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How stiff is space-time?

[“Quantum foundations” (field theory) series] [Draft]

Credit: Pixabay/CC0 Public Domain

If space-time is considered as a kind of material, then what properties make sense? Properties that can be observed, measured.

In the context of General relativity, (aggregate) properties related to curvature and rippling. Like elasticity, stiffness.

In the context of quantum physics, (discrete) properties related to quanta, energy density, charge, spin, entanglement. Like gradients, flux – “jitter and jive.” So, …


How stiff is space-time?

(quotes) … space-time is extremely stiff … even rapid motions involving large amounts of mass [e.g., mergers of two black holes] produce only tiny wiggles in space-time.[1]

Space-time can be distorted, but it’s very hard work.

Upon combining general relativity with quantum mechanics, we calculate that space is a kind of quivering Jell-O, in constant motion.

In gravitational waves, bending of space-time in some directions causes bending in others. – Wilczek, Frank. Fundamentals – Ten Keys to Reality. (Chapters 8, 2, 1, 8)

Well, what bends space-time?

(quote) Bending space-time requires energy, and energy causes space-time to bend. – ibid. Chapter 8.

The cosmological constant merely recognizes the possibility that space-time itself, a material that general relativity allows to bend, push, and shake, might also have inertia. – ibid. Chapter 9.

How does the interaction of quanta at a discrete level result in bending of space-time at an aggregate level? (Like pressure on a flexible membrane.)

How can we tell if space-time is bent?

[List of experiments]

Gravitational lensing

Time gradients [a better term?] [notable experiments]

Models and metaphors, levels of description

• “Times are legion” … “a different rhythm in every different place” – dynamic “spiderweb of times”

As Carlo Rovelli remarks in his book The Order of Time, “Times are legion: a different one for every point in space.” [2]

* Quanta Magazine > “An Ultra-Precise Clock Shows How to Link the Quantum World with Gravity” by Katie McCormick (October 25, 2021)

(quote) In a paper posted earlier this month to the scientific preprint server, researchers from the lab of Jun Ye, a physicist at JILA in Boulder, Colorado, measured the difference in the flow of time between the top and the bottom of a millimeter-tall cloud of [100,000 ultracold strontium] atoms.

* Science News > “An atomic clock measured how general relativity warps time across a millimeter” by Emily Conover (October 18, 2021)

Does the variation in time flux (the different rate at which time passes at locations in space-time) bend space-time, or does bending in space-time cause variation in time flux? [A better question is …]

• “The incessant flux of quantum Grid” – dynamic equilibrium

Wilczek’s characterization of space-time is predicated on a model which he calls the Grid, a multilayer structure, a dynamic medium (“aboil with virtual particles”), whose properties are quite different from the classical ether.

(quote) The entity we perceive as empty space is a multilayered, multicolored superconductor. What an amazing, astonishing, beautiful, breathtaking concept. Extraordinary, too. – Wilczek, Frank. The Lightness of Being (p. 97). Basic Books. Kindle Edition.

And his metaphor of the Grid as an exotic (cosmic) superconductor. With condensates. [3]

Like the gauge bosons, all these particles would be massless but for Grid superconductivity. But Grid superconductivity gives them mass [neutrinos are a special case] and also allows the heavier ones to mix with, and thereby decay into, lighter ones in complicated ways. – ibid. (p. 169)

Emergence in space-time

From the incessant flux of the Grid and the legion of times, meaningful patterns emerge. An order via confinement. A dynamic stability via chaos.

(quote) To get the most out of the big bang theory, we need to refine our assumption that the distribution of matter early on was completely uniform. Small deviations from uniformity will do, because they get amplified by gravitational instability. – Wilczek. ibid. Chapter 7.


[1] How might such stiffness be quantified? By sampling reality.

We can make a pretty reliable estimate of what it takes to knock loose a piece of the condensate responsible for the Grid’s (electroweak) superconductivity. The weak force is short-ranged, but not infinitely so. The W and Z bosons are heavy, but not infinitely so. The observed range of the force, and mass of the force carriers, give us good handles on the stiffness of the condensate responsible for those effects. Knowing the stiffness, we can estimate how much energy we need to concentrate in order to break off individual pieces (quanta) of the condensate … or whatever kind of new stuff … that makes the Grid a cosmic superconductor. – Wilczek, Frank. The Lightness of Being (p. 194). Basic Books. Kindle Edition.

