Combinatorial constructions of intrinsic geometries

04/10/2019 ∙ by Stanislaw Ambroszkiewicz, et al. ∙ 0

A generic method for combinatorial constructions of intrinsic geometrical spaces is presented. It is based on the well known inverse sequences of finite graphs that determine topological spaces. If a pattern of the construction is sufficiently regular and uniform, then the notions of geodesic and curvature can be defined in the space as the limits of their finite versions in the graphs. This gives rise to consider the graphs as finite approximations of the geometry of the space.

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1 Introduction

Blackmore and Peters (2007) [4]: ”Is there a unifying topological abstraction covering manifolds, non–manifolds and other possible geometric models that might be useful to improve algorithmic design for geometric computations?”

The geometric computations process finite combinatorial structures.

It seems that asymptotic finite approximations of geometrical spaces may answer the above question. That is, if a sequence of finite combinatorial structures (graphs) uniformly approximates a geometrical space, then all geometrical aspects of the space are encoded in the construction of the sequence.

It is well known that any compact metric space can be approximated by a sequence of finite polyhedra (i.e. realizations of finite simplicial complexes) since the works of Alexandroff (1937) [1] and Freudenthal (1937) [15].

For a compact metric space and its open covering there is a natural notions of graph denoted and nerve denoted . In both cases, the set of vertices is . An edge of the graph is defined as a pair of elements of with nonempty intersection. Analogously, a simplex of is represented as a subset of elements of such that their intersection is nonempty. For compact spaces, there are finite refinements of the covering, so that the corresponding graphs and simplicial complexes are finite structures.

A sequence of more and more refined uniform coverings along with some consistency conditions are the base to define coarse graphs and coarse simplicial complexes, see so called coarse geometry in Cencelj, Dydak, Vavpetič and Virk (2012) [8], Austin (2015) [3], Jensen [17] (2017), and Grzegrzolka and Siegert (2019) [16]. The coarse geometry is a general framework where coarse graphs and coarse simplicial complexes uniformly approximate metric compact spaces. The topological properties of the spaces are encoded in the coarse graphs and complexes.

A compact metric space can be tessellated (dividing onto uniform closed subsets) instead of being covered by open sets. The difference is that two tessellation elements are either disjoint or are adjacent, i.e. they have a joint part of their borders. The adjacency can be represented as a graph. If the divisions are refined step by step, this leads to the notion of inverse sequence. Along with some consistency condition, the inverse limit of the sequence with Hausdorff reflection gives a space homeomorphic to the original space. This idea was explored by Mardevsic (1960) [23], Smyth (1994) [27], and Kopperman et al. (1997, 2003)[19, 20].

Asymptotic finite approximations of continuous mappings between such spaces are also important. They were investigated in Charalambous (1991) [9], and Mardevsic (1993) [24]. The recent papers by Debski and Tymchatyn (2017) [11] and (2018) [10] present a comprehensive (and up to date) approach to discrete approximations of complete metrizable topological spaces and continuous mappings.

This is the topological point of view, whereas we are interested in asymptotic finite approximations of geometrical spaces, as well as asymptotic finite approximations of continuous and smooth mappings between such spaces. The geometry is understood here in the spirit of Busemann’s (1955) book The geometry of geodesics [6]. His approach is based on axioms that characterize the geometry of a class of metric spaces (called G-spaces), and may be summarized as follows. The spaces are metric and finitely compact, i.e any bounded infinite sequence of points has at least one limit point. Any two points can be connected by a geodesic.

The Busemann’s approach is general enough to include all complete Riemann and some Finsler spaces.

Dyer, Vegter and Wintraecken [13] [14]

(2015, 2019) proposed a generalization of Euclidean simplices to non-degenerate Riemannian simplices on a Riemannian manifold, and triangulations of the manifold constructed by such simplices. The notion of Riemannian manifold itself is complex and involves sophisticated notions of atlas and metric tensor.

Summarizing the short review of the state of the art, it seems that the mathematical framework for the constructive (and computational) approach to geometry has already existed for a long time, however, in an abstract topological setting, or based on the complex notion of Riemannian manifold.

Perhaps the right way is to find a generic method for constructing geometrical spaces instead of approximating already defined ones. The method should be fully computational, i.e. based on processing finite structures, and versatile enough to express sophisticated topological and geometrical concepts like: Continuum, continuity, metric, geodesic, curvature, dimension, homotopy type.

We propose a generic method for asymptotic combinatorial constructions of geometries. It is based on the idea of Busemann that geodesics determine a geometry. However, in our approach, the most primitive notion is adjacency relation (finite graph) that is supposed to approximate a geometry. Geodesics in such a graph may be defined as shortest paths between vertices.

An inverse sequence of such graphs (constructed inductively according to a pattern) and its inverse limit (together with Hausdorff reflection) provide more than a topological space. If the pattern is sufficiently regular, then there is also a geometrical structure of the space with a pre-metric encoded in the pattern. A geodesic in the space is the limit of corresponding geodesics (shortest paths) in the finite graphs. This simple concept of geodesic gives rise to derive sophisticated geometrical notions like curvature.

