The 'bed bug analogy' is what modern theoretical physicist uses to instill in the initiate's mind that a human being can cogently talk of 'higher dimensions' while fully trapped within 3d. Ask where the alleged eleventh dimension is located and they frankly tell you that they cannot even imagine where it is, let alone see one! You get totally lost because you thought science was about things that can be observed. Perhaps you go ahead and think that you are the only one 'floating' inside a room composed of 'eleven dimensional' so called 'geniuses', but you are mistaken! As you will see, you are not alone! These self proclaimed 'geniuses' are actually victims of the infamous 'emperor's clothes'!
Ask how you can possibly 'know' of a 'fourth dimension', and you are taken to tour in 'flat land', the land whose inhabitants are all 'bed bugs'. These 'bed bugs' can, nevertheless do maths! So you are told that there is another poor guy like you, who similarly think that there are only two dimensions, but it is only because he cannot imagine a third dimension. Welcome to flat land, the home of mathematical 'bed bugs'.
This analogy is a non stater because, as you can see, a bed bug is actually a squarely 3d creature. It is not a 2d analogy of a 3d anything. It is a genuine example of a 3d object. Here, they might tell you 'come on, be more imaginative'. So lets imagine one that is flat enough that it is, in fact, a 2d 'creature'. You find that one such a thing doesn't exist! A basic lesson we learned in kindagarten was that an object doesn't have to varnish from all angles in order to varnish completely. If it disappears from one angle, the whole thing disappears. So we realy can't construct anything by adding 'dimensions' bit by bit. We either have a 3d object or we have nothing. The whole idea of '1d, then 2d, then 3d, then 4d...'. is poppycock because there is no such a sequence of things such as 0d,1d,2d,3d,..that makes us extrapolate and say 'next we have 4d,...'. We only have either 3d or we have nothing.
Since obviously there is no such a 2d 'bed bug' capable of seeing things from within a '2d', flat 'tissue paper', the physicist actually ends up contemplating on one 3d scenario, calling it '2d' and another equaly 3d scenario, and call it '3d'. He reasons with what can only be seen by a 3d creature and uses this to try to tell you how 2d entities inhabiting 'flat land' will see things! In reality, however, such 'creatures' that are impossible to exist, living in an 'hammock' that is impossible to exist cannot possibly see anything! What is before us is two considerations both of which are based entirely on what a 3d creature sees, not one seen by a 2d creature and another one, by a 3d creature.
With this, you will now understand my main argument against the flat land analogy. We will not talk of 'what a bed bug will see', like they incongruously do it in theoretical physics. We will just talk of measuring things relative to different back drops, having in mind that both back drops are as seen by a 3d creature. Lets consider the circle drawn on a spherical surface. The reasoning applied is straight foward: the circumference of the circle is, of course, 2πr, but if we pretend that the 'radius' (call it r') is distance along the curved sphere, the circumference will, of course, be smaller than 2πr'. Taking the 'radiuses' as though to be along the curved surface in which the circle is drawn is (erroneously) said to be how the 'bed bug' living in the 'curved 2d surface' will take them, being unaware of the curvature as it is along another 'dimension'. In reality, of course such a 'bed bug' cannot exist and as such the question of how 'it' does geometry is simply meaningless and so such cannot be a meaningfull analogy of anything! Inexistent scenarios cannot be an analogy of anything existent!
The problem with the above 'analogy' is best highlighted if you consider the case where the spherical surface is made by 'hammering' an initially flat sheet. Lets say you had already drawn the circle with radius r. If there was a 'ruler' embedded in the sheet that was initially indicating radius r you can easily see that the process of 'hammering' the sheet into a spherical surface necessarily stretches the ruler the same way it stretches the sheet itself, so that the ruler still indicates that the circle is r and never r'! Yes the length along the curved surface will be longer, but the ruler will be similarly longer, and so there will be no change in the measured 'radius'.