Also, as Wilczek remarks, the Grid’s total density is likely small; otherwise, there would be no sentient beings to observe the universe. The fact that, sort of like fish, we “swim” in the Grid easily.

And, as noted in another comment, here’s a link to a paper (PDF) by Wilczek for the MIT Physics Annual 2009, “What is Space?” The paper is a useful recap of the history of space — so-called empty space, the void, the plenum, ether; and ends with a brief overview of a toy model of quantum reality.

[2] See Rovelli, Carlo. The Order of Time (pp. 14-18). Penguin Publishing Group. Kindle Edition.

But if different clocks mark different times, as we have seen above, what does ‘t’ indicate? When the two friends meet up again after one has lived in the mountains and the other at sea level, the watches on their wrists will show different times. Which of the two is ‘t’?

In a physics laboratory, a clock on a table and another on the ground run at different speeds. Which of the two tells the time? How do we describe the difference between them? Should we say that the clock on the ground has slowed relative to the real time recorded on the table? Or that the clock on the table runs faster than the real time measured on the ground?

The question is meaningless. … There is no “truer” time; there are two times and they change relative to each other. Neither is truer than the other.

But there are not just two times. Times are legion: a different one for every point in space. There is not one single time; there is a vast multitude of them.

Time has lost its first aspect or layer: its unity. It has a different rhythm in every different place and passes here differently from there. The things of this world interweave dances made to different rhythms.

[3] In particular, Chapter 8 in Wilczek’s book The Lightness of Being is all about the Grid. His Recapitulation at the end of the chapter lists key Grid properties. For example, a metric field for space-time rigidity; a universal (nonzero) density.

And the dynamics of interactions across Grid layers (quantum dimensions).

You can visualize the quantum dimensions as new layers of the Grid. When a particle hops into these layers its spin changes, and so does its mass. Its charges—electric, color, and weak—stay the same. – Wilczek, Frank. The Lightness of Being (p. 188). Basic Books. Kindle Edition.

11 thoughts on “How stiff is space-time?

  1. Here’s a sketch, my attempt at a simple visualization (cartoon) of Wilczek’s Grid. Grid dynamics interplay.

    Multilayered cosmic grid sketch

    Perhaps a visualization, a conceptualized framework, similar to a geographic information system (GIS)? A Grid “map” with overlays for spatio-temporal location and corresponding attribute information. Overlays (which may be toggled) for continuous fields and discrete objects. Thematic layers. Topological relationships.

    GIS layered grid

    Cf. Hydrological and Cartographic modeling with the dimension of time.


    An example of use of layers in a GIS application. In this example, the forest-cover layer (light green) forms the bottom layer, with the topographic layer (contour lines) over it. Next up is a standing water layer (pond, lake) and then a flowing water layer (stream, river), followed by the boundary layer and finally the road layer on top. The order is very important in order to properly display the final result. Note that the ponds are layered under the streams, so that a stream line can be seen overlying one of the ponds.

    Update 11-5-2021

    I like this visualization (below), although the article uses the diagram in a different context.

    • > “Quantum physics in proteins: AI affords unprecedented insights into how biomolecules work” by Deutsches Elektronen-Synchrotron (11-3-2021)

    Quantum wave packet between two layers

    (caption) Illustration of a quantum wave packet in close vicinity of a conical intersection between two potential energy surfaces. The wave packet represents the collective motion of multiple atoms in the photoactive yellow protein. A part of the wave packet moves through the intersection from one potential energy surface to the other, while the another part remains on the top surface, leading to a superposition of quantum states. Credit: DESY, Niels Breckwoldt

  2. Here’s my attempt to “package” the structure of Wilczek’s Grid, using the metaphor of the Grid as a layered superconductor.

    0. space-time substrate
    0.1 virtual energy stratum
    0.2 field-condensate strata [1]
    0.3 quasi-particle stratum


    [Wilczek, Frank. The Lightness of Being. Basic Books. Kindle Edition.]

    [1] As noted in Wilczek’s The Lightness of Being, “We discussed how empty space is presently filled with various material condensates” (p. 249).

