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My question is little bit philosophical. I would like to explain my ideas with a 2 dimensional universe model.

If we had lived in 2 dimensional universe like a plane, What could we observe when seeing a 3d object?

For example: If a square pyramid that is inside full of material comes to the plane universe in right angle, what could the people who live in 2d universe observe? Firstly, we could see small square and slowly it would enlarge and then it would dissappear suddenly. If the scientists who live in the 2d universe observed such event , All scientists could get shocked and they could check their physics formulas because energy is not conversed in that event. They had believed that Energy is conserved and nobody could create energy from nothing and finally they could understand that total observed universe energy can be much more what they calculated with their formulas.

My questions:

  1. What could we see a 4d pyramid comes to our 3d universe? is it correct that Firstly we would see small cube and it would enlarge and then suddenly disappears?

  2. Dark matter and Dark energy is related with such ideas? http://map.gsfc.nasa.gov/universe/uni_matter.html

  3. I know we do not observe such strange events till now but maybe scientists can get some results during micro experiments. Is there any such experimental results at least in micro universe (atomic level experiments in quantum mechanics)?

Sorry for your time if it was asked before.

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3 Answers 3

up vote 8 down vote accepted

This idea is explored at great length in the novella "Flatland", which is a classic of pedagogical mathematics. It was published in the late 19th century, and the themes reappear in H.G. Welles "The Time Machine", and in Einstein's relativity. The four-dimensional pyramid will, if it crossed our 3-dimensional universe, as a cube growing from a point, then suddenly vanishing.

With modern computers, it is very easy to simulate four dimensional objects in various 3d projections (then in 2d projection to the screen). These simulations are useful for building up the intuition for higher dimensional spaces.

But as physics, it is impossible to miss a fourth dimension: we would observe objects escaping our plane! This would reveal itself as missing energy in high energy collisions, assuming that it takes a large amount of energy to leave our three-dimensional plane, because we are bound to a surface. These types of ideas became current in variations of Randall Sundrum and Arkani-Hamed,Dvali,Dimopoulos models in the last decade. These are infinite extra dimensions scenarios, and they predict that gravity at long distances will fall off as a different power. This is generally impossible to reconcile with cosmology.

The large-extra-dimension ideas are ruled out with scientific certainty by the observations consistent with a high Planck scale. This includes tiny neutrino masses, lack of proton decay, absence of strong CP violation, absence of electroweak corrections like magnetic moments or non-renormalizable W and Z interactions, absence of direct lepton-quark coupling, and a few other things I can't remember. These issues are discussed here: Can black holes be created on a miniature scale? .

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Wouldn't a pentachoron give a tetrahedral and not rectangular projection? –  Manishearth Apr 20 '12 at 8:23
    
I assumed OP meant something like an Egyptian pyramid. You are right of course. –  Ron Maimon Apr 20 '12 at 8:27
    
Oh, square-pyramid-thingy. I think there can be two distinct square pyramids in 4D--one would have 6 tetrahedral and one cuboidal face, the other would 4 cuboidal and one tetrahedral face. Or something like that--I'm hopelessly confused :) –  Manishearth Apr 20 '12 at 8:31
    
I mentioned Egyptian pyramid that goes inside 2d universe. –  Mathlover Apr 20 '12 at 8:38
    
@Manishearth: I imagined the 4d linear point suspension of a cube, which gives the OP's intended construction--- 6 tetrahedrons, one cube. The other construction you give is impossible--- you can't attach cube to tetrahedron by matching faces. To make the general shapes in 4d, you use polyhedrons with pairwise gluing along equal shape faces, and embed the gluing in 4d locally, then globally (if you can)--- it's not so hard to do with pencil and paper, but the topology is tough slogging. –  Ron Maimon Apr 20 '12 at 9:15

Remember, people in a 2D universe cannot see "out of the paper" to see an object. They can see:

a) A projection of the object (like how you draw a cube on paper). Projections have lines(in 2D) and planes(in 3D) which look like they intersect, but they really don't. A projection is sort of like what a 2D guy would see if light was shined on a 3D object such that its shadow falls on paper

b) A cross section--which is what you've described above.

Here is a projection of a hypercube, or a 4D cube, in 3D space, later flattened to your 2D screen. The viewing angle is diagonal in 4D space.

enter link description here

If it was rotating, you would see this

There is also an analogue to the tetrahedron, known as the "pentachoron". You can see the projection there.

As projections, you can see what they look on the Wikipedia page. As "cross sections", the hypercube would show a cube that changes size. If the hypercube were at an angle, I think you would see a tetrahedron (just like cutting a cube at an angle gives a triangle)

What could we see a 4d pyramid comes to our 3d universe?

The pentachoron would give tetrahedra in 3D space, and these would enlarge.

Note that if, by pyramid, you mean "square pyramid"(like an Egyptian pyramid) you will see cuboidal slices(distorted on tilt)

Note that there's not really a "comes to"--Out 3D universe is a slice of a hypothetical 4D universe, and yes, you can think of the object "approaching" and "intersecting" our slice. But the terminology "comes to" can be interpreted differently, as if a dimension is a place (big misconception propagated by novels and games)

Dark matter and Dark energy is related with such ideas?

We knew that spacetime(four dimensional) existed way before we discovered dark matter/energy. All matter bends spacetime, including dark matter, so there's a mundane relation there. Dark energy is related to spacetime in a stranger fashion.

I know we do not observe such strange events till now but maybe scientists can get some results during micro experiments. Is there any such experimental results at least in micro universe

String theory predicts "curled up" dimensions, which are significant at extremely tiny scales. But I don't know about any experimental results. There's this and this, though--dealing with macro scales.

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Pyramid (so square based?) You would see a 5 sided square-- meaning the pyramid would have 5 faces and the base still would have 4 sides(square) yet somehow when you look at it every face lands on a side it's quite confusing to look at let alone make sense of. That would be my most simple answer to give since I see lots of long complicated answers I figure I would give an answer based upon appearance of what you would physically SEE with your eyes not lab equipment which would be necessary to explain the how's and why's to it all.

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