The current approach to digital land is disappointingly realistic. The digital world has no obligation to abide by the nature of the physical reality around us. Therefore, in my view, there is no reason to accept the same limitations of reality (and our ability to perceive it) as we make a transition into the digital world. Human perception is limited greatly by the brain. Our brains have developed over thousands of years to ensure our survival in a very specific set of environmental circumstances. Atmospheric pressure, temperatures between approximately -50 F and +120 F, an abundance of what we call “visible” light, uniform gravity, etc. One of our greatest limitations has been the perception of space as a three dimension volume in which we can move up, down, left and right, along with time, a somewhat different axis, which moves only forwards. We now know that this condition is by no means objective; time is not experienced uniformly by everything in the Universe and space likely consists of many additional dimensional axes that are not perceptible to human minds. Massless particles move at the speed of light and thus do not experience time at all in their own frame of reference, while hypothetical particles of imaginary mass like the Tachyon may move backwards in time due to their superluminal velocity. Computers today can act as an extension of the brain, simulating and abstracting these imperceptible parts of reality into forms that a human mind can grasp. Why not take advantage of this?

The word dimension is almost always misused by the general public. In Sci-Fi and fantasy, characters often travel to an “alternate dimension”, which is portrayed as a completely different world separate from our own. In these cases, “dimension” is used as a synonym for “realm”, “world” or “parallel universe”. In reality, a dimension is nothing more than an axis in a coordinate system. Length, width and height are the spatial dimensions that humans perceive, alongside time, which is not perceived by the eyes but felt intuitively. In mathematics, we are free to define as many dimensions as we’d like. Using linear algebra each axis can be represented as a vector coefficient. This allows for the same generalizable operations, like cross product and determinant, to be applied to calculations using an arbitrary amount of dimensions. This has applications to fields like Principle Component Analysis and Neural Networks. Physics predicts reality to contain many more than the four dimensions we can perceive ourselves. There are several not yet fully accepted theories that propose what the structure of these dimensions could be; from the single additional dimension used by Einstein and others to connect gravity and electromagnetism to the total 26 dimensions proposed by Bosonic String Theory. In any case, it will likely not be known for sure what the exact nature of reality is until larger particle accelerators can be built.

Three dimensional virtual worlds are nothing new. Second Life, Minecraft, Roblox, VR Chat, and many more have accomplished a ‘digital world’ without tokens, blockchains or any sort of distributed network. Furthermore, I see a fundamental issue with the way virtual worlds are approached within web3. In the web2 gaming world, virtual worlds became valuable because of what you can do within them, not because of the simple fact that one can enter them. In other words, land in the metaverse is not lindy and therefore we should not value the protocols and companies building these worlds based on this assumption.

Designers of today’s virtual worlds have an obsession with mapping the real world onto the digital realm. I think this is truly a disappointing approach to take, because frankly, the physical world is relatively boring in comparison to what could be possible with the aid of a computer. Taking full advantage of the additional computational resources afforded to us by power computers will allow for more engaging, novel metaverse experiences as well as introduce backdoors that make the often forced-in blockchain networks actually be useful to the development of the virtual world, rather than simply serving as a settlement layer for the sales of in-game NFTs.

When one cannot solve an integral in normal spatial coordinates (x,y,z), one typically looks for a new coordinate system in which the problem takes a simpler form. After mapping (x,y,z) to the new system, the integral becomes solvable in this simpler form. A common example of this is solving for the volume of a sphere. Doing this by integrating over (x,y,z) would be nearly impossible to do by hand so instead the mapping to spherical coordinates is performed first.

Then the integral is simplified and easy to solve:

My intention is not to explain high school calculus in this piece, but to suggest that the same principle of coordinate transform can and should be applied liberally to the metaverse to allow for novel experiences to emerge. The obsession with mapping the constraints of physical reality directly onto virtual worlds is a dangerous fetish that will continue to inhibit the mass adoption of these virtual worlds until it is left behind. Within new coordinate systems, movement becomes different and reality bends as a result. Consider the coordinates that are given by the mapping u=ln(x/y) and v=sqrt(xy) . The farther out on the line y=x you go, the longer each step you take gets. This system is referred to as ‘Hyperbolic Coordinates’.

This could have interesting implications for games, especially those where land is to be sold, as this system would allow for an infinitely large supply of land where there were inherent differences at the most fundamental level to different locations. Beyond this there are an infinity of different realities to be considered, along with an infinite number of axes to include. Some of these will be useful, others not. Another potential side effect immersing people into new spatial dimensions via VR headsets and a projection of these states into 3D could be (slightly) altered states of consciousness. It is well known that feeding humans irregular sensory data, through methods such as sensory deprivation or binaural beats, can cause temporary hallucinations / altered perception. Existing (virtually) within an alien system of coordinates would likely have a similar effect (or in the worst case scenario, just induce nausea).