    And the various (relative) densities of condensates (pp. 109-110):

    Here are simple estimates of the densities involved, as multiples of what the astronomers actually find:

    • Quark-antiquark condensate: 10^44
    • Weak superconducting condensate: 10^56
    • Unified superconducting condensate: 10^112
    • Quantum fluctuations, without supersymmetry: ∞
    • Quantum fluctuations, with supersymmetry: 10
    • Space-time metric: ? (The physics here is too murky to allow a simple estimate.)

    And stiffness (p. 194):

    We can make a pretty reliable estimate of what it takes to knock loose a piece of the condensate responsible for the Grid’s (electroweak) superconductivity. The weak force is short-ranged, but not infinitely so. The W and Z bosons are heavy, but not infinitely so. The observed range of the force, and mass of the force carriers, give us good handles on the stiffness of the condensate responsible for those effects. Knowing the stiffness, we can estimate how much energy we need to concentrate in order to break off individual pieces (quanta) of the condensate …

    And strata (p. 214):

    (In the language of Chapter 8 there are two condensates, and five distinctive kinds of field disturbances.) Things could be even more complicated. We don’t know. A major goal of the LHC project is to address these questions experimentally.

  3. Regarding visualization of the Grid’s multilayer structure – subspace.

    • Scientific American > “Could Gravity’s Quantum Origins Explain Dark Energy?” by Conor Purcell (October 28, 2021)

    (quote) According to Daniele Oriti, a co-author of the new paper, the core idea behind any theory of quantum gravity is that gravitation arises from a myriad of tiny, discrete, quantum objects that form a sort of hidden underworld, a deeper substructure beneath the familiar dimensions of space and time. “These quantum objects, which are very difficult to imagine,” Oriti says, “are essentially the building blocks of space itself. They do not exist in space, but are themselves the very stuff out of which space is made.

    Oriti explains that the model’s acceleration of the expansion of the universe, during the stage corresponding to today, is caused by interactions between the subspace quantum objects that make up gravity in the theory.

  4. Earlier this year, Ethan Siegel wrote an excellent recap on the reality of virtual particles: “… does the mere fact that we have quantum fields in our universe mean that empty space is actually filled with something physical?”

    The distinction between observable effects of the quantum vacuum and scattering off the quantum vacuum. The concept of vacuum polarization. Vacuum birefringence. The Casimir effect. Proton soup.

    • Forbes > “Ask Ethan: Do Virtual Particles Really Exist?” by Ethan Siegel, Senior Contributor (May 7, 2021) – The effects of virtual particles are real, but the particles themselves are not!

    (quote) This doesn’t mean that empty space itself is full of particles, but rather that you have quantum mechanical operators, including the “particle creation” and “particle annihilation” operators, acting on the vacuum state continuously. This is often visualized as “particle-antiparticle pairs popping in and out of existence,” but that part is just a calculational tool for visualizing what’s happening on a quantum level within empty space.

    The [virtual] particles, however, are not real, in the sense that we cannot collide or interact with them.

    However, if you have real particles — i.e., a non-vacuum state — then the same quantum field theory techniques you would use to calculate the quantum vacuum actually tells you about real, physical particles (and antiparticles) that can pop in-and-out of existence. For example, we normally think of a proton as being made of three quarks, held together by gluons. But when we perform high-energy collisions of these protons and probe their insides through deep inelastic scattering, we actually find all sorts of extra particles inside: extra quarks and antiquarks, an extreme density of gluons, and even leptons and additional bosons in there. Not only are the effects of virtual particles “real” in particle-rich environments, but the particles themselves are real, too.

    Interacting Grid layers?

    … the quantum vacuum itself will exhibit real, physical effects on matter and radiation that passes through them. The vacuum gets polarized, meaning it generates its own internal fields, and those internal fields — not just the external ones — affect the matter and radiation that passes through. However, there are no particles themselves in there to smash into, collide with, or scatter off of.

  5. • As noted in my June 12, 2019 comment for “Acceleration causes gravity, gravity causes acceleration” post:

    CR: And in fact, generally, it is presented in books by saying, “Well, [the] gravitational field is nothing else than spacetime.” But I like better to think of it the other way around. Spacetime, [is] nothing else than gravitational field.

    SC: So Einstein is saying that gravity is not a thing that lives in spacetime, it’s a manifestation of the nature of spacetime itself.

    CR: Exactly, it’s not one additional thing that lives in spacetime, it’s a manifestation of Newtonian space and time.

