# Does Time Smoothen Space? Implications for Space-Time Representation

## Abstract

**:**

## 1. Introduction

#### 1.1. Research Background

#### 1.1.1. Spatial Granularity in Theory

“Granularity is closely related, but not identical, to imprecision. Granularity refers to the existence of clumps or grains in information, in the sense that individual elements in the grain cannot be distinguished or discerned apart”. Duckham et al. [1]

“When we state that observables are not pure numbers but operators, and the observed values of these observables are the eigenvalues of these operators, that alone is sufficient to ensure that two operators which do not commute must be related by an uncertainty principle”. [21]

#### 1.1.2. Spatial Granularity in Practice

#### Intersect and Overlay

#### Visibility Analysis

#### Graph Partition and Granulation

#### Path Analysis

#### From Tilings to Lattices

## 2. Establishing Whether All Locations on a Map Can Be Uniquely Addressed

#### 2.1. Unique, Spherical Address Units

#### 2.2. Kepler Conjecture

”The packing density δ (Λ) of any sphere packing Λ in R3 does not exceed π √18 ≈ 0.74048”. [45]

#### 2.3. Interim Conclusions

## 3. Does Time Smoothen Space?

#### 3.1. Space-Time Representation

- Time is directional (the arrow of time);
- Chaotic behavior may amplify small uncertainties through time;
- Relative space and time are subjective, and neither exists independently;
- Both space and time are generally conceived as continuous, yet for purposes of objective measurement, they are conventionally broken into discrete units;
- Temporal data is often incomplete. Science has traditionally viewed incompleteness as something to be overcome rather than to be openly acknowledged or willingly incorporated into models as a basic characteristic.

#### 3.2. Time as Relative Dimension in Space

#### 3.2.1. Premises

A necessary constraint is ”to get rid of the continuum and build up physical [… read spatial…] theory from discreteness”. [60]

One should “concentrate only on things which, in fact, are discrete in existing theory and try and use them as primary concepts—then to build up other things using these discrete primary concepts as the basic building blocks. Continuous concepts could emerge in a limit, when we take more and more complicated systems”. [61]

”The most obvious physical concept that one has to start with…and which is connected with the structure of space-time in a very intimate way, is in angular momentum”. [60]

#### 3.2.2. Building Dimensions

#### 3.2.3. Building a Probabilistically Continuous Dimension from a Graph

^{n−2}as given by Cayley’s formula [67], but every connected graph contains at least one spanning tree [68], (theorem 4.12). It is thus always possible for such a connected graph to randomly form an ordered set [66] (e.g., a tuple, line, or tree, etc.) where every node has a unique matrix of graph distance to all other nodes. If such a graph describes the diffusion of information throughout a network, then the maximum “time” (in a number of moves) it could take for information to diffuse throughout the network is when the spanning tree is an Euler path. Simple paths between other nodes will therefore require some fraction of this maximum. The probability of a topological chain (a tuple) of length no greater than i to form a path between any two particular nodes is given by Equation (1):

^{th}node added to the path; n = total nodes. i.e., the initial chance of a direct connection (P = 2/n) plus the sequential sum of the chances that each further node added (v) to the chain will be the destination, each such chance being 1/n.

#### 3.2.4. Topological Rotation

_{j}.

#### 3.2.5. Conservation of Momentum

_{ij}follows distribution d defined by probabilities P (from Equation (2)), and total angular momentum M follows distribution D, and each subgraph n belongs to a set of points N.

#### 3.2.6. A Lattice of Communication

_{ij}. For all cases where this is so, the probability P of the path ij is such that ij has a greater probability than iv.

_{ij}becomes more probable than alternative paths from i to v. Figure 2 illustrates this, but by taking a node i, which projects to the same point in LCT space as the rest of G and a path from this node to j, which is reached via some structured route therefrom. This divergence “stretches” G from its projected point-like form into a vector. Other branches are added to illustrate that if structured paths exist from j to other points, these could also project to vectors which, in this simple case, are resolvable in 2D (Figure 2).

#### 3.2.7. Building a Probabilistically Continuous Volume

“a set of vectors is linearly independent if no one of them can be expressed as a linear combination of the others” [73], p. 116.

“F_{n}can’t contain more than n independent vectors. Our definition of dimension will in fact amount to saying that the dimension of an F-vector space V is the maximal number of independent vectors. This definition gives the right answer for the dimension of a line (one), a plane (two) and more generally F_{n}(n)”. [73], p. 121.

#### 3.2.8. Precision in Projected Space

**B**, which has the dimensions dx, dy (being less than or equal to radius r).

**B**can fill space, but this can be resolved by recognizing that there is indeed the possibility for every relative position in LCT, but not every vector can be embedded to a common absolute precision in ${\mathbb{R}}^{\mathrm{n}}$ simultaneously. Thus, if the graph topology to resolve the LCT requires ${\mathbb{R}}^{\mathrm{n}}$ for a continuous space, realizing this requires at least ${\mathbb{R}}^{\mathrm{n}+1}$. Another way of viewing this conclusion is to say that the lattice in ${\mathbb{R}}^{\mathrm{n}}$ represents the most complete (probable) mapping between LCT and ${\mathbb{R}}^{\mathrm{n}}$ achievable.

_{2}must be an element of the B

_{1,2}vector dual V in the next dimension in ${\mathbb{P}}^{}$.

