Euclid extended a point three times to get three linear dimensions, so would three orthogonal rotations around a point give three polar dimensions? It would, but after the first two rotations give a circle and a sphere, our space has no more dimensions to rotate into. Yet in mathematics, rotating a sphere in an orthogonal direction gives a hypersphere. A hypersphere is a sphere rotated in another dimension as a sphere is a circle rotated in another dimension, so its surface has three dimensions (Figure 2.5).

The possibility that our space is a surface was introduced by Bernard Riemann in 1857, as his Riemann Sphere is a hypersphere whose surface has three dimensions, just like our space. It still applies today because relativity lets space curve like a surface and cosmology lets it expand like a surface. It follows that our space could be a three-dimensional surface within a four-dimensional bulk:

“When it comes to the visible universe the situation could be subtle. The three-dimensional volume of space might be the surface area of a four dimensional volume” (Barrow, 2007) p180.

Davies makes the case even more clearly:

“… the shape of space resembles a three-dimensional version of the surface of a sphere, which is called a hypersphere.” (P. Davies, 2006), p45.

The mathematics is solid, but a hypersphere needs a fourth dimension outside physical space. Our   space doesn’t have four degrees of freedom but the network generating it could have. Just as a three-dimensional network can represent a circle and a sphere, so a four-dimensional network can represent a hypersphere. Our space would then be a three-dimensional surface, just as computer screens are two-dimensional surfaces. Our space behaves like a hypersphere surface so if it is generated, it could actually be so. A four-dimensional quantum network could create a three-dimensional surface that acts like our space, based on polar dimensions. Reverse engineering then suggests that our space is the surface of hyper-bubble expanding in the quantum bulk.

This conclusion explains why our space is expanding everywhere at once, not from a physical centre, as a linear space would. It also implies that our space is curved. Planets, stars, and galaxies are curved, so why not the universe? The earth was once thought flat because it seemed so, but now we know it isn’t, so is space the same? But if space is curved, why does it seem flat to us? One answer is that a hypersphere surface that has expanded for 14 billion years at the speed of light would seem flat to us, just as the surface of the earth once did. What else then does space as a hypersphere surface imply?

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A scalable system is one whose performance doesn’t degrade as it expands, however big it gets (Berners-Lee, 2000). As a network grows, it gets more demands, but if growth increases supply as well as demand, its performance doesn’t degrade. A network is scalable if each new point added increases supply as well as demand. The Internet was designed this way, as each new Internet Service Provider adds more processing to handle demands as well as more users to create demands.

One way to make a network scalable is to decentralize control by giving each point its own processing, so each new point adds processing power as well as load. When the Internet began, pundits expected it to collapse in chaos without central control, but instead it thrived. It was then realized that centralized networks collapse suddenly when stressed because an overload crashes the whole system, but decentralized networks degrade gracefully because an overload gives a local crash but the rest carry on. We think dictatorships are strong but nature knows they aren’t, so scientists expected brains to have a control center but they don’t, because decentralized networks are more reliable (Whitworth, 2008).

Is our space then scalable? The evidence suggests so because the laws of physics didn’t change as space expanded (Sutter, 2022). Space today behaves as it did when our universe was the size of a golf ball, so it is scalable. It follows that new points of space add processing to allow it to expand indefinitely. Each point of space then has a finite ability to process matter, and the evidence agrees:

“…recent observations favor cosmological models in which there are fundamental upper bounds on both the information content and information processing rate.” (Paul Davies, 2004), p13.

We call the upper limit of what space can hold a black hole, so that black holes exist also suggests that our space is scalable.

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If empty space is something not nothing, then what is it? Quantum simulations describe spacetime as follows:

“…we think of empty spacetime as some immaterial substance, consisting of a very large number of minute, structureless pieces, and if we let these … interact with one another according to simple rules … they will spontaneously arrange themselves into a whole that in many ways looks like the observed universe.” (Ambjorn et al., 2008), p25.