    CR: And why points lose their meaning in quantum gravity is because the gravitational field is also space. The gravitational field doesn’t live in space, but is space itself. So in the moment in which you look at the quantum properties of the gravitational field, you’re also looking at the quantum property of spacetime, of space.

    So, applying Rovelli’s notion that “The gravitational field doesn’t live in space, but is space itself” makes this NASA APOD caption more about spacetime energy-density (energy-momentum) and geometric stress – stretching and bending (as in fluid dynamics) – rather than attraction and repulsion.

    • NASA > APOD > “Dark Matter in a Simulated Universe” (October 31, 2021)

    (caption) Explanation: … The gravity of unseen dark matter is the leading explanation for why galaxies rotate so fast, why galaxies orbit clusters so fast, why gravitational lenses so strongly deflect light, and why visible matter is distributed as it is both in the local universe and on the cosmic microwave background. The featured image from the American Museum of Natural History’s Hayden Planetarium Space Show ‘Dark Universe’ highlights one example of how pervasive dark matter might haunt our universe. In this frame from a detailed computer simulation, complex filaments of dark matter, shown in black, are strewn about the universe like spider webs, while the relatively rare clumps of familiar baryonic matter are colored orange. These simulations are good statistical matches to astronomical observations. … dark matter – although quite strange and in an unknown form – is no longer thought to be the strangest source of gravity in the universe. That honor now falls to dark energy, a more uniform source of repulsive gravity that seems to now dominate the expansion of the entire universe.

  6. Here’s some further speculation about the Grid, particularly interactive processes between layers and the emergence of localized properties.

    • Inertia (the origin of inertia) as “knots” in spacetime flux-topology?

    Metaphorically as in fluid dynamics, visualize laminar (smooth) vs. turbulent (rough) flow within and between Grid layers.

    Asymmetries in “laminar” flow produce imperfections (defects) in energy flux of the quantum vacuum. Or, a disruption in flow between Grid layers (which must be treated probabilistically), with non-zero vorticity.

    And as to why neutrinos have such negligible inertia – the simplest of twists or knots.

    • Mach’s principle as a framework for spacetime geometry.

    Wiki: Essentially, we humans seem to wish to separate spacetime into slices of constant time.

    However, Einstein was convinced that a valid theory of gravity would necessarily have to include the relativity of inertia:

    So strongly did Einstein believe at that time in the relativity of inertia that in 1918 he stated as being on an equal footing three principles on which a satisfactory theory of gravitation should rest:

    1. The principle of relativity as expressed by general covariance.

    2. The principle of equivalence.

    3. Mach’s principle (the first time this term entered the literature): … that the gµν are completely determined by the mass of bodies, more generally by Tµν.

    In 1922, Einstein noted that others were satisfied to proceed without this [third] criterion and added, “This contentedness will appear incomprehensible to a later generation however.”

    It must be said that, as far as I can see, to this day, Mach’s principle has not brought physics decisively farther. IT MUST ALSO BE SAID THAT THE ORIGIN OF INERTIA IS AND REMAINS THE MOST OBSCURE SUBJECT IN THE THEORY OF PARTICLES AND FIELDS. Mach’s principle may therefore have a future – BUT NOT WITHOUT THE QUANTUM THEORY. — Abraham Pais, in Subtle is the Lord: the Science and the Life of Albert Einstein (Oxford University Press, 2005), pp. 287–288.

    Relational theory à la Smolin, and relational quantum mechanics à la Rovelli.

    Lee Smolin proposes a system of “knots and networks” such that “the geometry of space arises out of a … fundamental quantum level which is made up of an interwoven network of … processes.” [Fluxes, topological interactions?] Smolin and a group of like minded researchers have devoted a number of years to developing a loop quantum gravity basis for physics, which encompasses this relational network viewpoint.

    Carlo Rovelli initiated development of a system of views now called relational quantum mechanics. This concept has at its foundation the view that all systems are quantum systems, and that each quantum system is defined by its relationship with other quantum systems with which it interacts.

