Power set unions

Symbolic models for discrete spaces

Karen Dawn Colins

Atomic Energy of Canada Limited

Foundations for understanding the geometry of space

Continuum models have been the traditional foundation

Discrete models are a redesigned foundation, based on collections of discrete quanta: points, line segments, triangles, tetrahedra, and so on

"That scientific experience is not just heaped up as it comes but is built by combining the data of sense according to the plan of reason, was first clearly stated by Kant.  Although Kant knew full well that science has a history and even gave the exact dates at which several of its main branches came of age, he apparently never thought that reason's plan may undergo mutations, that the building of experience may have its very foundations redesigned while it is still being worked on."

Roberto Torretti, 1983

Specialization:  simpler and easier

Playing with simple models helps us understand the complex nature of space

Discrete models are among the simplest (and fun to play with)

"In dealing with mathematical problems, specialization plays, as I believe, a still more important part than generalization. Perhaps in most cases where we seek in vain the answer to a question, the cause of the failure lies in the fact that problems simpler and easier than the one in hand have been either not at all or incompletely solved. All depends, then, on finding out these easier problems, and on solving them by means of devices as perfect as possible and of concepts capable of generalization."

David Hilbert, 1900

Timeline of discrete models of space

1758:  Leonard Euler.  Proof of some Notable Properties with which Solids Enclosed by Plane Faces are Endowed.  Novi Commentarii Academiae Scientarium Petropolitanae.

1854:  Bernhard Riemann.  On the Hypotheses Which Lie at the Bases of Geometry.  Nature (Translation by WK Clifford, 1873).

1961:  T. Regge.  General Relativity without Coordinates.  Il Nuovo Cimento.

1999:  JW Barrett, M Rocek, RM Williams.  A Note on Area Variables in Regge Calculus.  Classical and Quantum Gravity.  (Areas of two-simplexes, instead of lengths of edges, used as dynamical variables of Regge calculus.)

Continuum models assume that points are regular

Of the lines passing through a regular point on a surface, the line having maximum curvature and the line having minimum curvature meet at a right angle

Here is a function giving a surface in which the point at the origin is regular only for a special case.

surface[n_]:=RevolutionPlot3D[(E^(-(Cos[2θ+n Cos[2θ]])(t^2))),{t,0,1},{θ,0,2π}];
Table[surface[n],{n,{0,1,2,3}}]

In the first plot, for the point at the origin, the two lines having minimum and maximum curvature meet at right angles; the assumption holds and the origin is a regular point

The last 3 plots show counterexamples to the assumption, demonstrating that continuum models aren't sufficiently general

Discrete models assume that points are vertices

Above is a random triangle tesselation of a region of a flat surface.  The degree of a vertex is the number of edges that meet at the vertex.

vertexDegrees={5,5,7,5,8,8,4,5,6,6,6,6,8,6,6,5,5,6,5,6};

Mean[vertexDegrees]//N

Terminology for discrete models

A polytope is the the general term of the sequence "point, line segment, polygon, polyhedron, ...,"

An n-simplex is the simplest polytope in n-dimensional space

Grid[{{"Dimension, n","Polytope","n-simplex",SpanFromLeft},
{0,"Point","Point",Graphics[{PointSize[Large],Point[{0.0,0.5}]},ImageSize->{120,40}]},
{1,"Line Segment","Line Segment",Graphics[{Thick,Blue,Line[{{0,0},{20,0}}],Black,PointSize[Large],Point[{0,0}],Point[{20,0}]},ImageSize->{120,40}]},
{2,"Polygon","Triangle",Graphics[{EdgeForm[Thick],Red,Polygon[{{1,0},{0,Sqrt[3]},{-1,0}}]},ImageSize->120]},
{3,"Polyhedron","Tetrahedron",Graphics3D[{EdgeForm[{Thickness[0.014
]}],PolyhedronData["Tetrahedron","Faces"]}]},
{4,"Polychoron","Pentachoron","⋯"},
{5,"⋯","⋯","⋯"}},Frame->All]

The great thing about n-simplexes is that for n>1, there is a relation between the content of the simplex and the lengths of its edges (not true of all polytopes)

Simplicial complexes are two or more distinct simplexes joined at one or more common points, line segments, triangles, tetrahedrons …