So you see that theoretical physicist's terrible fallacy is to use the unchanged 'out side' ruler and then insinuate that such is the ruler that the 'bed bug' uses, when in reality, the 'inside' ruler must bend, stretch and contract together with the 'space'. The mistake comes about because the theoretical physicist is actually considering two perspectives as seen by a 3d observer while erroneously insisting that one of them is what the '2d observer' sees, while the other is what the 'bed bug' thinks that the measurement 'should be'. By thinking this way, the physicists imposes the two perspectives onto the very same creature he tells us can contemplate on, and observe only one perspective! This leads them to the error of not noticing that there is only one 'ruler' (not 2) trapped in ths surface, and that ruler must bend, stretch or shrink alongside the surface, whenever the surface undergoes the 'distortions' due to 'curving'.
Finally, notice that this is very relevant to General relativity since 'space' is supposed to be 'distorted' by masses, which can move from place to place. The masses supposedly 'distorts' an originally 'flat' space, or at least alters the curvatures. Rullers donnot, in anyway, seems to exist 'out side of space' for them to help us tell whether or not 'space' is 'curved'! On the contrary, they must get distorted alongside the 'space' every time a massive object passes there. So the amount of change that the ruler undergoes is exactly the same as that which 'space' undergoes, rendering the ruler useless in informing us about how 'curved' the space is! The usual contemplation of 'alterance of Euclidean Geometry' supposedly observable from within the flat land comes from using an hypothetical 'out side of space' 'ruler' that never streaches nor 'contracts' together with 'space', erroneously taken to be 'a ruler trapped in space' just because it can be hypothetically bent to follow the curvature.
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A surface (actually a membrane) is not a 2d ANALOGY of a 3d object. It is a genuine EXAMPLE of a 3d object, or of a property of a 3d object.
So there is no way for even conceptual existance of a 2d creature seeing 1d lines and 0d dots, but not 2d surfaces. Why? because whatever we can see as 'line' is actually a 2d surface. we see something in front of us that has length and width, (but with a, be it small, dept that we are not seeing, due to perspective). It is only that the length is extremely larger than the width. It is extremely thin, that is, but it is still a genuine, 3d object, nevertheless.
You can then see that there is no such a scenario as '2d analogy of a 3d scenario' or similar so called 'analogies'. To say Y is an analogy of X means 'Y is similar to X in some way but different in other ways'. If Y cannot possibly exist, but X exists then Y is not similar to X in ANY WAY, and a such, a non-stater as 'analogy' for X.
It is this lack of 'anyway' similarity that bedevils the 'bed bug' analogy and get students lost when trying to use it to understand in what way a 3d scenario can be an 'analogy' of a 4d scenario. For one, everything you visualize 'inside' a membrane will be a genuine EXAMPLE of a 3d object. You then lack what is significantly different in the alleged 2d that you might use it to understand that 'a 4d scenario will be different in this way, from a 3D scenario', making the analogy useful. Specifically, we don't see in what way must there be a 'small thickness of a 3d object', the thickness of which 'extends into the forth dimension' just like there must be such a thickness in an alleged 2d 'object'. As an appropriate analogy, we would rather expect a 3d object to vanish entirely since we cannot visualize its alleged '4D extension' just like the way the 'bed bug' must vanish completely upon the vanishing of the 3d extension. In this way 2d would be truely analogous to 3d in that in both cases, the vanishing of an added 'dimensions' leads both objects to cease existing.
The theoretical physicist is bamboozled by the fact that though his tape measure is flexible enough to bend into any shape, it is rigid enough not to stretch or contract. However, a 2d surface in question must bend in all directions, a feat impossible without the surface shrinking and stretching at various regions within the surface. Therefore there is realy no such a ruler capable of being trapped inside the surface that behaves like the 3d tape measure in that it can be bent with the surface without being stretched and shrunk in the process, with the shrinking/stretching being by the same amount the surface itself is stretched/shrunk at the place the 'tape measure' is sitting on.
All this means that the claim made by GR, that some 'bed bugs' trapped in the surface can study and discover the non-Euclidean geometry 'of the surface' is ridiculous nonsense! Then the whole GR flashes down the toilet!