To justify the use of a blockchain in the creation of a virtual world in any capacity, there must be a compelling reason why blockchain makes the given system objectively better than what is available now. This typically comes down to one of the following reasons: 1) cryptoeconomics allow for computing something at a scale that is too expensive for individual entities, 2) the distributed nature of blockchains allowing one to evade local laws/regulations (including access bans), 3) privacy features allowing for one to hide from censorship / overreach of corporations/governments, 4) automatic coordination mechanisms and trustlessness of blockchains allowing for some kind of economic game to be played that is generally not feasible otherwise. An example of 1) would be any resource network like Livepeer, Arweave, etc, 2) would be Venezuelans preserving their wealth via BTC and USDC in the face of hyperinflation and capital controls, an example of 3) could be Monero or ZCash, and an example of 4) would be oracle or distributed key management systems. In my view, virtual worlds fall into none of these categories and are being built with a focus on blockchain/NFTs today because this technology is currently the most popular technology subvertical. Games are also seeing the sale of NFTs as way to increase their revenue, as this will allow secondary sales in which the game creator can extract a royalty fee. NFT inclusion in games will continue to accelerate and can be considered an example of reason 4) in the best case scenario, however. For example, in CryptoVoxels the parcels being NFTs enables community ownership over a plot of land; not a novel idea but certainly made easier/more efficient via the use of NFTs.

So when does a blockchain truly become relevant? Games are typically not illegal / censored in most free countries, so reasons 2) and 3) become irrelevant. Games may involve certain types of economic coordination mechanisms, but this not always the case and may be solved by the gameplay incentives themselves before the need for monetization. This leaves reason 1), computing at scale. If we unshackle ourselves from terrestrial thinking, we could easily imagine a world complex enough to require computing at a scale that only a blockchain could provide efficiently. Coordinates at the most simple level are represented in games by vectors. In code it may look something like:

character_one_position = [0, 1, 3];

Where 0 is the character’s x coordinate, 1 their y coordinate and 3 their z coordinate. Then, as this character moves around and interacts with the environment, their position updates itself by using vector addition and subtraction, scalar (dot) products, and vector (cross) products. In three dimensions these operations can be done by a computer in milliseconds and do not need to be executed on-chain. However, we could imagine a coordinate system complicated enough that it would potentially make sense to calculate this on-chain rather than on centralized servers. Livepeer, for example is at least 40% cheaper than AWS for GPU transcoding as of May 2021 (and is likely much cheaper than this now). It also so happens that complex vector operations are best done on GPUs so the cost savings would apply here as well. Imagine a coordinate system given by:

coordinate = [n_1, n_2, ... n_m]; 

where m is a large number. Then, consider some type of complex mapping to a new system of coordinates:

mapping: {z_1 = 0, z_2 = φ(z_2) +  i* φ(z_1), z_m =  φ(z_m) + i* φ(z_(m-1)}

In this case φ(z_m) is the Euler Totient Function, which is a product of the distinct semi-prime numbers that divide the number in question and is given by:

My choice of this function here is largely irrelevant to the overall point. It is, however, a function that becomes relatively complex for large numbers and is also used within the RSA encryption system. With significantly large enough m, the computation of simple movements within this world would become very expensive. Once the computation requirements cross a certain threshold, it begins to make sense to use a network like Render or Livepeer to facilitate the calculations, rather than buy the required resources directly from a cloud provider. Now, a reason for a metaverse token emerges. The game, if it is to support thousands or millions of players simultaneously, would need privileged access to the GPU networks. They could emit tokens to the transcoders willing to process their jobs first, creating a flashbots-like private incentivized mempool for their system. This would allow for the game to continue to enjoy the reduced fees of decentralized networks while not facing the risk of being left behind in favor of other jobs on the rendering networks. At the same time, the experience delivered by this game would go far above what is available to users in modern metaverse games today.

3D virtual worlds are generally boring. This is certainly reflected in the uptake of VR-based games over the last decade or so of the technology being available. The US VR market was valued at just $7 billion in 2021 while the total value of the gaming market was worth around $180 billion. Although the cost and weight of headsets has come down considerably, it is still a pastime that is only enjoyed by a small group of people. Taking the causal nature of normal gaming and replacing it with an immersive experience is not worth it to most if it means experiencing a less realistic, nausea-inducing version of the real world. Needlessly forcing a liquid token into this equation doesn’t make this experience any more appealing. The way out? Invite the mathematically inclined to invent new worlds with a completely different ‘reality’; coordinate systems and dimensional axes that would be incomprehensible in the real world. Concepts formerly confined to offices of math professors and lecture hall blackboards can now be unleashed into the metaverse and enjoyed by all. 3D worlds are thinking too small. Wen 5D, 10D, and 500D metaverses?

