A phase transition creates the geometry of the continuum from discrete space

TL;DR

This paper demonstrates that a phase transition in graph models of discrete space can produce emergent continuum-like geometry, satisfying key properties of Euclidean space at large scales.

Contribution

It introduces a disordered graph model that recovers continuum Euclidean geometry, including Pythagoras' theorem, through a phase transition in discrete space.

Findings

Graphs exhibit continuum-like behavior at large scales.

The model satisfies Hausdorff dimension, straight lines, and Pythagoras' theorem.

Disorder is essential for emergent Euclidean geometry.

Abstract

Models of discrete space and space-time that exhibit continuum-like behavior at large lengths could have profound implications for physics. They may tame the infinities that arise from quantizing gravity, and dispense with the machinery of the real numbers, which has no direct observational support. Yet despite sophisticated attempts at formulating discrete space, researchers have failed to construct even the simplest geometries. We investigate graphs as the most elementary discrete models of two-dimensional space. We show that if space is discrete, it must be disordered, by proving that all planar lattice graphs exhibit the same taxicab metric as square grids. We give an explicit recipe for growing disordered discrete space by sampling a Boltzmann distribution of graphs at low temperature. We then propose three conditions which any discrete model of Euclidean space must meet: have a Hausdorff dimension of two, support unique straight lines and obey Pythagoras' theorem. Our model satisfies all three, making it the first discrete model in which continuum-like behavior is recovered at large lengths.

Equations

$$⟨Z⟩=6-12/N.$$

$$⟨\kappa {⟩}_{\mathrm{pat}}=(6+2p-e)/N_{\mathrm{pat}}.$$

$$E_{\mathrm{walk}}=-\sum _i{w}_i|w_i|,$$

$$N_{\mathrm{geo}}\propto {\lambda }^{\gamma }\mathrm{with}\gamma \approx \mathrm{1.1.}$$

$$R_{\mathrm{sph}}(\Lambda )\equiv \frac{2}{\pi \Lambda }\arccos \left[\frac{\cos (\pi \Lambda )}{\cos (\pi \Lambda /2)}\right]=\sqrt{3}+O({\Lambda }^2).$$

$${\sqrt{3}}_{\mathrm{est}}=1.726\pm \mathrm{0.005.}$$

Full text

Geometry, Phase transition, Networks, Emergent space, Graph dynamics

I Introduction

The small-scale structure of space has puzzled scientists and philosophers throughout history. Zeno of Elea Hagar claimed that geometry itself is impossible because there is no consistent form this small-scale structure can take. He argued that a line segment, which can be halved repeatedly, cannot ultimately be composed of pieces of non-zero length, else it would be infinitely long. However, it also cannot be composed of pieces of zero length, for no matter how many are added together, the resulting line will never be longer than zero.

It is a lasting tribute to the optimism of researchers that work on geometry nevertheless carried on. It was not until the 19th century – nearly two and a half millennia later – that Cantor finally resolved the paradox by defining the continuum. He showed that the line must be composed not just of an infinite number of points, but of an uncountably infinite number, so that the second half of Zeno’s argument (a proof by induction) fails. This uncountable infinity is described by the mathematical machinery of the real numbers. The continuum is the basis for all descriptions of space and space-time, and therefore all of theoretical physics.

In the 20th century, Weyl Weyl further claimed that the continuum is the only possible model of space. He constructed a tiling argument, purporting to show that if space is discrete, Pythagoras’ theorem – or, equivalently, the Euclidean metric – is false. Weyl’s proof, however, contains an unstated assumption which turns out to be the key to its resolution.

Despite this long belief in the necessity of the continuum, researchers are actively pursuing discrete Bombelli ; Benincasa ; Hamma , or at least piece-wise flat Regge ; Regge2 ; Ambjorn ; Ambjorn2 ; Horava , models of space and space-time, as they offer the possibility to remove non-renormalizable infinities which arise in simple versions of quantum gravity. All these models can be thought of as graphs, where just the graph itself matters, not its embedding into another space. The only natural Leymarie metric in this case is graph geodesic distance: the distance between two nodes is the smallest number of edges joining them.