So that all topological features (invariants like homotopy type, covering dimension and embedding dimension), and the geometrical features (like flatness, elliptic or hyperbolic curvature) of the space can be deduced from the pattern of its construction. Such geometrical spaces are stand-alone constructions without an ambient space, and present the intrinsic point of view in geometry.

We are going to show that the class of geometries that can be constructed by such patterns is quite large including also intrinsic hyperbolic geometries. The crucial feature of the proposed approach is its computational aspect, i.e. the geometries are (by their constructions) uniformly approximated by finite graphs.

2 Intrinsic geometry of flat torus

Figure 1: The duals to the first two quadrilateral tessellation graphs of torus

Let us consider flat torus and flat Klein bottle. Constructions of finite approximations for these geometries are extremely simple for any dimension . We will consider only 2-dimensional case that is sufficient to explain the basic idea. Also finite approximations of Euclidean spaces are simple and important. The unit interval and the unit square will serve as examples.

Classic torus, as a subset of , can be tessellated by convex quadrilaterals, however, not by flat squares because of the curvature. It is possible to tessellate torus with 4 quadrilaterals.

The next finer tessellation is done by dividing each of the 4 quadrilaterals into 4 smaller quadrilaterals. Then, each of the smaller quadrilaterals may be divided into 4 quadrilaterals, and so on.

Actually, any tessellation net (consisting of edges and vertices) is a graph. The dual (to this tessellation graph) is the graph where the vertices correspond to quadrilaterals, and edges correspond to the adjacency of the quadrilaterals.

The first two duals are shown in Fig. 1. The dual of the first tessellation is a multi-graph because the edges are duplicated. Note that the graphs are an abstraction, i.e. they are independent of any concrete torus.

We may construct the infinite sequence of finer and finer tessellations and corresponding dual graphs. For any of the graphs, the geodesics are defined as the minimal paths. Let the length of the geodesics be normalized by the graph diameter. Then, the inverse limit of the graphs with Hausdorff reflection (to be defined formally in the Section 2.1) determines a flat metric space (known as flat torus) that locally looks like Euclidean space , i.e. for any point, there is an open neighborhood of the point that is isomorphic to an open subset of the Euclidean space . The flat torus is homeomorphic to torus. By the famous Nash embedding theorem, a flat torus can be isometrically embedded in with the class 1 smooth mapping (see Borrelli et al. (2012) [5]) but not with the class 2.

Figure 2: The dual of the first quadrilateral tessellation graph of Klein-bottle
Figure 3: The dual of the second quadrilateral tessellation graph of Klein-bottle. It is composed of the duals of two Möbius strips tessellation graphs (where edges are solid lines) by joining (by edges shown as dashed red lines) the borders of the strips

Also Klein bottle can be tessellated by convex quadrilaterals. The simple argument is that the bottle can be constructed by joining the borders of two Möbius strips, see Fig. 2 and Fig. 3. So that the infinite sequence of graphs (dual to finer and finer tessellations) can be constructed. The sequence determines the geometry known as flat Klein bottle. Although it cannot be embedded in (as a topological space), it is a concrete geometrical space that can be approximated by finite graphs. What is the specific structure of these finite graphs that determines the -embedding dimension of the Klein bottle?

Let denote the set of natural numbers without zero. For the flat torus, let us define explicitly the dual graphs denoted by . It is convenient to use the dot notation (ASN.1) to denote the sub-quadrilaterals resulting from consecutive divisions. The quadrilaterals (vertices) of , see left part of Fig. 1, are denoted by the letters and . The quadrilaterals (vertices) of are denoted by where and belong to the set .

The next finer tessellation results in quadrilaterals (vertices in ) denoted by labels of the form , where , and belong to the set .

In general case, the vertices in the graph are denoted by finite sequences of the form . Let denote the set of all such of length . So that, is defined as the set of vertices of the graph . The edges of are determined by the adjacency relation between the quadrilaterals. The sequence is called an inverse sequence.

For our purpose it is convenient to consider the adjacency relation (denoted ) instead of the set of edges of the graph .

For less or equal to the length of , let denote the prefix (initial segment) of of length .

Note that for any of length , the sequence () may be interpreted as a sequence of nested quadrilaterals converging to a point on the torus if goes to infinity.

The relation is symmetric, i.e. for any and in , if , then . It is also convenient to assume that is reflexive, i.e. any is adjacent to itself, formally .

Let denote the union of the sets for . The relations (for ) can be extended to , i.e., to the relation in the following way. For any and of different length (say and respectively, and ), relations and hold if there is of length such that is a prefix (initial segment) of , and holds, i.e. is of the same length as and is adjacent to .

2.1 Topology and geometry

Let us consider the flat torus again, and change a bit the notation. Let denote the pattern of construction the inverse sequence of graphs corresponding to the flat torus. Let the sequence be and denoted ,    be denoted , and be denoted .

