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
3.2.4. Topological Rotation
3.2.5. Conservation of Momentum
3.2.6. A Lattice of Communication
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.
“Fn 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 Fn (n)”. [73], p. 121.
3.2.8. Precision in Projected Space
3.3. Stability of the LCT
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
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
References
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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
Sang N. Does Time Smoothen Space? Implications for Space-Time Representation. ISPRS International Journal of Geo-Information. 2023; 12(3):119. https://doi.org/10.3390/ijgi12030119
Chicago/Turabian StyleSang, 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