In two dimensions, toy models of ‘quantum graphity’ Konopka aim to define Hamiltonians over all graphs with a fixed number of nodes, from which an approximation to a smooth manifold might emerge at low temperature. Recent attempts have aimed to produce planar graphs made up of triangles, but, so far Conrady ; Chen , the low-temperature phases contain defects, conical singularities and multiply-connected topologies, so are unlike simple, smooth manifolds. A basic feature of such graphs is their Hausdorff dimension, which in this context is the power with which the number of nodes in a ball of radius $r$ grows with $r$. If one deliberately restricts the ensemble of graphs under consideration to triangulations of the plane, a further problem with graph models of two-dimensional space is encountered: Completely random triangulations (i.e. typical graphs chosen at random from this ensemble, and termed ‘Brownian maps’) do not even have Hausdorff dimension two. They are so crumpled that the number of nodes in a disc of radius $r$ scales as $r^{4}$, not $r^{2}$ leGall .

In light of these difficulties, the prospects for building a consistent discrete model of even the Euclidean plane seem poor. In this Article, we show that it is in fact possible to discretize space. We do three things. First, we prove that any discrete model of two-dimensional space must be disordered, by showing that all planar lattice graphs have a taxicab-like metric Taxicab . Order is the hidden assumption in Weyl’s proof of the impossibility of discrete space. Second, we describe a local, statistical process, with an associated temperature, which provides an explicit recipe for growing disordered graphs. Third, we propose three tests which any model of Euclidean space must pass. We find that graphs grown by our thermal process, at low temperature, achieve the required properties: they have a Hausdorff dimension of 2, support the existence of unique straight lines, and satisfy Pythagoras’ theorem.

II Lattice graphs are taxicab graphs

The natural way to measure the distance between two nodes on a graph is to count the edges in the shortest path which separates them. A shortest path of this kind is called a geodesic. It is well known that with this measure of distance, the square grid graph has a taxicab geometry Taxicab , where the distance between two nodes is the sum of the magnitude of the differences of their Cartesian coordinates (Figure 1). On this graph there are typically many geodesics between two nodes a distance $\lambda$ apart, each resembling an irregular staircase. Together these form a geodesic bundle comprising $N_{\rm geo}\propto\lambda^{2}$ nodes. More complex lattice graphs show a similar phenomenon (Figure 2a).

In general, any doubly-periodic planar graph must belong to one of the wallpaper groups, familiar from crystallography, and be composed of unit cells containing one or more nodes. We prove that all doubly-periodic planar graphs have the taxicab metric, regardless of the complexity of the unit cell. That is to say, geodesics in all but a finite number of directions form broad, parallelogram bundles (the number of exceptional directions may, however, be large for sufficiently complex unit cells Fritz ). Such graphs therefore do not satisfy Euclid’s axiom of a unique straight line between two points, nor Pythagoras’ theorem. Our proof is in two parts, which we call geodesic composition and geodesic rearrangement. We sketch the proof here, and give full details in the Methods section.

Sketch of the proof

If we have a geodesic on a graph between two nodes, it is clear that cutting it in two yields two paths, each of which is a geodesic between its respective end nodes (were that not so, it would be possible to create a shorter path between the original two nodes). Even in classical geometry, however, putting two geodesics (straight lines) end-to-end does not always give a geodesic: they need to be parallel. The situation with graphs is more interesting still.

Equivalent nodes in different unit cells are said to be of the same type. We first construct a geodesic between two nodes of the same type, which are separated by a vector distance of $(m,n)$ unit cells. If we choose the node type so that this is the shortest of all such geodesics (or one of the shortest, if the choice is not unique), then we are able to prove that many copies of this path can be concatenated end-to-end, and the result is still a geodesic between the now widely-separated end points. We call this the geodesic composition property. It is not trivial, since it relies on the assumption of planarity; a non-planar counterexample is shown in Figure 2c.

