Example Graphs and Styles

[LLM Generated]

Ready-made example substrates and ambient styles used across the Infra* project notebooks (SyntheticInfrageometry, InfraCausality, InfraGaugeTheory, InfraAnalysis). All generators and styles are defined in the initialization cells of the Definitions section; each gallery cell below renders one substrate type at the three sizes "Small", "Medium", "Large" with the matching ambient styles, ready to copy.

Definitions

Install the Infrageometry paclet from the Wolfram Cloud; it supplies the mesh, tessellation, and uniform-length generators.

PacletInstall["https://www.wolframcloud.com/obj/hajek_pavel/Infrageometry.paclet",ForceVersionInstall->True];Needs["WolframInstitute`Infrageometry`"]

Ambient styles render a substrate as a faint backdrop so that highlighted constructions stand out; splice one into any generator through its Graph options.

ambientGraphStyles=<| "Default"->{}, "GrayFaint"->{EdgeStyle->Directive[StandardGray,Opacity[0.15]],VertexStyle->Directive[StandardGray,Opacity[0.3]]}, "GrayOpaque"->{EdgeStyle->Directive[StandardGray,Opacity[0.4]],VertexStyle->Directive[StandardGray,Opacity[0.6]]}, "Gray"->{EdgeStyle->StandardGray,VertexStyle->StandardGray}|>;

A triangulated rectangle with only interior vertices kept; the discrete stand-in for a flat patch of the plane.

rectangleGraph[size_,opts : OptionsPattern[Graph]]:=(SeedRandom[1]; InteriorMeshGraph[ DiscretizeRegion[Rectangle[], MaxCellMeasure->(size /. {"Small"->0.02,"Medium"->0.008,"Large"->0.003}), PrecisionGoal->Infinity],opts]);

Square and cubic grids restricted to their interior vertices.

gridGraph[size_,opts : OptionsPattern[Graph]]:=With[ {n=size /. {"Small"->7,"Medium"->12,"Large"->25}}, InteriorGraph[GridGraph[{n,n}],opts]];gridGraph3D[size_,opts : OptionsPattern[Graph]]:=With[ {n=size /. {"Small"->4,"Medium"->10,"Large"->14}}, InteriorGraph[GridGraph[{n,n,n}],opts]];

Final-state hypergraphs of Wolfram-model rewriting, flattened to undirected graphs; four built-in rules of varying homogeneity.

wolframUniverseHypergraph[size_String,opts : OptionsPattern[Graph]]:=wolframUniverseHypergraph[1,size,opts];wolframUniverseHypergraph[i_Integer,size_,opts : OptionsPattern[Graph]]:=wolframUniverseHypergraph[i,size,opts]=With {rule=<| 1->{{x,y},{x,z}}->{{x,z},{x,w},{y,w},{z,w}}, 2->{{x,y},{z,y}}->{{x,y},{z,w},{y,w},{x,w}}, 3->{{1,2,2},{1,3,4}}->{{4,5,5},{5,3,2},{1,2,5}}, 4->{{1,1,2},{1,3,4}}->{{4,4,3},{2,5,3},{2,5,3}}|>[i], init=<|1->{{0,0},{0,0}},2->{{1,1},{1,1}},3->{{0,0,0},{0,0,0}},4->{{0,0,0},{0,0,0}}|>[i], gen=size /. <| 1->{"Small"->6,"Medium"->8,"Large"->13}, 2->{"Small"->8,"Medium"->10,"Large"->15}, 3->{"Small"->100,"Medium"->500,"Large"->1000}, 4->{"Small"->200,"Medium"->500,"Large"->1000}|>[i]}, evolved=

[◼]	WolframModel

[rule,init,gen,"FinalState"], UndirectedGraph

[◼]	HypergraphToGraph

[evolved],opts;

Regular and uniform tessellations by Schlafli-type symbol: planar, hyperbolic, and Archimedean.

tesselatedSurface[type_,size_,opts : OptionsPattern[Graph]]:=TessellationGraph[ type /. {"Triangular"->{3,6},"Square"->{4,4},"Hexagonal"->{6,3},"Hyperbolic"->{3,7}, "Trihexagonal"->{3,6,3,6},"TruncatedSquare"->{4,8,8},"RhombiTrihexagonal"->{3,4,6,4},"TruncatedTrihexagonal"->{4,6,12}}, size /. (type /. { "Triangular"->{"Small"->{8,8},"Medium"->{16,16},"Large"->{20,15}}, "Square"->{"Small"->{8,8},"Medium"->{16,16},"Large"->{20,15}}, "Hexagonal"->{"Small"->{8,8},"Medium"->{16,16},"Large"->{20,15}}, "Hyperbolic"->{"Small"->1,"Medium"->2,"Large"->3}, "Trihexagonal"->{"Small"->5,"Medium"->5,"Large"->5}, "TruncatedSquare"->{"Small"->5,"Medium"->5,"Large"->5}, "RhombiTrihexagonal"->{"Small"->5,"Medium"->5,"Large"->5}, "TruncatedTrihexagonal"->{"Small"->5,"Medium"->5,"Large"->5}}),opts];