The current approach to digital land is disappointingly realistic. The digital world has no obligation to abide by the nature of the physical reality around us. Therefore, in my view, there is no reason to accept the same limitations of reality (and our ability to perceive it) as we make a transition into the digital world. Human perception is limited greatly by the brain. Our brains have developed over thousands of years to ensure our survival in a very specific set of environmental circumstances. Atmospheric pressure, temperatures between approximately -50 F and +120 F, an abundance of what we call “visible” light, uniform gravity, etc. One of our greatest limitations has been the perception of space as a three dimension volume in which we can move up, down, left and right, along with time, a somewhat different axis, which moves only forwards. We now know that this condition is by no means objective; time is not experienced uniformly by everything in the Universe and space likely consists of many additional dimensional axes that are not perceptible to human minds. Massless particles move at the speed of light and thus do not experience time at all in their own frame of reference, while hypothetical particles of imaginary mass like the Tachyon may move backwards in time due to their superluminal velocity. Computers today can act as an extension of the brain, simulating and abstracting these imperceptible parts of reality into forms that a human mind can grasp. Why not take advantage of this?

The word dimension is almost always misused by the general public. In Sci-Fi and fantasy, characters often travel to an “alternate dimension”, which is portrayed as a completely different world separate from our own. In these cases, “dimension” is used as a synonym for “realm”, “world” or “parallel universe”. In reality, a dimension is nothing more than an axis in a coordinate system. Length, width and height are the spatial dimensions that humans perceive, alongside time, which is not perceived by the eyes but felt intuitively. In mathematics, we are free to define as many dimensions as we’d like. Using linear algebra each axis can be represented as a vector coefficient. This allows for the same generalizable operations, like cross product and determinant, to be applied to calculations using an arbitrary amount of dimensions. This has applications to fields like Principle Component Analysis and Neural Networks. Physics predicts reality to contain many more than the four dimensions we can perceive ourselves. There are several not yet fully accepted theories that propose what the structure of these dimensions could be; from the single additional dimension used by Einstein and others to connect gravity and electromagnetism to the total 26 dimensions proposed by Bosonic String Theory. In any case, it will likely not be known for sure what the exact nature of reality is until larger particle accelerators can be built.

Three dimensional virtual worlds are nothing new. Second Life, Minecraft, Roblox, VR Chat, and many more have accomplished a ‘digital world’ without tokens, blockchains or any sort of distributed network. Furthermore, I see a fundamental issue with the way virtual worlds are approached within web3. In the web2 gaming world, virtual worlds became valuable because of what you can do within them, not because of the simple fact that one can enter them. In other words, land in the metaverse is not lindy and therefore we should not value the protocols and companies building these worlds based on this assumption.

Designers of today’s virtual worlds have an obsession with mapping the real world onto the digital realm. I think this is truly a disappointing approach to take, because frankly, the physical world is relatively boring in comparison to what could be possible with the aid of a computer. Taking full advantage of the additional computational resources afforded to us by power computers will allow for more engaging, novel metaverse experiences as well as introduce backdoors that make the often forced-in blockchain networks actually be useful to the development of the virtual world, rather than simply serving as a settlement layer for the sales of in-game NFTs.

When one cannot solve an integral in normal spatial coordinates (x,y,z), one typically looks for a new coordinate system in which the problem takes a simpler form. After mapping (x,y,z) to the new system, the integral becomes solvable in this simpler form. A common example of this is solving for the volume of a sphere. Doing this by integrating over (x,y,z) would be nearly impossible to do by hand so instead the mapping to spherical coordinates is performed first.

Then the integral is simplified and easy to solve:

My intention is not to explain high school calculus in this piece, but to suggest that the same principle of coordinate transform can and should be applied liberally to the metaverse to allow for novel experiences to emerge. The obsession with mapping the constraints of physical reality directly onto virtual worlds is a dangerous fetish that will continue to inhibit the mass adoption of these virtual worlds until it is left behind. Within new coordinate systems, movement becomes different and reality bends as a result. Consider the coordinates that are given by the mapping u=ln(x/y) and v=sqrt(xy) . The farther out on the line y=x you go, the longer each step you take gets. This system is referred to as ‘Hyperbolic Coordinates’.

This could have interesting implications for games, especially those where land is to be sold, as this system would allow for an infinitely large supply of land where there were inherent differences at the most fundamental level to different locations. Beyond this there are an infinity of different realities to be considered, along with an infinite number of axes to include. Some of these will be useful, others not. Another potential side effect immersing people into new spatial dimensions via VR headsets and a projection of these states into 3D could be (slightly) altered states of consciousness. It is well known that feeding humans irregular sensory data, through methods such as sensory deprivation or binaural beats, can cause temporary hallucinations / altered perception. Existing (virtually) within an alien system of coordinates would likely have a similar effect (or in the worst case scenario, just induce nausea).