Next, we show that a long concatenation of this single type of geodesic can, apart from short tails at the ends, be broken down into many alternating copies of two different geodesics. The proof uses Dedekind’s pigeonhole principle Pigeon , applied to the number of nodes in the unit cell. If $m$ and $n$ are relatively prime, these two geodesics are not parallel. They therefore perform the role of the coordinate directions in the square grid graph and, in the same way, can be re-arranged in any order to produce many irregular staircase-like geodesics, all of the same length. The set of these geodesics forms the broad geodesic bundle, with an area proportional to the square of its length – a complete contrast to the narrow lines required by Euclidean geometry.

III Growing disordered graphs

In light of the impossibility of generating Euclidean geometry from planar lattice graphs, we turn to disordered graphs which triangulate the 2-sphere. Triangulations here are graphs composed of triangles which, when embedded in the 2-sphere, are planar Lutz . We also require that they contain no tetrahedra, so that we only need to keep track of nodes and edges, not faces. As a seed graph, we start from the octahedron (Figure 4), a simple triangulation of the 2-sphere. All triangulations of the 2-sphere are known to be transformable into one another by Steinitz moves Steinitz , illustrated in Figure 3, which are local, and add (‘push’) or remove (‘pop’) nodes while preserving the property of being a triangulation.

We grow the seed graph to a size of $N$ nodes through push moves, and then apply $8N$ alternating push and pop moves to ensure equilibration. This equilibration stage is necessary, since some graph properties (such as mean node eccentricity – see section IV) change slightly, converging (presumably to thermal equilibrium values) after around $4N$ alternating push-pop moves moves.

Let $Z_{i}$ be the degree of node $i$, and $\langle Z\rangle$ the mean degree of all nodes in the triangulation. Because every triangular face has three edges, and every edge belongs to two triangles, Euler’s polyhedron theorem Euler implies that

[TABLE]

Since the integrated Gaussian curvature over a smooth, closed surface is $4\pi$ Bonnet , we see that $\kappa_{i}\equiv 6-Z_{i}$ is a natural measure of the local, discrete equivalent of Gaussian curvature for the triangulation, up to a constant factor. If we consider a patch of the graph consisting of $N_{\rm pat}$ nodes, with $e$ exiting edges, and with a simple closed-path perimeter of length $p\geq 3$ edges, then we find the Euler characteristic implies the average discrete curvature over all nodes in the patch is

[TABLE]

This can be shown by considering a new triangulation, formed from two copies of the patch, and identifying nodes and edges on the perimeters, then correcting for the exiting edges. Thus a Steinitz push move locally decreases $|\langle\kappa\rangle_{\rm pat}|$, and a pop move increases it.

To create an ensemble of graphs, we first define an energy $E$ for every graph. We then repeatedly select a random node as a candidate for a push or pop move, and calculate the energy change $\Delta E$ that would result. We perform the move with a probability given by the Metropolis algorithm Metropolis with an associated temperature $T$. Thus, the move is always accepted if $\Delta E$ is negative, and accepted with probability $\exp(-\Delta E/T)$ if $\Delta E$ is positive.

Curvature model

The most obvious choice of energy to reduce curvature fluctuations at low temperature is $E_{\rm curv}=\sum_{i}\kappa_{i}^{2}$, where the sum is over all nodes $i$. As shown in Figure 4 and also considered in Aste , this does indeed drive the local curvature to zero almost everywhere at low temperature, but it does so by creating a branched polymer phase consisting of thin tubes with curvature trapped at their ends and junctions (Figure 4b). The result of this ‘curvature model’ is far from flat. We attribute this to the energy functional failing to sufficiently penalize small curvatures spread over large areas.

Walker model

To address the deficiency of the curvature model, we introduce a second statistical process by putting walkers on the graph. Walker models have previously been used to create both local-cluster structure Ikeda ; Vazquez ; Toivonen , as well as scale-free Krapivsky graphs from local rules Saramaki ; Caravelli ; but here we are interested in Euclidean behavior. At each time step, we add $\kappa$ walkers of type $+1$ to every node with curvature $\kappa>0$, and $|\kappa|$ walkers of type $-1$ to every node with $\kappa<0$. Additionally, 12 walkers of type $-1$ are added to random nodes to maintain the mean walker number [this requirement can be seen from eq. (1)]. The walkers then diffuse by moving to a random neighboring node. Whenever a $+1$ and a $-1$ walker occupy the same node, both walkers annihilate. Walker moves alternate with push-pop moves. To define the dynamics, we replace $E_{\rm curv}$ with a new energy $E_{\rm walk}$ for the graph under push-pop moves:

[TABLE]

where $w_{i}$ is the net number of walkers on node $i$. At low temperatures, this energy tends to shrink regions of positive curvature and grow regions of negative curvature. We call this new evolution scheme, which biases the graph towards flatness on long length scales, the ‘walker model’.