Consider the infinite sequences of the form . Let denote the set of such sequences. By the construction of (), any such infinite sequence corresponds to a sequence of nested quadrilaterals (on an Euclidean 2–torus) converging to a point. However, there may be four such different sequences that converge to the same point of the torus.

For an infinite sequence, denoted by , let denote its initial finite sequence (prefix) of length .

The set may be considered as a topological space (Cantor space) with topology determined by the family of clopen sets such that where is the length . Note that the adjacency relation is not used in the definition.

We are going to introduce another topological and geometrical structure on the set determined by .

Two infinite sequences and are defined as adjacent if for any , the prefixes and are adjacent, i.e. holds. Let this adjacency relation, defined on the infinite sequences, be denoted by . Note that any infinite sequence is adjacent to itself. Two different adjacent infinite sequences may have a common prefix.

The transitive closure of is an equivalence relation denoted by . Let the quotient set be denoted by , whereas its elements, i.e. the equivalence classes, be denoted by and . Actually, is the inverse limit of the graph sequence, whereas is the Hausdorff reflection of the limit relatively to the equivalence relation .

There are rational and irrational points (equivalence classes) in . For the flat torus, each rational equivalence class has four elements, whereas the irrational classes are singletons. Each rational class corresponds to a vertex of a tessellation graph, and equivalently to a face in graph .

By the construction of , any point of the initial Euclidean 2-torus corresponds exactly to one equivalence class (a point in ) and vice versa.

The relation determines natural topology on the quotient set .

Usually, a topology on a set is defined by a family of open subsets that is closed under finite intersections and arbitrary unions. The set and the empty set belong to that family. Equivalently (in the Kuratowski style), the topology is determined by a family of closed sets; where finite unions and arbitrary intersections belong to this family.

Let us define the base for the closed sets as the collection of the following neighborhoods of the points of . For any equivalence class , i.e. a point in , the neighborhoods (indexed by ) are defined as follows.

Note that may be equal to .

A sequence of equivalence classes of (elements of the set) converges to , if for any there is such that for all ,    .

The set of rational classes is dense in .

The topological space is a Hausdorff compact space. The sequence of graphs may be seen as finite approximations of the space , more exact if is grater. By the construction, the space is homeomorphic to Euclidean 2-torus. Actually, the graph sequence contains the geometric structure (specific to the flat torus) that is much more rich than the topology. The structure allows to define geodesics in many ways.

The normal metric and corresponding geodesics are defined in the following way. Let the -geodesics be defined as the shortest paths in between two vertices. The distance between two vertices and of is denoted by , and defined as the length of a geodesic between and divided by the diameter of graph .

Pseudo-metric is a generalization of a metric where the distance between two distinct points can be zero.

Pseudo-metric on is defined as follows.

It is clear, that if iff and only if .

So that the metric, denoted , is defined on as follows.

Let the pseudo-metric be called normal.

Note that the above definitions are for the concrete and specific inverse sequence corresponding to the flat torus. The general definitions are presented in the Section 4.

Geodesic metric space means that for any its two points, there is a geodesic path between them whose length equals the distance between the points.

Hence, is a geodesic metric space known as flat torus. What does it mean the flatness? What is specific in the construction pattern that determines this very flatness and the curvature of the space in general? There questions and much more will be answered in the following sections.

Different metrics may be also defined for .

2.2 Summary of the example

Figure 4: The first four consecutive graphs of the unit interval
Figure 5: The first four consecutive graphs of the flat circle

The presentation of the example was a bit informal. In the Section 4, a general and rigorous framework will be introduced.

Borelli et al. (2012) [5] constructed class 1 isometric embedding mapping of flat -torus into Euclidean space . Although, the embedding is interesting by itself, it does not explain (in a simple way) the nature of intrinsic geometry of the flat torus. The pattern of inductive construction of the finite graphs approximating the geometry of the flat -torus is extremely simple. It is even simpler than the pattern of Euclidean -cube. A flat torus of dimension may be taken as a basis for constructing intrinsic hyperbolic and elliptic geometries, see Section 6.

The essence of the asymptotic combinatorial construction of the flat 2-torus presented above, is the sequence of graphs (). Actually the topology and the geometry were defined on the basis of the sequence alone without reference to an Euclidean -torus that serves only as a helpful intuition. The same method can be applied to the unit interval (Fig. 4), circle (Fig. 5), the unit square (Fig. 6) as well as to -cube for arbitrary . Also dimensional torus and Klein bottle can be approximated by such sequences of finite graphs.

Actually, without correspondence to a metric space, we may construct a graph sequence () according to a regular and uniform pattern, see the Section 4. Then, the graph may be considered as an approximation of a geometry (may be a novel one), more exact if is approaching infinity. Hence, the example presented above gives rise to define a generic method (to be defined in the Section 4) for constructing new intrinsic geometries.

Figure 6: The duals of the first three quadrilateral tessellation of the unit square

3 Intrinsic geometry of 2-sphere

Let us present the second working example. We are going to construct an inverse sequence of graphs that corresponds to intrinsic geometry of -sphere.

