Drawing

h=Sqrt[(2^2)-(1/4)(((4-3))^2)]

ShowLabeledGraph[g=ChangeVertices[Graph["TetrahedralGraph"],v={{-2,0},{2,0},{-3/2,h},{3/2,h}}]]

v=Vertices[g];
Show[{GraphPlot[g,Method->None],Graphics[Text[" "<>ToString[Norm[Subtract@@v[[#]]]]<>" ",Mean[v[[#1]]],Background->White]&/@Edges[g]]}]

Norm/@Subtract@@@(Edges[g]/.Thread[Range[4]->Vertices[g]])

ToCommonEdges[g,"TetrahedralGraph"]

GetGraphData["TetrahedralGraph","File"]

Properties

Matrices

GraphicsRow[GraphMatrixPlot[GraphData["TetrahedralGraph"]]]

Properties

TableForm[GraphDataTable["TetrahedralGraph",x],TableDepth->2]//TraditionalForm

GraphData

GraphDataClassString

GraphDataClassString["TetrahedralGraph"]