This approach is compatible with the idea that space is a network of indivisible points that interact with each other according to simple rules, as proposed here. What then are the services that space provides in order to look like our space? The following are suggested:

• Locations. Points that define when objects are in the same locality and so interact.
• Directions. The neighbors a point can interact with define possible directions from that point.
• Dimensions. The number of independent degrees of freedom that the space can extend into.

Locations, directions, and dimensions are services that a space that looks like ours must provide. For example, in Wilson loop networks, each point is a volume of our space, and in Penrose spin networks, points interact in events that have input and output directions (Penrose, 1972). The need for dimensions is illustrated by their use in geometry. About two thousand years ago, the Greek Euclid created geometry by defining the structure of space as follows. He began with a point location of no dimensions that extended in one direction to make a line, that then extended at right angles to give a plane, that again extended to give a cube. This defined our space as having three linear dimensions, so every point could be represented by three number coordinates (x, y, z).

The resulting Cartesian System worked so well for geometry that it became the standard, but it doesn’t suit all cases. For example, war-gamers adapt Euclid’s space to the two dimensions of a game board by using hexagons not squares, as they give more directions of attack. Yet most simulations still assume a Cartesian or linear space, including loop quantum gravity (Smolin, 2001), cellular automata (Wolfram, 2002), and lattice simulations (Case, Rajan, & Shende, 2001), but to work on a network, a Cartesian space needs:

• A maximum size: The size of the space must be known in advance, to allocate coordinate memory.
• A zero-point origin: The space has to have absolute centre, or (0,0,0) point, within itself.

However, our space behaves in a way that satisfies neither of these requirements.

The maximum size requirement arises because the memory of linear coordinates depends on the network size. For example, a point stored as (2,9,8) in a 9-unit cube, must be stored as (002,009,008) in a 999-unit cube, so it needs more memory. This limit caused the Y2K bug that experts worried would crash our systems in the year 2000. The problem was that early computers stored years as two digits, to save memory, so 1949 was stored as “49”, but that meant that the year 2000 was stored as “00”, just as the year 1900 was. Even now, airline booking systems can mistake a 101-year-old woman for a baby for this reason.

If our space is a generated Cartesian space, its maximum size had to be set before the first event to avoid a Y2K bug. But our space has been expanding at the speed of light for billions of years and still is, with no end in sight, so its final size is undefined. A Cartesian space the size of our universe should have crashed by now, but it hasn’t, so our space can’t have linear dimensions.

The zero-point origin requirement arises because a Cartesian space needs a (0,0,0) point within itself, so when Hubble showed that every star and galaxy is receding from us, assuming a Cartesian space made our Earth the origin of the universe! But our planet only began recently, so it can’t be so! The discovery that our space is expanding everywhere at once wityhout an absolute center within itself also means it can’t have linear dimensions.

In general, linear coordinates work well for small, fixed spaces but not for a huge space like ours, that is expanding indefinitely from no point within itself.

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If matter exists, and space is just its absence, is it nothing at all? The greatest minds of physics have wondered whether space exists, and in particular:

If all the matter in the universe disappeared, would space still exist?

If space is something it would, but if it is nothing at all, it wouldn’t. For Newton, space was the canvas upon which God painted, so without matter it would exist as an empty canvas. In contrast, Leibniz didn’t believe that God would make what had no properties, so he defined space relative to matter, just as distance is defined by two marks on a platinum-iridium bar in Paris. It followed that objects only move with respect to each other, so without matter there would be no space.

Newton’s reply to Leibniz was a spinning bucket of water (Figure 2.2). At first the bucket spins not the water, then the water also spins and presses up against the side to make a concave surface. If the water spins with respect to another object, what is it? It can’t be the bucket because initially, when it spins relative to the water, the surface is flat, and later, when it is concave, the bucket and the water spin at the same speed. In a universe where all objects move relative to other objects, a spinning bucket should be indistinguishable from one that is still. Another example is an ice skater spinning in a stadium whose arms splay out by the spin. If this movement is relative the stadium, why doesn’t spinning the stadium make the skater’s arms splay? Such examples suggest that the skater is spinning relative to space, not matter (Greene, 2004), p32.