A -sphere is a perfectly round surface with a constant curvature. In Analysis, is defined as a subset of three-dimensional Euclidean space . If a radius (say ) and a center (say ) is fixed, then is the set of all points such that . Without , it is not easy to define Euclidean 2-sphere as a stand-alone intrinsic geometrical space. It may be represented as a Riemannian manifold. However, necessary and sufficient conditions, for the manifold to be isomorphic to a sphere, are quite complex, see [25].

In the similar way as for torus in the previous section, we are going to construct an inverse sequence that determines an intrinsic geometrical space isomorphic an Euclidean 2-sphere. The flat torus from the previous section was only homeomorphic to an Euclidean 2-torus.

Triangulations seem to be natural candidates for constructing such inverse sequence.

A triangulation on a sphere is a simple graph embedded on the sphere so that each face is triangular. The graph may be represented as a geodesic polyhedron. Triangles can then be further subdivided for new geodesic polyhedra. So that, finer and finer triangulations (graphs) may be considered as getting better and better approximations of 2-sphere. Although the idea is clear, constructions of such triangulation is not easy. Note that this triangulation should be abstract and independent of any concrete 2-sphere, and pure combinatorial without numerical calculations. There are some related investigations in Popko (2012) [26], and Thurston (1998) [28]. Despite these efforts, the problem is still open.

Figure 7: Class I operator (a), the kiss operator (class II) (b), a kiss–like operator (c)

The triangular Goldberg–Coxeter operators (see [12]) may be applied for constructing finer triangulations. There are three classes of such operators. Class I operators give a simple division with original edges being divided into sub-edges, see Fig. 7 (a). For Class II operators, triangles are divided with a center point; as an example, see the kiss operator in Fig. 7 (b). Class III may be viewed as a askew combined version of the I and II classes.

The class I operators correspond to the flat surfaces. It seems that the kiss operator may be appropriate to construct finer sphere triangulations.

Figure 8: (a) Octahedron, (b) disdyakis dodecahedron projected into sphere, (c) Archimedean truncated cuboctahedron. Source CC BY-SA 4.0 [29]

The five Platonic solids (Tetrahedron, Cube, Octahedron, Dodecahedron and Icosahedron) are (by projection) regular tessellations of the sphere.

Octahedron (see Fig. 8 (a)) has a nice property; its finer triangulation (by applying the kiss operator) results in equilateral spherical triangles when projected into sphere, see Fig. 8 (b). The triangulation forms the disdyakis dodecahedron that is a Catalan solid with 48 faces, and its dual is the Archimedean truncated cuboctahedron, see Fig. 8 (c). Duality means that triangle faces correspond to vertices in the dual, and adjacency of the faces corresponds to edges.

A sphere is a perfectly round surface. Any two points of the sphere are locally isomorphic (indistinguishable). For this very reason, in the sequence of finer and finer triangulations, vertices can not be distinguished from each other. That is, all vertex neighborhoods (in the limit) are the same, i.e. as subgraphs, they have the same structure.

It seems that Octahedron (see Fig. 8 (a)) may be considered as the first basic triangulation on the sphere. The next finer triangulations are constructed by by applying the kiss operator. Each triangulation may be considered as a graph where vertices and edges are the vertices and edges of the triangles. In the inverse sequence, we consider the graphs that are dual to the triangulation graphs. Each vertex of the dual graphs is of degree 3, whereas the faces of the dual graphs are polygons each one having or edges. We are going to show that the inverse sequence of such dual graphs is the basis for constructing an interesting intrinsic geometric space related to Euclidean 2-sphere.

3.1 Construction of inverse sequence

Let be the triangulation corresponding to Octahedron. The triangle faces are denoted by numbers 1,2,3,4,5,6,7, and 8. Let be the dual to . Then, vertices of are numbers from 1 to 8, whereas the edges correspond to the adjacency between the faces. It is the graph of the net of Cube.

The second finer triangulation graph corresponds to disdyakis dodecahedron, see Fig. 8 (b). Here, the spherical triangles are also equilateral. Let be dual to this graph; then corresponds to the Archimedean truncated cuboctahedron, see Fig. 8 (c).
has 6 square faces. is composed of 6 octagon faces, 8 hexagon faces, and 12 square faces. consists of six 16-gon faces, eight 12-gon faces, 12 octagon faces, 48 hexagon faces, and 72 square faces.

We may continue the divisions, getting the infinite sequence of finer and finer triangulations (graphs ), and their dual graphs (, where the vertices represent triangle faces, and edges represent face adjacency. For any , each vertex of is of degree 3.

The inductive construction of the sequence of graphs is as follows. Suppose that and corresponding dual triangulation graph are already constructed. The graph is constructed by applying the kiss operator (see Fig. 7 (b)) to each triangle face in . Let be the dual to the .

The consecutive graphs have faces with edge number of the form , or . Faces from become (in ) twice longer (in the number of edges), and new square-faces and hexagon faces are created in .

Let denote the pattern of the above a bit informal construction of the sequence (). Let be denoted by for any .