  7. Video #13 in Sean Carroll’s “Biggest Ideas” chat series includes an introduction to the Riemann Curvature Tensor.

    Also, in topology, winding numbers, topological defects (in cosmology) in homotopy groups:

    π 0 -> Domain Walls (field theory)
    π 1 -> Cosmic String
    π 2 -> Monopoles
    π 3 -> Textures

    • YouTube > Sean Carroll > The Biggest Ideas in the Universe > #13 “Geometry and Topology” (June 20, 2020)

    (description) This is Idea #13, “Geometry and Topology.” Yes that’s two ideas, and furthermore they’re from math more than from science, but we’ll put them to good use. In particular we look at Riemannian (non-Euclidean) geometry, and a kind of topological invariants called “homotopy groups.”

    (from transcript) … that’s the integers modulo two it’s a set with only two elements and that’s what this disk has it has two different ways of mapping a circle into it topologically either the completely trivial way or the way the wraps around exactly once

    so you don’t have to be able to classify every topological space in terms of winding numbers that are integers okay you can have more subtle situations like this one we’re winding twice as the equivalent winding zero times and then you can … you can calculate pi two of things how you map spheres into different topological spaces how do you map three spheres into different topological spaces

    one of the things that this is very useful for is something called topological defects in cosmology

    if you ever heard of cosmic strings or magnetic monopoles these are classified by these homotopic groups

    so π 0 the number of connected components is related in field theory to things called domain walls we’ll explain later why this is true hopefully

    π 1 helps you calculate whether or not there are cosmic strings

    π 2 tells you whether or not there are monopoles which is kind of fun

    there’s even something called textures which are classified by π 3

    textures are not quite as respectable as the other one so naturally because they can sort of dissolve away in physical fact they’re not stable like the other ones are and so of course I wrote several papers about textures in fact I wrote a whole paper doing nothing but calculating the homotopic group pi/3 of different spaces so you can figure out whether or not different field theories gave rise to textures

    okay, so this is a long video. sorry about that, but I wanted to squeeze in two topics at once because, as fun as these are, both geometry and topology … are mathematical facts not facts about the physical world. so, there are mathematical ideas. the ideas are absolutely big. they’re absolutely crucial, and it’s important to do them, and I wanted to do things in a slightly different way. like you may have heard before that the parallel postulate is not true in non Euclidean geometry. you may have heard before that coffee cups can be deformed into donuts. but I wanted to give you a little bit of the feeling of how actual physicists and mathematicians make use of these facts. how they quantify them. how they turn them into equations. they turn them into things that you can then calculate and then put to use


    Parallel transport

    Riemann Curvature Tensor

    Topological defect (topological soliton)

    In mathematics and physics, a topological soliton or a topological defect is a solution of a system of partial differential equations or of a quantum field theory homotopically distinct from the vacuum solution.

    Domain wall

    A domain wall is a type of topological soliton that occurs whenever a discrete symmetry is spontaneously broken. Domain walls are also sometimes called kinks in analogy with closely related kink solution of the sine-Gordon model or models with polynomial potentials. Unstable domain walls can also appear if spontaneously broken discrete symmetry is approximate and there is a false vacuum.

    Cosmic string

    Cosmic strings are hypothetical 1-dimensional topological defects which may have formed during a symmetry-breaking phase transition in the early universe when the topology of the vacuum manifold associated to this symmetry breaking was not simply connected.

    Texture (cosmology)

    In cosmology, a texture is a type of topological defect in the order parameter (typically a scalar field) of a field theory featuring spontaneous symmetry breaking. They are not as localized as the other defects, and are unstable. No textures have been definitively confirmed as having been detected, but their existence is compatible with current theories and observations of the universe.

  8. In my original post “How stiff is space-time?” (October 25, 2021), I noted two articles on preprint research which measured “the difference in the flow of time between the top and the bottom of a millimeter-tall cloud of [100,000 ultracold strontium] atoms.”

    Here’s another article on that research published in the February 22, 2022, issue of Nature.

    • Science Mews for Students > “A new clock shows how gravity warps time — even over tiny distances” by Kendra Redmond (June 2, 2022) – Using an elaborate optical lattice single-clock design (which spans two rooms), JILA researchers measured time dilation within a tiny cloud of strontium atoms over a distance of one millimeter.

    Jun Ye led the research group … at JILA in Boulder, Colo. (That institute was once known as the Joint Institute for Laboratory Astrophysics.) It’s run by the University of Colorado and the National Institute of Standards and Technology.

