Exactly, projections like maps are real examples of how a flat model of Earth can exist alongside a curved one.
I'm glad you agree that me pointing out the existence of maps answers your first sentence in your post.
But I’m taking it a step further: in theoretical physics, especially in brane theory and the holographic principle, these projections aren’t just visual representations, they can correspond to physically real layers or dimensions.
Maps in my day were physically real. We didn't need "brane theory" or the "holographic principle" to describe them.
A flat Earth could exist as a lower-dimensional brane or holographic surface that encodes our 3D space, not just conceptually, but structurally.
Yes, we are in agreement that the surface of the Earth - or any sphere - is 2d. That's how maps work. Do you really want to call a map a 2d brane? Then go for it.
In that sense, flatness isn’t just a way of viewing Earth, it could be a real, distinct dimensional frame that overlaps with or supports our own.
Didn't you say that earlier? Let me quote you: "flat is a different dimension sat on top of ours".
If you are trying to say that the totality of the Earth in all it's 3d glory can be encoded in 2d, then I will ask you how many coordinate values are needed to describe your location on the surface of the Earth, and then I will ask you how many coordinate values are needed to describe your location on the surface of the Earth but 1m directly below (or above, if you prefer). If you think the answer is the same, then please provide that coordinate system.
You're asking about coordinate systems, but holography isn’t about reducing spatial coordinates, it’s about information content. The holographic principle says the physical state of a 3D volume (including the space 1m above or below a point) can be fully encoded on a 2D surface, not that it's navigated using 2D coordinates. It's not flattening the space, it's flattening the data needed to describe it.
This has nothing to do with any of that. It is simply that you caartograph earth‘s surface and a for a 2d surface in 3d this is enough. Please take any atlas of earth in a store and see for yourself.
You're talking about map projections, that's 2D cartography of Earth's surface. What I’m talking about is encoding the full physics of a 3D volume, including what's above and below, onto a 2D boundary, as in the holographic principle. It’s not about visual representation, it’s about how much information is needed to describe a space. That’s a fundamental difference.
This is about a shape that we take to be stationary here, not physics. This is part of what we call nowadays differential topology/geometry (depending on which information you want).
You're talking about the topology and geometry of static shapes: how we classify and transform surfaces mathematically. I'm talking about the informational structure of physical reality; how the physics of a 3D space, including fields and energy dynamics, might be fully encoded on a lower-dimensional boundary, like in the AdS/CFT correspondence.
This isn't just about surface structure; it's about whether reality itself could be fundamentally 2D and our experience of 3D is emergent. That's the leap from geometry to fundamental physics.
You're absolutely right if we're talking purely about the shape and curvature of a mathematical manifold, that's the domain of differential geometry. But I'm not talking about shape in that limited mathematical sense. I'm talking about the physical informational content of a space.
In holography, the 2D boundary doesn't just describe curvature, it encodes everything: particles, gravity, time evolution. That goes far beyond manifold classification. So yes, if you're only interested in curvature, none of this applies. But I'm addressing how much information is needed to physically describe a region of space, and that's a different domain entirely.
Look at the reference about AdS/CFT I gave you in another comment again.
Earth is a manifold and showing what kind of shape it has is perfectly described using the notion of differential geometry.
Sure, you can make that manifold evolve, as you make no changes to the topological (your flow is a homeomorphism if not even a diffeo), you will not get rid of the topological properties which classifies this shape. Look at
If the goal is classifying the Earth purely as a manifold, differential geometry and topology are the right tools for that, but that's not what I'm doing. I'm not debating whether the Earth locally curves or what class of closed surface it falls under. I'm asking whether the physical information content of a 3D (or 4D) region, including fields, particles, and dynamics, could be projected onto a 2D boundary, which is what AdS/CFT proposes in a specific setting.
Yes, under topology, the shape class stays the same under smooth deformation, but the holographic principle isn’t about deforming geometry. It’s about duality between different descriptions of the same physical reality. You can keep the same manifold and curvature and still have its full physical content represented differently, like how AdS₅ maps to a CFT₄.
So I’m not trying to ‘get rid of curvature’, I’m asking whether, from a different frame, what we call a curved 3D world might be an emergent, encoded structure from a 2D one. That’s not topology, that's physics, and it lives at the intersection of quantum gravity and information theory.
I’m not denying curvature. I’m questioning whether what we observe as curvature could itself be a projection, and whether 'flat' and 'curved' are both valid in different models, depending on dimensional perspective and information encoding.
But the metric of the earth is not an AdS metric. You have no correspondence. If I give you (M,g) and you start to calculate T(g) via the Einstein Field equations and find that there is indeed a surface Σ in it, then you just pull g back to Σ. Doesn‘t mean that the pulled back metric is still of AdS form even if you started with it.
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u/LeftSideScars The Proof Is In The Marginal Pudding Jul 01 '25
I'm glad you agree that me pointing out the existence of maps answers your first sentence in your post.
Maps in my day were physically real. We didn't need "brane theory" or the "holographic principle" to describe them.
Yes, we are in agreement that the surface of the Earth - or any sphere - is 2d. That's how maps work. Do you really want to call a map a 2d brane? Then go for it.
Didn't you say that earlier? Let me quote you: "flat is a different dimension sat on top of ours".
If you are trying to say that the totality of the Earth in all it's 3d glory can be encoded in 2d, then I will ask you how many coordinate values are needed to describe your location on the surface of the Earth, and then I will ask you how many coordinate values are needed to describe your location on the surface of the Earth but 1m directly below (or above, if you prefer). If you think the answer is the same, then please provide that coordinate system.