Let us define explicitly the inverse sequence of graphs . Let us again use the dot notation (ASN.1) to denote the sub-triangles resulting from consecutive divisions. The triangles (vertices) of are denoted by the numbers 1,2,3,4,5,6,7, and 8. The triangles (vertices) of are denoted by , , , , , and ; where .

The next finer triangulation results in triangles (vertices in ) denoted by labels of the form , where , and and belong to the set . Fig. 9 shows the first and the second triangulation, and the labeling. In the right part of Fig. 9, for any two adjacent triangles and , the labels for their sub-triangles are distributed in the symmetric way relatively to their common edge. Although, the triangles 3, 4, 7, and 8 are not visible, their division and labeling are the same.

Figure 9: The first two triangulations. In the middle and the right parts, the triangles 2 and 6 have one common edge; this is denoted by the arrow
Figure 10: A finer triangulation

In general case, the vertices in the graph are denoted by finite sequences of the form .

denotes the set of all such of length , and is the set of vertices in the graph .

Let us recall that for less or equal to the length of , denotes the prefix (initial segment) of of length .

The vertices and edges of are defined in the following way. Suppose that the label distribution was already defined for -th triangulation. For a single triangle its sub-triangles are labeled in the clockwise cyclic order as shown in the left part of Fig. 10 where the black circle denotes the mass center of the triangle. The very black circle is the orientation point of any of the sub-triangle for labeling in clockwise order. In the right part of Fig. 10, the triangle was chosen (as an example) for the label distribution in the next finer triangulation.

The label distribution determines the adjacency relations between triangles, and equivalently also the edges in graph for any .

So that, we have constructed the inverse sequence of planar graphs .

For any , let denote the set of vertices of the graph , and denote the adjacency relation corresponding to the edges of the graph. The relation is extended to be reflexive.

3.2 Topology and geometry

Let . The adjacency relation can be extended to on in the following way. For any and of different length (say and respectively, and ), relations and hold if there is of length such that is a prefix (initial segment) of , and holds.

Consider the infinite sequences of the form . Let denote the set of such sequences.

Any such sequence corresponds to a sequence of nested triangles (on the Euclidean 2-sphere) converging to a point. However, there may be infinite many such sequences of nested triangles that converge to the same point of the space .

Let denote the transitive closure of . Then, any equivalence class of is either a singleton, or it has two elements if it correspond to an edge of a graph in the inverse sequence. The space is a compact closed Hausdorff space with the covering dimension equal to 1. Although it is an interesting topological space, its topology is different than the topology of Euclidean 2-sphere.

3.2.1 Sierpinski pattern

Similar construction of an inverse sequence of graphs can be carried out on the basis of Sierpinski graphs, see Fig. 11.

Figure 11: Sierpinski graphs

Let the pattern of the construction of the graphs be called Sierpinski pattern and denoted .

The first graph consists of vertices denoted by , , , and . Any two of them are adjacent.

For any , the vertices of are of the form where is element of the set whereas for ,    is element of the set .

The adjacent relation on is defined inductively as follows. For any , the vertices , , and are pairwise adjacent. Also and are adjacent, and are adjacent, and are adjacent.
Vertices and are adjacent, if is a suffix consisting of elements that all are (that is, ).
Vertices and are adjacent, if is a suffix consisting of elements that all are .
Vertices and are adjacent, if is a suffix consisting of elements that all are .
If consist of the same elements, i.e. either or or , then vertices and are adjacent where is the length of .

So that, the inverse sequence of planar graphs () is constructed.

The adjacency relation is similar to the “adjacent-endpoint relation” for Lipscomb spaces, see Lipscomb (2009) [21].

The space is homeomorphic to , and it seems that it determines the similar intrinsic geometry.

3.3 Metric

Pseudo-metric means that the distance between two distinct points may be zero. We are going to define a pseudo-metric that reflects our intuition of Euclidean 2-sphere encoded in the inverse sequence ().

For any , the graph is planar. Let each face (i.e. induced cycle) of graph be assigned one and the same unit length defined as

where is the diameter of the triangulation graph dual to .

Consider an edge (denoted ) of . There are exactly two faces (say and ) of that have in common. Let and denote the numbers of edges of and respectively. The length of edge (denoted by ) is defined as the arithmetic mean of and .

The length between two vertices and of is denoted by , and defined as the length of the shortest path (relatively to ) between and .

For any and elements of , the pseudo–metric is defined as follows.

Note that for some ,   .

Let us define the equivalence relation on the set .

iff .

Let denote the metric on the quotient set .

The space (denoted by ) is a geodesic metric space.

There are rational and irrational points (equivalence classes) in . Each rational equivalence class has infinite (countable) number of elements, whereas the irrational classes are singletons. Each rational class corresponds to a vertex of degree 4 or 6 in a triangulation graph (for some ), and equivalently to a square face or hexagon face in .

By the construction of , any point of the initial Euclidean 2-sphere corresponds exactly to one point in , and vice versa.

The set of rational classes is dense in .

The inverse sequence of graphs may be seen as finite approximations of the space , more exact if is grater.

Is the space isomorphic to an Euclidean 2-sphere? If yes, then what is the radius of the sphere?