This seemed to settle the matter, so space is something, until Einstein showed that the movement of objects actually is relative. Mach then resurrected Leibniz’s idea to be that the water in Newton’s bucket rotated with respect to all the matter of the universe, so in a truly empty universe it would stay flat and a spinning skater’s arms wouldn’t splay. This theory can’t be tested, because we can’t empty the universe, but his willingness to speculate without evidence shows how disturbing some physicists find the idea that space is:

“…substantial enough to provide the ultimate absolute benchmark for motion.” (Greene, 2004), p37.

The current verdict of physics is that “space-time is a something” (Greene, 2004), p75., so it could be a quantum network output, but how would it register object collisions? Computing suggests two options: 

1. Centralized. A central processor registers every object’s absolute position and compares them every cycle to deduce a collision for those at the same point. To its inhabitants, this space would seem to be continuous and to have no existence in itself, but the processing needed increases geometrically with the number of objects as each is compared to every other. For the atoms and electrons in our universe, the load is enormous, so a central processor could overload and collapse the whole system.

2. Decentralized. Each network point is allocated a finite processing capacity to handle local events, so a collision occurs when one gets more processing than it can handle and overloads. To its inhabitants, this space would seem discontinuous and to exist apart from the objects in it. This approach wastes processing on empty space but means that the system as a whole never fails.

Current computing prefers the decentralized option because then the whole system never fails. Our Internet is decentralized for that reason. Given that our universe has run for fourteen billion years without failing, for a virtual space, the second option is expected, that each point of space has a finite capacity to handle whatever passes through it. When nothing does, that processing still runs but gives a null result. Empty space is then null processing not nothing, so if every object disappeared, it would still exist, just as a screen still exists even when it is blank.

It follows that empty space isn’t the passive canvas of Newton, because null processing is active not passive, nor is it the nothingness of Leibniz, because null processing is something not nothing.

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In language, a continuous thing exists without gaps, and in mathematics, a continuous line has no breaks. Space is then continuous if it has no gaps or breaks in it, however small. Yet continuity makes movement illogical, as Zeno’s paradoxes (Mazur, 2008) illustrated two thousand years ago:

1. If a tortoise running from a hare sequentially occupies infinite points of space, the hare can never catch it, because every time it gets to where the tortoise was, it has moved a little further on; OR

2. If space and time aren’t continuous, there must be an instant when the arrow from a bow is at a fixed unmoving position. If so, how can many such instants beget movement?

To deny the first paradox exposes one to the second, and vice-versa. Zeno’s paradoxes resurface today as infinities in field equations. For example, continuity requires an electron to exist at a point with no size, which makes its mass infinitely dense. Physics avoids these infinities by a mathematical trick called renormalization, which attributes all particle interactions to other particles. Dirac described this tactic as follows:

“Sensible mathematics involves neglecting a quantity when it turns out to be small – not neglecting it just because it is infinitely great and you do not want it!” (Kragh, 1990), p. 184.

Feynman was even blunter:

“No matter how clever the word, it is what I call a dippy process! … I suspect that renormalization is not mathematically legitimate.”  (Richard Feynman, 1985), p128.

Physicists ignore the infinities of continuity because they want to, not because there is a reason to do so:

“… although we habitually assume that there is a continuum of points of space and time this is just an assumption that is … convenient … There is no deep reason to believe that space and time are continuous, rather than discrete…” (Barrow, 2007), p57.

There is no reason to believe that space is continuous, but there is a reason to think it isn’t. The quantum of quantum theory means that shorter wavelengths of light only have more energy up to the limit of Planck’s constant. This limit, called the Planck length, is then, according to quantum theory, the shortest length possible.