The walker model generates a triangulation which, at low temperature and long lengths, appears qualitatively to have minimal curvature (Figure 4c). To establish that these graphs satisfy Euclidean geometry at long length scales, we subject them to three tests: a Hausdorff dimension of 2; geodesic confinement; and the Pythagorean theorem.

IV Testing our graphs

Euclidean geometry is defined through five axioms. These are neither as logically primitive as they first appear, nor do they readily translate into conditions for discrete models of space. We therefore propose three conditions for any discrete model, including ours, purporting to capture Euclid’s geometry at large lengths. The first, Hausdorff dimension, sits outside the original axioms, since they concerned the plane. The second condition is the appearance of straight lines in the large length limit, which we call geodesic confinement. The third is the Euclidean metric itself, commonly known as Pythagoras’ theorem, which is a synthesis of all the axioms.

Hausdorff dimension

If the number of nodes in a ball of radius $r$ scales as $N\propto r^{d_{H}}$, then $d_{H}$ is the Hausdorff dimension of the graph. As we noted earlier, planarity is not indicative of dimension: random triangulations of the 2-sphere have $d_{H}=4$ as they converge, in the large node limit, to ‘Brownian maps’ leGall . To calculate the dimension of our graphs, we define the half-circumference $H$ of a graph as the average over all nodes of the node eccentricity, where the eccentricity of a node is the greatest geodesic distance between it and any other node in the graph. If nodes are a measure of area, then we would expect a graph which approximates a smooth spherical surface with $d_{H}=2$ to satisfy the scaling $H\propto N^{1/2}$. This is not the case for the curvature model (Figure 6a), but is true for the walker model in the low temperature limit for a large number of nodes (Figure 6b). The upwards curvature of the solid gray lines in Figure 6b shows evidence that this $d_{H}=2$ phase survives to temperatures above zero.

Geodesic confinement

In a doubly-periodic graph, the total number of nodes $N_{\rm geo}$ in the geodesic bundle between two nodes a distance $\lambda$ apart scales as $N_{\rm geo}\propto\lambda^{2}$. From Figure 6cd, we see that the scaling of $N_{\rm geo}$ with $\lambda$ also approximates a power law for the low-temperature walker model, but with a different exponent:

[TABLE]

An exponent $\gamma<2$ implies qualitatively different behavior to the doubly-periodic lattice case, and in the limit $N\rightarrow\infty$, it is consistent with the narrow geodesics (‘straight lines’) familiar from Euclidean geometry. We call the collapse of the broad, $N_{\rm geo}\propto\lambda^{2}$ geodesic bundles ‘geodesic confinement’ (Figure 2b), by analogy to the flux tubes and color confinement seen in strong-force interactions ColorConfinement .

Pythagoras’ theorem

Finally, we consider the validity of Pythagoras’ theorem on graphs generated by the walker model. Although this can be proved in general for Euclidean geometry, on graphs we test it empirically by calculating the length of the perpendicular of an equilateral triangle. If Pythagoras’ theorem holds, this will be $\sqrt{3}$ times half the side length.

Because we are generating approximations to a spherical surface, rather than a plane, we want to make use of as much of the graph as possible, rather than a small patch on which statistics will be poor. We therefore perform the analogous calculation using spherical, rather than plane trigonometry. If we draw an equilateral spherical triangle on a smooth 2-sphere, with side-length $\Lambda$ times the half-circumference, the ratio of the length of the perpendicular of the triangle to half its side length will be

[TABLE]

The same ratio $R$ can be calculated for a graph formed by the low-temperature walker model (Figures 6ef, 7), and although the fluctuations are significant, they appear to be unbiased, so that performing linear regression of $R$ against $R_{\rm sph}$ gives an estimate for $\sqrt{3}$ which is only one standard deviation from the traditional value:

[TABLE]

We believe this is a non-trivial result, unlikely to emerge accidentally, and so we take it as strong evidence that Pythagoras’ theorem is satisfied in general for the low-temperature walker model. Since the straightedge and compass operations of drawing circles of any radius, and drawing and measuring (but not extending) lines are simple operations on graphs, many other constructions of classical geometry may readily be tested.