Note that the similar construction of metric can be done for Sierpinski pattern . The resulting geometrical space (let it be called Sierpinski sphere) is also interesting.

4 Abstraction and generalization

We are going to generalize the notions introduced in the previous sections.

Let () be a sequence of graphs and mappings generated by an arbitrary pattern of inductive construction (denoted by ) such that:

  • denotes the set of vertices of . For , the sets and are disjoint.

  • Mappings , for , are surjective, i.e. the image of the domain and the codomain are equal.

Let denote an infinite sequence such that its -th element belongs to . Let denote -th element of .

A thread is defined as an infinite sequence such that for any ,   .

Let denote the set of all threads.

If for any and any vertex of , the number of threads such that is infinite, then the sequence () is called an inverse sequence.

Let denote the union .

Inverse sequence is a special case of inverse systems (see [20], [27], and [10]) where topological spaces are substituted for finite graphs, and directed set is substituted for linear order.

Note that the inverse mappings determine a tree-like partial order on the set .

For and , let denote the “prefix” of of length such that ,   , and so on. It is convenient to define the tree-like partial order (i.e. being a prefix denoted by ) determined by the inverse mappings. That is, if is a prefix of or .

It is an informal use of the notion of prefix as an initial segment of a sequence. However, it is a convenient notation that operates on the tree-like partial order instead of the inverse mappings, and is similar to the notion of prefix from the preceding sections.

We say that and are of the same length if they belong to the same . Let the adjacency relation be defined in the very similar way as in the previous sections.

For a thread , let , as an element of , be also called the initial segment (“prefix”) of length of the thread . It is justified by the fact that for any ,    is uniquely determined by the inverse mappings.

For these new (more abstract and general) notions, to have the intended meaning, some restrictions are necessary on the patterns of inductive construction of the inverse sequences.

4.1 Regular patterns

Pattern is called regular if it satisfies the following two conditions.

  1. Connectedness: If and (belonging to , for some ) are adjacent, then there are two different elements of (say and ) such that and , and and are adjacent.

  2. Consistency: If of length , and of length are adjacent, then for any and any :    and are adjacent.

For any regular pattern , the adjacency relation for threads and (elements of ) is defined as follows.

iff for any :

Its transitive closure is an equivalence relation. So that, the Hausdorff reflection of relatively to , i.e. , is a compact Hausdorff space, where the topology is given by the following collection of the neighborhoods of the points of . For any equivalence class , i.e. a point in , the neighborhoods (indexed by ) are defined as follows.

For the flat torus it works well, and also for strongly uniform patterns as a generalization of the pattern of the flat torus; see the next Section.

For weakly uniform patterns, as a generalization of the pattern of a 2-sphere (see the next Section), the relation is too weak to capture our geometrical intuitions, see the Section 3.3 where a special pseudo-metric was introduced instead of .

For the both examples (flat torus and -sphere), the corresponding pseudo-metrics ( and ) are the spacial cases of the notion of asymptotic pseudo-metric defined below.

4.2 Asymptotic pseudo-metric

Let be an arbitrary regular pattern, and let for any , be a metric (determined by predefined edge lengths, and called -metric) on the set of vertices of graph .

If for any threads and in the following limit exists

and the convergence is uniform, then

is called asymptotic pseudo-metric. It is clear that the triangle inequality is satisfied.

The asymptotic pseudo-metric will be identified with the sequence .

Hence, it is reasonable to say that the sequence approximates the corresponding geometrical space , where is the metric determined by . Let the space be called an asymptotic metric space and denoted .

In the Sections 5 and 7, we will show that the notion of asymptotic pseudo-metric is enough to define geodesics and curvature in a constructive way.

4.3 Uniform patterns

We are going to abstract from the examples of flat torus and -sphere, and define a general class of patterns with corresponding asymptotic pseudo-metrics.

Let us recall some standard notions form graph theory.

A chordless cycle in a graph, also called a hole or an induced cycle, is a cycle such that no two nodes of the cycle are connected by an edge that does not itself belong to the cycle.

A subgraph of is induced, if the all edges in between vertices of are also in .

Union of two subgraphs and of is defined as the subgraph such that the set of its vertices is the union of the sets of vertices of and of , and the set of edges is the union of the set of edges of and of .

We are going to define the notion of -cell as an elementary building block of a geometrical space. Let the graph be fixed.

  • 1-cell is an edge in .

  • 2-cell, denoted , is the set of edges of an induced (chordless) cycle in the graph . It is the cycle.

  • 3-cell, denoted , is a minimal subset of the set of all -cells of satisfying the following conditions.

    • For any , and any -cell in , there is exactly two -cells in that has this very -cell in common. The cell is one of them.

    • For any two of -cells in , they have at most one -cell in common.

  • Let us assume that -cells have been already defined in .
    (n+1)-cell, denoted , is a minimal subset of -cells of satisfying the following conditions.

    • For any , and any -cell of , there is exactly one -cell in , different than , that has also this very -cell.

    • For any two of -cells in , they have at most one -cell in common.