To justify the use of a blockchain in the creation of a virtual world in any capacity, there must be a compelling reason why blockchain makes the given system objectively better than what is available now. This typically comes down to one of the following reasons: 1) cryptoeconomics allow for computing something at a scale that is too expensive for individual entities, 2) the distributed nature of blockchains allowing one to evade local laws/regulations (including access bans), 3) privacy features allowing for one to hide from censorship / overreach of corporations/governments, 4) automatic coordination mechanisms and trustlessness of blockchains allowing for some kind of economic game to be played that is generally not feasible otherwise. An example of 1) would be any resource network like Livepeer, Arweave, etc, 2) would be Venezuelans preserving their wealth via BTC and USDC in the face of hyperinflation and capital controls, an example of 3) could be Monero or ZCash, and an example of 4) would be oracle or distributed key management systems. In my view, virtual worlds fall into none of these categories and are being built with a focus on blockchain/NFTs today because this technology is currently the most popular technology subvertical. Games are also seeing the sale of NFTs as way to increase their revenue, as this will allow secondary sales in which the game creator can extract a royalty fee. NFT inclusion in games will continue to accelerate and can be considered an example of reason 4) in the best case scenario, however. For example, in CryptoVoxels the parcels being NFTs enables community ownership over a plot of land; not a novel idea but certainly made easier/more efficient via the use of NFTs.

So when does a blockchain truly become relevant? Games are typically not illegal / censored in most free countries, so reasons 2) and 3) become irrelevant. Games may involve certain types of economic coordination mechanisms, but this not always the case and may be solved by the gameplay incentives themselves before the need for monetization. This leaves reason 1), computing at scale. If we unshackle ourselves from terrestrial thinking, we could easily imagine a world complex enough to require computing at a scale that only a blockchain could provide efficiently. Coordinates at the most simple level are represented in games by vectors. In code it may look something like:

character_one_position = [0, 1, 3];

Where 0 is the character’s x coordinate, 1 their y coordinate and 3 their z coordinate. Then, as this character moves around and interacts with the environment, their position updates itself by using vector addition and subtraction, scalar (dot) products, and vector (cross) products. In three dimensions these operations can be done by a computer in milliseconds and do not need to be executed on-chain. However, we could imagine a coordinate system complicated enough that it would potentially make sense to calculate this on-chain rather than on centralized servers. Livepeer, for example is at least 40% cheaper than AWS for GPU transcoding as of May 2021 (and is likely much cheaper than this now). It also so happens that complex vector operations are best done on GPUs so the cost savings would apply here as well. Imagine a coordinate system given by:

coordinate = [n_1, n_2, ... n_m]; 

where m is a large number. Then, consider some type of complex mapping to a new system of coordinates:

mapping: {z_1 = 0, z_2 = φ(z_2) +  i* φ(z_1), z_m =  φ(z_m) + i* φ(z_(m-1)}

In this case φ(z_m) is the Euler Totient Function, which is a product of the distinct semi-prime numbers that divide the number in question and is given by:

My choice of this function here is largely irrelevant to the overall point. It is, however, a function that becomes relatively complex for large numbers and is also used within the RSA encryption system. With significantly large enough m, the computation of simple movements within this world would become very expensive. Once the computation requirements cross a certain threshold, it begins to make sense to use a network like Render or Livepeer to facilitate the calculations, rather than buy the required resources directly from a cloud provider. Now, a reason for a metaverse token emerges. The game, if it is to support thousands or millions of players simultaneously, would need privileged access to the GPU networks. They could emit tokens to the transcoders willing to process their jobs first, creating a flashbots-like private incentivized mempool for their system. This would allow for the game to continue to enjoy the reduced fees of decentralized networks while not facing the risk of being left behind in favor of other jobs on the rendering networks. At the same time, the experience delivered by this game would go far above what is available to users in modern metaverse games today.

3D virtual worlds are generally boring. This is certainly reflected in the uptake of VR-based games over the last decade or so of the technology being available. The US VR market was valued at just $7 billion in 2021 while the total value of the gaming market was worth around $180 billion. Although the cost and weight of headsets has come down considerably, it is still a pastime that is only enjoyed by a small group of people. Taking the causal nature of normal gaming and replacing it with an immersive experience is not worth it to most if it means experiencing a less realistic, nausea-inducing version of the real world. Needlessly forcing a liquid token into this equation doesn’t make this experience any more appealing. The way out? Invite the mathematically inclined to invent new worlds with a completely different ‘reality’; coordinate systems and dimensional axes that would be incomprehensible in the real world. Concepts formerly confined to offices of math professors and lecture hall blackboards can now be unleashed into the metaverse and enjoyed by all. 3D worlds are thinking too small. Wen 5D, 10D, and 500D metaverses?