V Methods

Our proof that all planar lattice graphs satisfy the taxicab metric is in two parts, which we call geodesic composition and geodesic rearrangement.

Proof of geodesic composition

Consider a doubly-periodic planar graph made up of identical unit cells, each of which comprises $\omega$ distinct nodes. Equivalent nodes in different unit cells are said to be of the same type. Let ${\cal G}_{pp}({\rm\bf v})$ denote a particular geodesic between two $p$-type nodes separated by $\mathbf{v}=(m,n)$ unit cells.

We first prove that for any displacement $\mathbf{v}$, for at least one node type $p$, the concatenation ${\cal G}_{pp}(k{\rm\bf v})$ of $k$ copies of ${\cal G}_{pp}({\rm\bf v})$ is also a geodesic (Figure 8a–d). Let $p$ be the node type which minimizes ${\cal G}_{pp}({\rm\bf v})$. Call this the optimal node assumption. Let $p_{0}p_{1}$ of length $|p_{0}p_{1}|=\lambda$ be a geodesic between $p_{0}$ and $p_{1}$, which are both of type $p$, but displaced ${\rm\bf v}$ units cells from one another (Figure 8a). Call this the $\mathbf{v}$-geodesic assumption. Let $p_{0}p_{1}p_{2}$ be two copies of $p_{0}p_{1}$ placed end-to-end.

Now suppose there is a path $p_{0}abp_{2}$ with length $|p_{0}abp_{2}|<|p_{0}p_{1}p_{2}|=2\lambda$ (Figure 8b); because the graph is planar, nodes $a$ and $b$ exist. Then $|ab|<\lambda$ or $|p_{0}a|+|bp_{2}|<\lambda$. If the former, then we contradict the optimal node assumption. If the latter, we contradict the $\mathbf{v}$-geodesic assumption. Therefore $p_{0}p_{1}p_{2}$ is a geodesic between $p_{0}$ and $p_{2}$. That is to say, ${\cal G}_{pp}(2{\rm\bf v})$, which is the concatenation of 2 copies of ${\cal G}_{pp}({\rm\bf v})$, is a geodesic. Call this the $2\mathbf{v}$-geodesic property.

We now show that the $(k-1)\mathbf{v}$-geodesic property implies the $k\mathbf{v}$-geodesic property (Figure 8c for $k=3$). Suppose there is a path $p_{0}abp_{k}$ with length $|p_{0}abp_{k}|<|p_{0}p_{1}\ldots p_{k}|=k\lambda$. Then $|ab|<\lambda$ or $|p_{0}a|+|b\,p_{k}|<(k-1)\lambda$ (Figure 8d for $k=3$). If the former, then we contradict the optimal node assumption. If the latter, then we contradict the $(k-1)\mathbf{v}$-geodesic property. Therefore $p_{0}p_{1}\ldots p_{k}$ is a geodesic between $p_{0}$ and $p_{k}$. This completes the first part of the proof.