The requirement for the cells to be minimal (relatively to the inclusion of sets of vertices) is essential. Let denote the sub-graph of determined by the -cell . Note that the sub-graph is induced.

Let us consider the inverse sequence (), corresponding to the flat 2-torus, as an example.

For any , each of the -cells in graph , that contributes to the surface of the torus, has four vertices. There is one special -cell that consists of all such -cells. Let these cells be called regular.

There are also different (than the regular ones) -cells and -cells. Any of such irregular -cells has more that 4 vertices, and may be viewed as a cross section of the torus. Any of such irregular -cells may be vied as a slice consisting of two neighboring cross sections and regular -cells between them.

4.3.1 Uniform pattern and its cell structure

Let be a regular pattern, and be its inverse sequence. The pattern is called uniform if the following condition is satisfied. There is natural number such that for any :

  • There is at least one -cell, and there are no -cells in .

  • There is a -cell (denoted ) consisting of -cells in , and covering the whole graph , i.e.

    such that

    1. The number of elements of is unbounded if .

    2. Let denote the set of vertices of graph .
      Then, for any , the image is either the set of vertices of the sub-graph of generated by a -cell in , or it is a vertex in .

Note that the sequence need not to be unique.

Let the sequence be denoted more precisely by , and be called a cell structure of uniform pattern .

The number is called the order of the uniform pattern .

If the size of the cells in is not bounded, then the cell structure is called weak, otherwise it is called strong.

Note that the pattern corresponding to the flat 2-torus, and the pattern corresponding to -sphere, are uniform.

For the pattern , there are at least two cell structures. The one consisting of regular cells is strong, and the one consisting of slices is weak.

For the pattern , the cell structure used to define pseudo-metric is weak, as well as its other cell structure.

The notion of uniform pattern is restrictive. Patterns that correspond to unit interval, square, cube, Euclidean -cubes, and spaces that are not closed in general, are not uniform.

Also spaces that are not homogeneous, like two 2-spheres that have one common point, are not uniform.

4.3.2 Normal metrics

Let be the order of a uniform pattern and be its cell structure.

We are going to introduce a natural -metric, for the set of vertices of graph , determined by predefined edge lengths.

The metric, for flat torus defined in Section 2.1), may be seen as an example of such metric where the length of each edge is , and the distance id defined as the length of minimal path divided by the diameter of .

A more general method for constructing the metric was introduced in Section 3.3 for pattern corresponding to an Euclidean -sphere and its concrete cell structure.

We are going to generalize this method. The key problem is to choose, for any , an appropriate measure unit for .

It is reasonable (according to intuition) to consider -cell of the cell structure as elementary building block of the geometry corresponding to of order .

For any , let us define graph such that its vertices are -cells belonging to , and the edges are determined by the adjacency between the -cells. The adjacency of two -cells means that the cells have a -cell in common.

Let the diameter of be denoted by . It will be a normalizing factor. Any -cell in is assigned the -unit of measure equal to .

Let denote a -cell in , and let be the diameter the graph .

The length of an edge of (relatively to ) be denoted ) and defined as follows.

The normalized length of an edge of graph , denoted , is defined as the arithmetical mean of all such that is an edge of . Note that is the parameter here.

For any vertices and of , let be defined as the length of the shortest path between and where the length of edges is determined by . Let be called normal -metric.

Let us consider the set of all threads . A pseudo-metric, denoted , is defined as follows. For any threads and :

It is clear that the triangle inequality holds. Let be called the normal pseudo-metric for uniform pattern and its cell structure . It is also clear that, for uniform patters, the limit always exists. So that, is an asymptotic pseudo-metric.

The normal pseudo-metric determines the following equivalence relation (denoted ) on .

iff .

Let denote the metric on the quotient space determined by . Let the metric be called normal for the uniform pattern and its cell structure .

The normal metrics will be identified with the sequence .

Hence, a uniform pattern with a cell structure determines the metric space denoted .

Note, that the number , as the order of uniform pattern with , is the covering dimension of the metric space .

It is reasonable to say that is the dimension of each of the graphs in the inverse sequence relatively to the cell structure and -metric .

4.4 Summary of the Section

All introduced notions are computational in the sense that they are approximated be their finite counterparts in graphs , for .

In the next sections, we will show that the notion of finite asymptotic metric space is enough to define geodesics and curvature in a constructive way.

What are relations of the geometries with normal metrics (and their generalizations) to the notion of Riemannian manifold? Evidently, the uniformity and the -order of pattern correspond to local homeomorphism to the Euclidean space of dimension , whereas the normal pseudo-metric corresponds to the metric tensor.

Still the notion of curvature is to be defined in our framework. So that the (fully computational) notions, corresponding to tangent space with an inner product, are needed.

For this very purpose, a special notion of geodesic (interesting and important for its own) is introduced in the next section.

5 Geodesics

Let be an asymptotic metric space, see Section 4.2.

Let be the corresponding sequence of -metrics on graphs for .