It follows that repeatedly dividing distance gives a smallest length that can’t be divided further. Just as closely inspecting a TV screen gives irreducible dots, closely inspecting our space gives irreducible Planck lengths, so it is digital not continuous. Not only does repeatedly dividing space give a pixel that can’t be split, repeatedly dividing time gives a cycle that can’t be paused, so it is also digital. This then answers Zeno’s paradoxes as follows:

There is indeed an instant when the arrow is in a fixed, unmoving position, but there is still movement because the next cycle generates the next physical position. Equally the hare cannot get closer to the turtle forever, because there is a minimum pixel distance that can’t be divided, so the hare catches the turtle.

A digital world of irreducible pixels and indivisible ticks makes the infinities of continuity disappear, like ghosts in the day, because denying the infinitely small avoids the infinitely large. If our world is a digital reality, its resolution is 10-33 meters and its refresh rate is 10-43 seconds, or Planck length and time respectively. The quantum network is very fine, but not infinitely so, and quantum processing is very fast, but not infinitely so either.

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This section considers how a quantum network could create a space like ours but to reverse engineer our space, its properties must be specified. It might seem strange to talk of the properties of space, but different virtual worlds have different requirements. For example, our world needs a space that can expand indefinitely without problems.

QR2.2.1   Is Space Continuous?

QR2.2.2   Is Space Nothing?

QR2.2.3   Is Space Linear?

QR2.2.4   Is Space Scalable?

QR2.2.5   A Polar Space

QR2.2.6   Three Polar Dimensions

QR2.2.7   Space Has Gaps

QR2.2.8   Space is Contained

QR2.2.9   Space Vibrates

QR2.2.10 Space Transmits

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Clearly a photo of me isn’t me, nor is a movie of me, and even a perfect physical copy of me isn’t me but my twin. Copying matter doesn’t copy reality but if quantum events cause physical events, as quantum theory suggests, why not copy them? Can quantum cloning save and reload reality?

Unfortunately, the quantum no-cloning theorem explicitly excludes this (Wootters & Zurek, 1982). To copy a quantum state, it must first be observed but according to quantum theory, observing a quantum wave restarts it at a point, which destroys it. Observing a quantum state makes it disappear, so quantum cloning is impossible.

To understand why, consider a laptop with a central processing unit (CPU) plus memory registers to store data. The CPU can copy the data in its memory registers but not its own state, because the act of doing so changes that state before the copy. It can input and output data but it can’t copy itself, because trying to read itself, changes itself. It follows that a network of CPUs only with no memory registers, as quantum theory describes, can’t copy itself by the logic above. Quantum theory then implies the no-cloning theorem because it describes a network of processors without data storage!

The quantum network proposed has no storage because it is constantly active. The quantum waves upon it also never stop, as they are either expanding, or restarting to expand again, in the observations we call physical events. It is like a star that constantly shines without pause, so it can’t save or reload itself.

Our Internet network uses memory buffers to handle overloads but a quantum network doesn’t have this luxury. Cell-phones save and reload physical states but the quantum network can’t store or reload quantum states by the no-cloning theorem. It must run by itself alone, with no backups, buffers, or saves to fall back on if it fails, so it doesn’t operate as our networks do.

This means that McCabe’s argument against virtualism doesn’t apply to a quantum network that isn’t based on information. Information needs physical states to exist but quantum events don’t. They can create physical events because they don’t depend on them, as quantum processing has no physical context.

One might expect processing powerful enough to generate physical events could save a copy of itself, but it can’t. It follows that all talk of uploading or downloading universes, minds, or ourselves is just wishful thinking. The quantum network has nowhere to store anything, so we live in a world of events that can’t be saved, not things that can (Seibt, 2024).

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If our world is information, can it be saved and restored like a game world? To do that requires a decoding context based on our world, which is circular as McCabe explains:

“All our digital simulations need an interpretive context to define what represents what. All these contexts derive from the physical world. Hence the physical world cannot also be the output of such a simulation.” (McCabe, 2005).