Proof of geodesic rearrangement

We next prove that for most displacements $\mathbf{v}$, for at least one node type $p$, the geodesic ${\cal G}_{pp}(k{\rm\bf v})$ consists of three parts: a tail at each end, which joins the nodes $p_{0}$ and $p_{k}$ to copies of some other type of node $q$, and between the tails, $k-1$ alternating copies of ${\cal G}_{qq}({\rm\bf u})$ and ${\cal G}_{qq}({\rm\bf u^{\prime}})$ for some displacement vectors ${\rm\bf u}$ and ${\rm\bf u}^{\prime}$ (Figure 8ef). We now only consider displacement vectors ${\rm\bf v}=(m,n)$ such that $m$ and $n$ are relatively prime (which occurs Hardy for random $m$ and $n$ with probability $6/\pi^{2}\simeq 0.61$) and large enough so that $\lambda>2\omega$, where $\omega$ is the number of distinct nodes in the unit cell. By Dedekind’s pigeonhole principle Pigeon , since $\lambda/\omega>2$, ${\cal G}_{pp}({\rm\bf v})$ must pass through at least two nodes of some other type $q$ different from type $p$ (Figure 8e). Therefore we can define a sub-geodesic ${\cal G}_{qq}({\rm\bf u})$ within ${\cal G}_{pp}({\rm\bf v})$, and a second geodesic ${\cal G}_{qq}({\rm\bf u}^{\prime})$ between the node $q$ in adjacent copies of ${\cal G}_{pp}({\rm\bf v})$ (Figure 8f).

Because $m$ and $n$ are relatively prime, ${\rm\bf u}$ and ${\rm\bf u}^{\prime}$ cannot be parallel. To see why, let the displacement ${\rm\bf u}$ be $(i,j)$ and the displacement ${\rm\bf u}^{\prime}$ be $(i^{\prime},j^{\prime})$ and assume $i^{\prime}\geq i$. Since ${\rm\bf u}\parallel{\rm\bf u}^{\prime}$ implies $i/j=i^{\prime}/j^{\prime}$, $(m,n)=(i+i^{\prime},j+j^{\prime})=(1+i^{\prime}/i)(i,j)$, where $i^{\prime}/i$ is an integer, contradicting $(m,n)$ being relatively prime.

The $k-1$ alternating geodesics can be rearranged in any order, forming a set of staircases between the end $q$ nodes (Figure 8f). Therefore the geodesic bundle occupies an area of $m\,n(k-1)^{2}$ unit cells. This completes the proof.

Computer code

See Supplemental Material at [URL will be inserted by publisher] for the computer code (in the C progamming language) used to generate the data in Figure 6, as well as the pctures in Figures 4, 5 and 7. There is also a video showing the different network models’ dynamics.

VI Discussion

We have shown that discrete space and Euclidean space, thought by many to be at odds, are indeed compatible. We avoid Zeno’s paradox because we do not require our model to be infinitely divisible. We avoid Weyl’s tiling argument because our model is disordered. Weyl’s argument is in fact an observation that certain non-planar lattices display the taxicab metric, which is unsurprising given our proof that all planar lattice graphs do.

Our model shows the emergence of Euclidean space on long length scales, from a local statistical process that employs only the intrinsic geometry of a graph. Beyond providing the first discrete model of the Euclidean plane, we believe our results draw together several disparate fields of physics, and prompt many intriguing questions that merit further investigation.

No embedding space

Smooth surfaces which are discrete at an atomic scale frequently arise in nature, such as liquid menisci or crystal surfaces Burton . These atomic systems are embedded in a background manifold, consisting of ordinary, flat, three-dimensional space. This embedding manifold allows distance on the surface to be defined in the usual Euclidean manner, and also means that normals to the surface exist. The system energy can then depend on extrinsic curvature (the spatial gradient of these normals), as well as intrinsic (Gaussian) curvature. Our graphs, by contrast, do not live in a background space. Instead, our measures of distance and curvature are only intrinsic, defined in terms of edges (distance) and node degree (curvature) that are properties of the graph itself. No normal vectors to our graph manifolds exist, nor can they be defined. Our graphs are themselves the space, and their edges are the quantum of distance.

We note that because our graphs approxmate boundary-less two-dimensional manifolds, we can approximate the plane only by forming a very large 2-sphere (since a large, smooth 2-sphere is locally close to a plane), or by forming a topological 2-torus. Our model in its current form is not able to grow any manifold with a boundary, such as a disc.

Phase transition

Phase transitions which create or destroy smoothness are well known in physics. A roughening transition Burton can turn flat crystal facets into smooth, curved surfaces, as measured with the metric of the embedding space. However, this embedding space is needed to define what smooth curvature means in this case.

More strikingly, the crumpling transition of membranes Bowick turns flat crystalline membranes into crumpled objects, confined in a small region of space. However, the irregular, jagged curvature of the crumpled phase is entirely extrinsic: a function of its embedding in three-dimensional space. The intrinsic, ordered, taxicab geometry of the membrane itself is unchanged through the crumpling transition.