    At the clock’s heart is a cloud of 100,000 strontium atoms. They’re stacked vertically and held in place by another laser. That laser effectively chills the strontium atoms into optical molasses — a cloud of atoms that are almost completely frozen in place.

    To test Einstein’s prediction over the smallest height difference yet, the JILA team split the new clock’s stack of atoms in two. The top and bottom stacks were separated by one millimeter. That allowed the scientists to see how fast the clock’s main laser ticked at two different — but very close — heights. This, in turn, revealed how fast time passed at both places.

    Atomic wave
    Credit: Pixabay/CC0 Public Domain

  9. uncrumpled paper

    Regarding my theme of what’s changed in the last 50 – 100 years in mathematical physics …

    Modeling the profile of spacetime: A question of the scalar curvature of three-dimensional spaces with steadily decreasing amounts and concentrations of energy-mass.

    Thinking like a physicist, what details can be ignored? Quantifying flatness. Mapping spaces. Segmentation. Dimensional reduction.

    [Image: “the [2D] bowling ball on a bedsheet” analogy – a “down” gravity well, with hand holding tweezers placing placing a small particle on the grid.]

    • Quanta Magazine > “A Century Later, New Math Smooths Out General Relativity” by Steve Nadis (November 30, 2023) – Mathematicians prove a theorem that illuminates the geometry of universes with tiny amounts of mass.

    … it was not until around 1980 that mathematicians proved, as part of a major theorem in general relativity, that an isolated physical system, or space, without any mass in it must be flat.

    This left unresolved the question of what a space looks like if it is almost a vacuum, having just a tiny amount of mass. Is it necessarily almost flat?

    In a recent paper, Conghan Dong, a graduate student at Stony Brook University, and Antoine Song, an assistant professor at the California Institute of Technology, proved that a sequence of curved spaces with smaller and smaller amounts of mass will eventually converge to a flat space with zero curvature.

    [Illustrating the approach to denuded spaces vs. utter flatness] As a proxy for a space with small mass, he suggested, we might imagine a crumpled sheet of paper that, after being smoothed out again, still has sharp creases and folds. You could use a hole punch to remove the most prominent irregularities, leaving a slightly uneven piece of paper with some holes in it. As the size of those holes shrinks, so will the unevenness of the paper’s terrain. At the limit, you might say, the holes would shrink to zero, the mounds and ridges would disappear, and you’d be left with a uniformly smooth piece of paper — a genuine stand-in for flat space.

  10. contour dynamics

    Forget visualizations of “particles,” no matter how small, moving apart from some medium.

    This article by Paul Sutter … got me imagining a 4D wave function of interacting fields and wavepackets … interfere, constructively and not … so that dynamic bendiness is innate in 4 dimensions … because we’re talking about waves from which wave dynamics emerge

    “‘Straight’ only in the sense that every step forward is always placed directly in front of the previous.”

    “Mass and energy bend the shape of spacetime” – rather spacetime energy dynamics have topology.

    “Bending of spacetime [affects] the paths and motions of everything else” – rather wave interactions follow the principle of least action.

    • > “How to think about a four-dimensional universe” by Paul M. Sutter, Universe Today (November 13, 2023)

    Spacetime is not a fixed stage, but a flexible membrane that suffuses the entirety of existence. It is a thing, an object and entity in its own right, a dynamical actor in the cast of characters in the universe. The universe consists of the usual assortment of particles and radiation and all their wonderful interactions, and those interactions don’t happen on top of spacetime but along with spacetime, which has now joined their ranks.

  11. Re my two October 25, 2021 comments for post “How stiff is space-time?” – on re-reading Wilczek’s book, I was reminded of his multilayer Grid figure.

    The primary ingredient of reality is alive with quantum activity. … The primary ingredient of reality also contains enduring material components. These make the cosmos a multilayered, multicolored superconductor.

    I will use the word Grid for the primary world-stuff. That word has several advantages [vs. ether, space-time, quantum field]: We’re accustomed to using mathematical grids to position layers of structure, … – Wilczek, Frank. The Lightness of Being: Mass, Ether, and the Unification of Forces (p. 74). Basic Books. Kindle Edition.

    Grid – the primary world-stuff

    (caption) Figure 8.1 Grid, old and new. a. A grid is often used to describe how various things are distributed in space. b. The Grid, which underlies our most successful world-model, …

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