The physical world can’t both cause information and be caused by it, so we can’t define information in physical terms then call physical events an information output. This follows from information theory as defined by Shannon and Weaver, but to understand it, imagine our universe frozen at a point in time, as many physical states. What then could load and restart it? A physical state has no information in itself, so who or what could decode it? Not us, as we would be frozen too, along with the universe! A simulation based on physical events can’t create those events as information, as McCabe says, nor can it reload a frozen universe. In contrast, a laptop can save and reload a game, because its operation doesn’t depend on game events in any way.

Matter can’t be information if information depends on matter, so some suggest that minds are information encoded by brains, so emulating a brain would clone it (Sandburg & Bostrom, 2008), so we could copy our mind to a younger body, or live forever as a hologram. Yet there is no evidence at all that computers experience events as minds do. A computer can create a hologram that acts and talks like a dead person, but no-one is there experiencing those events. Emulating a dead relative doesn’t resurrect him or her, any more than a video or photo of them does, so saving and reloading the information of past events doesn’t recreate them.

But if a person’s brain was copied exactly, atom for atom, wouldn’t that copy their consciousness? Nature provides the answer because it has already done it. Identical twins are essentially physical clones of the same egg, but they are different people with different experiences. It follows that a physical clone of me doesn’t make another me, but another person entirely, as I can’t experience their life, nor they mine. Chapter 6 critiques the silicon chip speculation in more detail (6.3.11).

Something is wrong with the idea of reloading reality, and it is the belief that only matter is real. If that was true, a perfect physical copy of me would be another me, but it isn’t. Instead, it is like reloading a game for another player. Even if one day we managed to copy atoms perfectly, cloning a person to another location would create their twin, not them, so if the source atoms remained, there would be two people in two bodies, not one. To really reload a physical event, we must copy what causes it, so can quantum reality be saved and reloaded?

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The nature of information was specified by information theory when Shannon and Weaver defined it in terms of binary choices between physical states [1] (Shannon & Weaver, 1949). One choice from two states is a bit of information, two choices is two bits, and so on. Eight bits, or a byte, is eight choices, so eight electronic on/off switches can store one byte. Eight switches have 256 possible combinations, so one byte can store about one text character. All our kilobytes and megabytes are based on choices between physical states.

If only physical states exist, then all information depends on them, so there is no software without hardware. Information stored as choices between matter states needs matter to exist, so to think that information can also create matter is like thinking a daughter can give birth to her mother, which is impossible.

A choice of one state, or no choice at all, is zero bits, so anything fixed one way contains no information. It follows that a physical book contains no information in itself because it only exists one way. This seems wrong but it isn’t, as hieroglyphics that no-one can read do indeed contain no information. Symbols only contain information if we can read them, which requires a decoding context. For example, the text you are reading now requires the decoding context of the English language, so if you don’t know English, you get no information. If you change the decoding context, like reading every 10th letter, the result will be different information from the same text.

Information theory defines the decoding context of a physical signal as the number of physical states it was chosen from. This number defines the amount of information sent, so one electronic pulse sent down a wire can represent one bit, or it can be one byte as ASCII “1”, or as the first word in a dictionary it can be many bytes. The amount of information in a signal depends not only on the signal itself, but also on its decoding context. If it weren’t so, data compression couldn’t store the same data in a smaller signal, but it can by better decoding. In general, the information in a physical signal is undefined until its decoding context is known. The transfer of information between a sender and receiver requires an agreed decoding context, so a receiver can only extract the information a sender put in a signal if they know how to read it.

Given the above definition, processing can be defined as the act of changing information by making new choices. Writing a book is then processing, as it can be written in many ways, and reading a book is also processing, as it can be read in many ways. Processing lets us save data in a physical state and reload it later, given a decoding context. Information is then static, while processing is a dynamic activity.

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[1] Mathematically, Information I = Log₂(N), for N choice options.