Simplicial complexes are models of discrete spaces

New concept:  power sets as symbolic simplexes

The power set of a given set is the set of all its subsets

verts[v_Integer]:=RandomSample[
CharacterRange["a","z"]⋃CharacterRange["A","H"]⋃CharacterRange["J","Z"]⋃CharacterRange["α","ξ"]⋃{"ρ"}⋃
CharacterRange["σ","ω"]⋃{"ϑ","ϒ","ϕ","ϖ"}⋃{"ϟ","ϡ"}⋃{"Γ","Δ","Θ","Ι","Λ","Ξ","Π","Σ","Τ"}⋃{"Φ","Ψ","Ω","℧"}⋃CharacterRange["ℵ","ℸ"]⋃{"ל","ץ"}⋃{"⌘","Þ"},v
];
graph2subset[arbitrarySpace_List]:=GraphPlot[Apply[Rule,Cases[arbitrarySpace,{_,_}],{1}],VertexLabeling->True];
Module[{s0,s1,s2,s3,ksets,simplexes,simgraphs},
{s0,s1,s2,s3}=Table[verts[n],{n,0,3}];
ksets={s0,s1,s2,s3};
simplexes=Subsets[#]&/@ksets;
simgraphs=Table["",{4}];
simgraphs[[2]]=Show[GraphPlot[{s1[[1]]->s1[[1]]},SelfLoopStyle->None,VertexLabeling->True],ImageSize->{185,40}];
simgraphs[[3]]=Show[graph2subset[simplexes[[3]]],Graphics[Text[simplexes[[3,4]],{0.5,0.05}]]];
simgraphs[[4]]=Show[graph2subset[simplexes[[4]]],
Graphics[Text[simplexes[[4,7]],{0.5,0.93}]],
Graphics[Text[simplexes[[4,6]],{0.1,0.45}]],
Graphics[Text[simplexes[[4,5]],{0.9,0.45}]],
Graphics[Text[simplexes[[4,8]],{0.5,0.55}]]
];
Grid[Append[Prepend[Transpose[{Range[0,3],ksets,simplexes,simgraphs}],{"n","Set","Power Set","Simplex"}],{"⋯","⋯","⋯","⋯"}],Frame->All]
]

There is a one-to-one correspondence between the elements of a power set and the elements of a simplex

Therefore we may define a symbolic k-simplex as the power set of a set of n=k+1 elements

New concept:  power set unions as symbolic discrete spaces

Define a symbolic discrete space as the union of the respective power sets of the elements (Subscript[Ψ, i]) of a set Ψ.

space[setOfSets_List]:=Apply[Union,Map[Sort,Subsets[#]]&/@setOfSets]

Tools for playing with space[□]

A function to generate a list of random symbols of v vertices:

verts[v_Integer]:=RandomSample[
CharacterRange["a","z"]⋃CharacterRange["A","H"]⋃CharacterRange["J","Z"]⋃CharacterRange["α","ξ"]⋃{"ρ"}⋃
CharacterRange["σ","ω"]⋃{"ϑ","ϒ","ϕ","ϖ"}⋃{"ϟ","ϡ"}⋃{"Γ","Δ","Θ","Ι","Λ","Ξ","Π","Σ","Τ"}⋃{"Φ","Ψ","Ω","℧"}⋃CharacterRange["ℵ","ℸ"]⋃{"ל","ץ"}⋃{"⌘","Þ"},v
]

A function that graphs the 2-subsets of an arbitrary space:

graph2subset[arbitrarySpace_List]:=GraphPlot[Apply[Rule,Cases[arbitrarySpace,{_,_}],{1}],VertexLabeling->True]

Some symbolic discrete spaces generated by space[□]

space[a particular set of 4-sets of 5 elements]

triangularDipyramid=space[{{"Ω","y","Y","C"},{"Ω","y","C","k"}}]
graph2subset[triangularDipyramid]

space[#]& /@, {all possible sets of 2-sets of 4 elements}

Clear[Ψ,Σ,x,y]
Ψ[v_Integer,t_Integer]:=Module[{x,y},
x=verts[v];
y=Subsets[Subsets[x,{t}],{2,Binomial[v,t]}];
Select[y,Apply[Union,#]==Union[x]&]
];
Module[{psi,sig},psi=Ψ[4,2];sig=Map[space,psi];
Grid[Transpose[{psi,sig,Map[graph2subset,sig]}],Frame->All]]

space[a particular set of 4-sets of 12 elements], giving the simplest genus-1 surface in 3D

genus1space=space[{{"a","b","c","d"},{"b","c","d","e"},{"c","e","f","d"},
{"c","g","e","f"},{"e","f","g","h"},{"e","h","g","j"},
{"h","i","j","g"},{"g","j","i","k"},{"i","j","k","l"},
{"b","j","k","l"},{"b","a","k","l"},{"c","b","k","a"}}]

graph2subset[genus1space]

space[a trivial set of one 3-set of 3 elements]

triangleSpace=space[{verts[3]}]
graph2subset[triangleSpace]

New discoveries based on patterns of vertex cardinalities in discrete spaces

Average vertex degree for a triangle-tiled surface of arbitrary genus

(Subscript[(OverBar[v]), 2])=6+((12(g-1))/v),

where v is the number of vertices, g is the genus of the surface, and v≥3.

Note that for a genus 1 surface, (Subscript[(OverBar[v]), 2]) does not depend on v.  In fact, the value is (Subscript[(OverBar[v]), 2])=6, the same as for an infinite plane.

For g=0, and v∈{4,6,12}, the formula yields (Subscript[(OverBar[v]), 2])∈{3,4,5}, respectively, corresponding to the tetrahedron, octahedron, and icosahedron.  In the limit as v->∞, (Subscript[(OverBar[v]), 2])->6.

Average vertex degree for a tetrahedron-tiled 3D space of arbitrary genus

(Subscript[(OverBar[v]), 3])=12+((40(g-1))/v),

where v is the number of vertices, g is the genus, and v≥5.  In the limit as v->∞, (Subscript[(OverBar[v]), 3])->12.

Summary

A union S of the respective power sets of the elements (Subscript[Ψ, i]) of a set Ψ is a symbolic model for a discrete space

S and the associated set of integer-number values of its vertex degrees constitute a quantized "symbolic geometry" for a discrete space

Assigning a set of real-number values to the 2-subsets of S then provides the "metric geometry"