# If I squeeze something really hard, will it ever become two-dimensional?

A mosquito just wanted to bite me! Päng - and it stuck to my hand, hardly recognisable anymore. I said to my girlfriend: "Just reduced the dimension of the mosquito by one!"

Therefore the question: If I squeeze a three-dimensional body really hard, will it ever become two-dimensional?

Two-dimensional means that it has a thickness of 0, so 1 atom layer is not 2D.

edit: I'd like to make this question a bit more general: Are there any real two dimensional objects or phonemes in a 3D (or 4D) world?

• Define two-dimensional body. (This is not meant to be flippant, but we sometimes talk about "two-dimensional systems* in the context of "one atom layer thick" system since they can be modeled as such, although they, in the real world, clearly still possess "thickness") – ACuriousMind Sep 22 '14 at 17:55
• If you squeeze something hard enough, you could make it a black hole, which is zero-dimensional. – Ben Crowell Sep 23 '14 at 0:02
• Quick question: Would you consider a line a 2D or 3D object? How would you define 'a line'? – Ali Razeghi Sep 23 '14 at 0:34
• Conversely, how would you turn a 2D object into a 3 dimensional one? And then work backwards. – Michael Sep 24 '14 at 1:37
• Phonemes? What do they have to do with it? – LarsH Sep 25 '14 at 2:27

From a mathematical point of view you will never make something two dimensional by squeezing it because it will always have a thickness greater than zero. The limit would be something like graphene that is a single atom thick. This is pretty thin, but it still has a non-zero thickness so it's still 3D.

However in the quantum world it is possible to produce structures that behave as if they are two dimensional. Particles like electrons have a wavelength, and if you can make a sheet thinner than the wavelength of the electrons then electrons in the sheet will behave as if the sheet really is just two dimensional. Indeed this is why graphene is often described as a 2D material. It's thin enough that conduction electrons behave as if they are restricted to a two dimensional manifold. There is more on this in the question How is graphene a 2D substance?

Note that this sort of system is only two dimensional for a range of energies, because if you increase the energy you reduce the particle wavelength and at some point the wavelength reduces enough that it falls below the thickness of the sheet.

• what if you squeeze it down to 1 Planck length of thickness? – robert bristow-johnson Sep 23 '14 at 1:37
• @robertbristow-johnson: how would you confine any system to a thickness of a Planck length? What physical mechanisms exist that could do this, even in principle? – John Rennie Sep 23 '14 at 5:01
• Since Atoms can be seen as waves, couldn't you reduce it to a 2-dimensional-wave which could be seen as two-dimensional? – Falco Sep 23 '14 at 8:22
• What about a Black hole? Doesn't that reduce matter to a singularity? i.e. if that can reduce matter to 1D then you can envisage something that reduces matter to 2D. – Snowcrash Sep 23 '14 at 14:58
• @SnowCrash: a black hole reduces everything to zero dimensions i.e. a point. There is no known black sheet that reduces everything to two dimensions. – John Rennie Sep 23 '14 at 15:04

By your own definition, "one atom thick" is not two dimensional.

In that case, you would have to squish something so hard that the atoms stop existing. In which case it is not two dimensional any more, either.

As John pointed out, graphene is often considered a model for a 2D material - it is SO much thinner in the thickness dimension that we have to re-think our physics for it (and some win Nobel prizes doing so).

But as stated - your mosquito didn't quite get to two dimensions.

"Just reduced the dimension of the mosquito by one!"

And you might say that "time stopped for this little fella". In that case, you did reduce him from a 4D to a 3D mosquito... and you were right.

Word of warning. It's not always worth it to "be right" when there's a girlfriend involved. That's not a law of physics, but it is a law of nature.

The users above have answered this question adequately, but there is an interesting class of materials where you actually may do the opposite! These are layered quasi-2D materials (see the image for example). In these materials, the individual layers act almost as if they are decoupled from the other layers above and below them because they are bonded very weakly. Usually one can cleave these kinds of materials with scotch tape. When one applies enough pressure, though, the electron wavefunction overlap gets more and more substantial, and eventually makes the quasi-2D material act more like a 3D material! Counter-intuitive, but interesting nonetheless!

• Does graphite fall into this class or were you thinking of something more esoteric? – Floris Sep 23 '14 at 12:03
• Yes, by and large graphite falls into this category. There are a few technicalities to worry about, e.g. a non-zero hopping matrix element between the layers. But to a first approximation, I'd say graphite falls into this category. – Xcheckr Sep 23 '14 at 19:14

## protected by Qmechanic♦Sep 24 '14 at 17:10

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