#### 3.3. Stability of the LCT

**T**be a tuple with a Gaussian probability of describing the shortest path over the field $\mathcal{F}$ (from Equation (6)). Let V be a linear vector in $\mathcal{F}$ having the same start x

_{a}and endpoint x

_{b}as

**T**(Equation (9)):

**V**and the elements of the Tuple

**T**is not empty (and thus the path taken was not linear) is equal to the probability that the integral path dt of the vector of a free particle moving from x

_{a}to x

_{b}over $\mathrm{\tau}$ convolutions, minus the straight-line vector

**V**= [x

_{a}x

_{b}] does not equal zero given a potential set of paths $\mathcal{D}\mathrm{xy}$. $P$ must have a value equal to or less than 0.5 since, for a Gaussian distribution, a linear route is more probable than an indirect route.

^{t}≃ 4D)

**.**This “topological time” is conjugate with the three spatial dimensions; it is not an independent “arrow” of time but Rovelli [79] shows the traditional sense of time as an independent dimension directing “progression” could be emergent from partial ordering effects at each spatial scale.

#### 3.4. Summary

## 4. Discussion: From a Dynamic LCT to a Computable Lattice

#### 4.1. Implications for Space-Time Representation

- Ideally, use discrete units which can be “smoothly” space-filling.
**(d,e)**; - Ideally, represent incompleteness and non-commutativity.
**(a,c,e)**; - Ideally, use isometric grains.
**(b)**; - Ideally, represent granular space-time conjugation explicitly.
**(c)**; - Ideally, represent local uncertainty in causality while respecting causality globally.
**(a)**(Galton et al. [80] observe that “granularity effects may often confound an attempt to derive strict causation”).

#### 4.2. FCCP for Space-Time Representation

## 5. Conclusions

#### 5.1. On the Representation of Time

**Computable representation of space-time:**How does the selection of an optimal grain size reflect the fact that there may be a non-linear uncertainty between a measurement made too early or too late, e.g., as to the position of a shadow at noon, which itself varies over seasons, scales and locations? Consider also how spatio-temporal conjugation affects the examples from Section 1. Intersect becomes intercept, at the grain precision in measuring speed must offset precision in estimating location. Visibility Analysis becomes not just a matter of where there is a line of sight but when and for how long. Graph granulation is commonly applied to space-time applications, but doing so with vague location [35] means spatial and temporal uncertainty are not independent. Goodchild’s [40] early work on path analysis must surely come into play, but whether 4D lattice paths tend to the Euclidean length in the same asymptotical manner is an open question;

**Incompleteness:**Hales’ proof [46] undermines the principle that absolute vs. discrete space is a matter of differentiation alone; precision-independent incompleteness needs further exploration;

**Spatio-Temporal Scale(s):**Although Peuquet agrees, “it usually does not make sense to measure time in meters or feet”, [53] the point has been moot since 1960 when the meter was defined in wavelengths of light. Perhaps the distinction between meters and the time by which they are measured over-complicates space-time for 4D models? One would not, after all, model a terrain’s height on a different mathematical base to its horizontal distance. On the other hand, seconds are not fundamental scientific units either, one might ask “seconds per what?” (Smart, see Harrison [59]).

#### 5.2. On Granularity and the Emergence of Spatial Smoothness

## Supplementary Materials

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Figure 7a (left) and Figure 8 (right) from Shortridge and Goodchild [12] (reproduced by permission, license 5104250425992).

**Figure 2.**Projecting the graph G into 2D. All nodes with equal probability of connecting project to point G. Nodes with a higher probability of connecting to each other than to nodes in point G project to sub-graphs with geometric position based on relative path length in units of fractions of the total momentum of the system M.

**Figure 3.**From Song and Miller [76] Visit probability using random walk theory, reproduced by permission license number 5104220298977.

**Figure 4.**Steps for the emergence of fuzzy space-time from a discrete graph: (

**a**) a triad of three nodes; (

**b**) a simple path chaining together triads; (

**c**) an orbit (topological “clock”); (

**d**) clustering of the graph; (

**e**) latent vectors over the graph (

**f**) uncertainty in the relative position of nodes, represented geometrically as a circular complement to the position (but not in projected space); (

**g**) packing of isometric uncertainty complements in projected space (B1 and B2 represent solutions to paths ABD and EADCBAEBD respectively); (

**h**) geometric projections of the graph (

**i**) motion as a path integral through a hexagonal lattice, yellow spheres represent possible positions for midway points between green spheres (i.e., the central node in each triad).

**Figure 5.**Representations of a trajectory in space-time. (

**A**) Implicit time (e.g., animation). (

**B**) One vertical side of a space-time cube. (

**C**) FCCP, black dot in yellow circle = time step, red dot in grey circle = inter-time step.

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Sang, N.
Does Time Smoothen Space? Implications for Space-Time Representation. *ISPRS Int. J. Geo-Inf.* **2023**, *12*, 119.
https://doi.org/10.3390/ijgi12030119

**AMA Style**

Sang N.
Does Time Smoothen Space? Implications for Space-Time Representation. *ISPRS International Journal of Geo-Information*. 2023; 12(3):119.
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**Chicago/Turabian Style**

Sang, Neil.
2023. "Does Time Smoothen Space? Implications for Space-Time Representation" *ISPRS International Journal of Geo-Information* 12, no. 3: 119.
https://doi.org/10.3390/ijgi12030119