Combinatorial balls of a tessellation around a vertex: flat, spherical, or hyperbolic local geometry.

tessellatedNeighborhood[type_,size_,opts : OptionsPattern[Graph]]:=tessellatedNeighborhood[type,size,opts]=InteriorGraph[ TessellationNeighborhoodGraph[ type /. {"Square"->{4,4},"Triangular"->{3,6},"Hexagonal"->{6,3},"Hyperbolic"->{3,7},"Spherical"->{3,5}}, size /. {"Small"->3,"Medium"->4,"Large"->5}],opts];

Meshes whose edges all have approximately the same Euclidean length, on a disc, an ellipsoid, and a hyperboloid; curvature regimes flat, positive, and negative.

uniformLengthDisc[size_]:=uniformLengthDisc[size]=UniformLengthGraph[Disk[], size /. {"Small"->100,"Medium"->300,"Large"->600}];uniformLengthEllipsoid[regime_,size_]:=uniformLengthEllipsoid[regime,size]=UniformLengthGraph[ BoundaryDiscretizeRegion[Ellipsoid[{0,0,0},regime /. {"Round"->{1,1,1},"Protruse"->{1,1,3},"Triaxial"->{1,2,3}}]], size /. {"Small"->100,"Medium"->300,"Large"->600}];uniformLengthHyperboloid[regime_,size_]:=uniformLengthHyperboloid[regime,size]=With[ {abc=regime /. {"Round"->{1,1,1},"Protruse"->{1,1,2},"Triaxial"->{1,2,2}}}, UniformLengthGraph[ DiscretizeRegion[ImplicitRegion[x^2/abc[[1]]^2+y^2/abc[[2]]^2-z^2/abc[[3]]^2==1&&-2<=z<=2,{x,y,z}]], size /. {"Small"->100,"Medium"->300,"Large"->600}]];

Meshed regions

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Throughout the gallery, sizes pair with styles: "Small" with "Gray", "Medium" with "GrayOpaque", "Large" with "GrayFaint".

Row[{Labeled[rectangleGraph["Small",Sequence@@ambientGraphStyles["Gray"],ImageSize->180],Text["Small"]],Labeled[rectangleGraph["Medium",Sequence@@ambientGraphStyles["GrayOpaque"],ImageSize->240],Text["Medium"]],Labeled[rectangleGraph["Large",Sequence@@ambientGraphStyles["GrayFaint"],ImageSize->300],Text["Large"]]}]

Small

Medium

Large

Grids

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The interior square grid and its cubic cousin.

Row[{Labeled[gridGraph["Small",Sequence@@ambientGraphStyles["Gray"],ImageSize->180],Text["Small"]],Labeled[gridGraph["Medium",Sequence@@ambientGraphStyles["GrayOpaque"],ImageSize->240],Text["Medium"]],Labeled[gridGraph["Large",Sequence@@ambientGraphStyles["GrayFaint"],ImageSize->300],Text["Large"]]}]

Small

Medium

Large

Wolfram-model universes

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One cell per rewriting rule: rules 1 and 2 grow homogeneous binary universes, rules 3 and 4 sparser ternary ones.

Tessellated surfaces

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Flat tessellations by triangles, squares, and hexagons, plus the hyperbolic {3, 7} surface.

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The four Archimedean tessellations come at one fixed patch size.

Tessellated neighborhoods

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Combinatorial balls with flat, hyperbolic, or spherical local geometry.

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The spherical {3, 5} ball closes up into the icosahedron at every size.

Uniform-length meshes

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Meshes with edges of equal Euclidean length; the disc is flat, the ellipsoid positively curved, the hyperboloid negatively curved.

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The generators take no style options; restyle by wrapping in Graph.

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Ellipsoid regimes "Round", "Protruse", "Triaxial"; one cell per regime.

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Hyperboloid regimes "Round", "Protruse", "Triaxial"; one cell per regime.

Learn more

These substrates pair with the Infra* paclets: SyntheticInfrageometry, InfraCausality, InfraGaugeTheory, InfraAnalysis, and Infrageometry itself. All are actively developed, experimental research code.