Let us consider the graph for a fixed . A path between vertices and , in the graph, is defined in the usual way as a finite sequence of vertices such that and , and for any , there is edge between and . Note that the adjacency relation is reflexive, so that there is loop (edge from vertex to itself) for each vertex in .

For any path its contraction is defined as the longest sub-path such that for any , the consecutive vertices and are different.

A -geodesic between two vertices in is defined as a shortest path (relatively to the -metric ) between these two vertices.

A -geodesic is called an extension of -geodesic (where ), if:

  • the contraction of path is the same as the path , where is the prefix of of length .

Let a sequence of geodesics , where is a -geodesic, satisfies the following conditions:

  • for any ,   is a -geodesic, and is an extension of .

If such minimal number exists, then let it be denoted by . Then, the sequence is denoted by , and called a -geodesic.

Note that may be viewed as a linear sub-graph (let it be denoted by ) of .

Let denote the set of vertices of graph , for .

Let .

Inverse mapping is constructed in the following way. For any vertex in let

such that .

Hence, the inverse sequence has been constructed and determines the intrinsic stand-alone geometric space that corresponds to . Let the spac ebe denoted by , and let it be identified with the geodesic . It is homeomorphic to the unit interval.

It is important to note that the extrinsic geometry of a -geodesic in the ambient space is the same as the intrinsic geometry of the geodesic. This gives rise to define a new notion of curvature different than the sectional curvature for Riemannian manifolds. This is done in the next sections.

It seems that the above definition of geodesic captures the geometrical intuition of shortest path. It is a bit different than the classic definition of geodesics in Riemannian manifolds.

Let us summarize this section. For an asymptotic metric space , we have defined a fully computational notion of geodesic. Hence, asymptotic metric spaces are G-spaces introduced by Busemann (1955) [6].

Examples of combinatorial constructions of an intrinsic geometry of flat -torus (in Section 2), and an intrinsic geometry of -sphere (in Section 3) support the view that the proposed approach is generic. In the next section, we are going to present constructions of intrinsic geometries of hyperbolic circle and elliptic circle.

6 Flat, hyperbolic and elliptic circles

Figure 12: Circles. From the top: flat, elliptic at one point, finite hyperbolic at one point, and infinite hyperbolic at one point

The unit interval can be represented as an intrinsic geometrical space in the simple way shown in Fig. 4. Circle can be made from the unit interval just by identifying the ends of the interval, see Fig. 5. It is homeomorphic to an Euclidean circle embodied in . However, as an intrinsic geometrical space it is the well known flat circle.

A common view is that Euclidean spaces are flat. Our intuition fails to imagine a flat circle, a flat torus, and a flat Klein bottle.

Circle is one of the simplest geometrical spaces. A circle in the Euclidean plane is elliptic. There is also a hyperbolic circle embodied in the hyperbolic plane , where the circumference of circle grows exponentially relatively to the radius.

What is an intrinsic geometry of an elliptic circle? What is an intrinsic geometry of a hyperbolic circle? The spaces are of dimension . If they are represented as Riemannian manifolds, the sectional curvature can not be defined for them.

It is clear that the notion: flat circle, elliptic circle and hyperbolic circle refer to the notion of curvature.

In order to grasp the intrinsic meaning of elliptic and hyperbolic curvatures of circles, we are going to introduce a new notion of curvature, different than the classic notion of sectional curvature in Riemannian manifolds.

For an asymptotic metric space , a curvature at a point and direction determined by a geodesic, may be defined on the base of the following simple idea.

Elliptic curvature at a point means that the normalized length of a -geodesic ending at this point is getting shorter if goes to infinity. This may be interpreted as gravitation force attracting to this point. The idea is consistent with the Einstein’s theory identifying gravity with space curvature.

Hyperbolic curvature at a point means that the normalized length of a -geodesic ending at this point is getting longer if goes to infinity. This, in turn, may be interpreted as repulsive force from this point, and interpreted as anti-gravity or the electromagnetic repulsion.

In order to explain the idea, let us consider the intrinsic geometrical space corresponding to the flat circle, see Fig. 12, the first picture from the top. Let its pattern be denoted by . Let the sequence () denote the corresponding inverse sequence. The graph consists of two vertices and . The set of vertices of graph is . The inverse mappings are defined as follows. For any vertex of graph ,   .

The space (also denoted ), equipped with the normal metric (see Section 4.3.2 for definition), is homeomorphic and isometric to an Euclidean circle. However, it is not isomorphic because it is flat like the flat torus presented in Section 1 and the unit interval from Fig. 4.

Let us choose the point that corresponds to the two threads , where each element is , denoted by , and , where each element is , denoted by . Note that and are adjacent, and the equivalence classes and are equal. and denote respectively the initial segment of and of length .

6.1 One-point elliptic curvature

Let us transforms the inverse sequence () into the inverse sequence denoted (), see Fig. 12, the second picture from the top, in the following inductive way.

Let denote the length of , i.e. the number such that . Since the sets and for are mutually disjoint, the length is well defined.

For any of length , let be the set of all , such that .

For any the set of verices of is defined inductively as follows.

  • Let be equal to , and be equal to .

  • For

  • For

  • For