In contrast, the phase transition we find at low temperature in the walker model changes the intrinsic metric of the graph from a crumpled, non-Euclidean ‘Brownian map’ leGall into smooth, Euclidean space. It is unclear, at present, whether the phase transition occurs at finite or infinite temperature. A renormalization group analysis of the model may shed light on this question.

Walker model

The phase transition which creates continuum geometry is driven by a statistical walker process. The motivation for this comes from the naïve curvature model, which minimizes the sum of the squares of the local discrete curvature $\kappa$, but disappointingly gives rise to a ‘Medusa’ phase (Figure 4b). This pathological behavior is consistent with previous investigations of triangulations, which lead to branched polymer phases and other exotic geometries rather than smooth, homogenous space Aste ; Gurau . The pathologies are due to concentrations of discrete curvature in confined regions; in other words, large, local curvature fluctuations. Our walker process – which solves a discrete version of Poisson’s equation, with the charge being the curvature $\kappa$ – is sensitive to small curvatures on large length scales, and so, through the energy functional $E_{\rm walk}$, ultimately acts to spread these fluctuations over the whole graph.

A background for simulations

A practical application of our Euclidean graphs is as a background for simulations. Lattices, such as the square grid, are intrinsically anisotropic, so special care is often needed when designing simulations to run on them. The rotational symmetry of our graphs therefore make them suitable spaces on which to perform algorithms such as lattice gas cellular automata LGCA .

Higher dimensions

We have built a discrete, graph model that behaves like two-dimensional Euclidean space at large lengths. Can the same be done for higher dimensions? While more computationally intensive, our walker model should generalize naturally to dimensions greater than two. In three dimensions, the key step is extending the Steinitz moves in Figure 3 to add and subtract tetrahedra, rather than triangles, as nodes divide and fuse. Whether the resulting graph will be Euclidean is, however, unknown. Our tests for geodesic confinement and the applicability of Pythagoras’ theorem are benchmarks for this and any other discrete models attempting to capture Euclidean geometry at large lengths.

We conjecture that the absence of geodesic confinement carries over to higher dimensional lattices, as it clearly does for the three-dimensional regular cubic grid. Unfortunately, the proof does not readily follow from our theorem in two dimensions, which relies on planarity, since all three-dimensional lattices are non-planar. Figure 2c gives an indication of the subtlety. It shows a non-planar, two-dimensional lattice not satisfying geodesic composition, a key step in our proof (see Methods).

Closely related results have been proved for more general cases. It is known that for asymptotically large radii, balls around a node in a lattice graph of any dimension can never be ellipsoids, but are rational polytopes Fritz . Isotropic behavior of geodesics is therefore impossible, and if these polytopes were exact, rather than having finite length whiskers in certain directions (the eventuality of Figure 2c) broad geodesic bundles would follow from the partial incidence of the faces of polytopes of radius $r$ around two points separated by $2r$.

The Minkowski metric

We have shown how to grow graphs with a Euclidean metric, that is, that satisfy Pythagoras’ theorem, $d^{2}=x^{2}+y^{2}$, where $d$ is distance and $x$ and $y$ orthogonal directions. What about other metrics? The most important is the Minkowski metric of special relativity, the two-dimensional analog of which is $d^{2}=t^{2}-x^{2}$, where $t$ is a time direction. How to represent this as a graph is an open question, because nodes must be intricately connected at large coordinate displacements. Taking an approach similar to causal set theory Bombelli ; Benincasa , but with neighbors separated by unit proper time, would suggest that the degree of each node diverges with the logarithm of the volume of space-time (or worse, as a power, for higher dimensions). Furthermore, unlike Euclidean space, where the square grid graph at least models a 4-fold rotational symmetry, it is not possible to construct a lattice graph which is symmetric under even a discrete version of the Lorentz transformation. Thus, it remains to be seen whether some variant of the walker process can be defined to probe and engender the fabric of space-time.

Acknowledgements.

The authors thank Prof. Yang-Hui He and Dr Ilia Teimouri for useful discussions.