How does one explain, clearly and simply, that we live in a three-dimensional world?
The explanation has to be understandable for a twelve year old child.
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As celeriko took the what I consider as what "ambient" space the object lies in approach (I.e., Whittney's embedding), I'll consider the degrees of freedom of movement.
Consider the normal directions you would have someone move when they are standing straight up. Have the child walk back and forth in a straight line. We can consider this the forward-backward freedom as the first dimension. Then have them shuffle sideways left or right without turning. We can consider this the left-right freedom as the second dimension. Put together, we get the usual ability to walk around on the surface of the planet (a 2-manifold!). But what if we jump in a rocket or dig through the planet? Or even if we just moved up or down in space? We can call that the up-down freedom giving us the usual third dimension. If you want to get crazy, have her stand still for 5 minutes (or like 30 seconds, I couldn't keep still that long). While you are not moving, time slides forwards. While you can't go back in time (as far as we know!), many people consider this to be the fourth dimension!
It'll be fun and give you a chance to run around a bit while exploring dimensions. Also you can get into a little bit of coordinate systems (move forward 10 feet and left 3 feet!).
Unfortunately, dimensionality is a very tough concept to truly understand, even for adults. I know that you are trying to find a clear and simple method, which what I have typed below is certainly not. My hope is you can use the information below to get ideas on how to develop your own scaffolded method for your student. What I would suggest would be a scaffolded approach by which you start with the very basics of dimensionality, showing how to "construct" higher dimensions using objects of a lower dimension, and progressing on once the student demonstrates they understand the previous dimension.
Individual points exist as objects with 0 dimensions. They do not have a size, they do not have a length/width/height. They merely exist at one specific spot in space. Points do not express dimensionality because they cannot be traversed or "moved through" in anyway. "Moving through" a point would just lead to another, wholly different point. Make sure to enforce the fact that the "points" that we draw and talk about are not true points because the graphite/ink they are drawn with has (albeit very minuscule) length, width, and height
To move from the 0th dimension to the 1st dimension, one need only draw any two points and connect them with a line.
True lines (linear/straight NOT curved or bending) exist as objects with 1 dimension, which we can call "length". "Moving through" a line can exist in only one way, along the length of the line. Trying to move somewhere not along the line will take you out of that one dimension because you will no longer be on the line.
To move from the 1st dimension to the 2nd dimension, one need only draw two lines and connect them with additional lines.
Polygons, circles, shapes all exist as objects with 2 dimensions. they still have the same first dimension as lines, "length", but they now have a second dimension, which we can call "width". Having two dimensions now allows us to move in two distinct directions, "left/right" and "up/down". If you draw a shape on a piece of paper you can see that you can move anywhere inside that shape or along the edges, and still only be in two dimensions. What you CANNOT do is raise off of the paper, because the shape does not exist above the paper, it only exists on the paper.
To move from the 2nd dimension to the 3rd dimension is a little tricker because it cannot be done on a single sheet of paper. A simple way to get around this is to draw two squares on two separate pieces of paper and hold one above the other. In the same way as before, we can draw two, 2D objects and "connect" them between the two pieces of paper.
Pretty much everything that we interact with daily are objects that exist in 3 dimensions. They contain the same first dimension as lines, "length", the same second dimension as a shape, "width", but now they have risen off the paper, so to say, into a third dimension, which we can call "height". You can verify this by standing up and stepping to the left/right (1st Dimension), stepping forward/backward (2nd Dimension), and jumping up and down (3rd Dimension). This can also be verified by picking up a book or a box and seeing that it, in fact, has three, distinct, dimensions.
I hope this helps some, this is the way that dimensionality was explained to me when I was younger. It also made it very intuitive for me to understand higher dimensions (4th, 5th, etc). Another idea might be to have your student try to imagine life as a 2D object (a square on a piece of paper, etc) and have them think about what life would be like, ala the novel Flatland, good luck!
Flatland. W. Abbott, 1884 Here's the Project Gutenberg link to read or download in your preferred e-version: https://www.gutenberg.org/ebooks/97
As an advanced student in elem school, my engineer single-parent mother gave me a copy while I was surviving a two year nothing-to-learn transition into public school from lifelong private. I believe I was 11 when I read it, and it entirely sorted dimensionality for me. In about 1962.
All the above good answers aside, I think that this question misses the mark in so far as the real problem is to internalize a good notion of what 'dimension' means. In particular, someone that age who familiar with the science-fiction notion of 'dimensions' referring to parallel universes (eg, traveling between dimensions with a magical vortex or portal) would be in a very rough spot trying to reconcile this notion with the the suggestion that our own world is three dimensional.
Try this out as a definition for your student(s):
The number of dimensions of a space is the number of pieces of information it takes to determine the location of something. For our world, you can (relatively) locate something by
1) turning the head left or right a certain amount
2) tilting the head up or down a certain amount
3) naming the distance of the object
That is, things can be up or down, left or right, and near or far. Each of those is a dimension in our physical space. Give someone any two of these and locating something is hopeless (or at least difficult/time consuming). Give them all three and they know exactly where it is.
I think that this spherical-coordinate, self-centered approach is probably easier for kids to grok in terms of 'number of dimensions' than the notion of us all floating around in a Cartesian 3-space.
You'll want to keep it simple and visual. Use easy, real-world examples.
I would say a dimension is similar to a "direction you can move in".
On a string you can only move left and right. It has one dimension. On a piece of paper, you can move left and right, but you can also move up and down--two dimensions. In a fish tank you can move up, down, left, right, but now also forward and backward--three dimensions.
Ask the twelve year old to imagine a video game where you can only move left and right, up and down (a side scroller, like Mario). That would be what the world would be like if it was only two dimensional. It would be a cartoon.
I'm seeing that my answer is very similar to Chris C's... Sorry. I'll post it anyway since I made it this far.
I have no experience with this age group, so this suggestion might be bonkers.
One reason it might be hard about dimensions is that there are so few examples (point, line, plane, 3-space is usually it), so it is hard to see exactly what property of these examples you are "abstracting" with the number "dimension".
One way to rectify this dearth of examples, is to give a bunch of interesting "nonspatial" examples of spaces, and try to figure out their dimensions.
I am sure you can come up with many more examples. This should convey the concept of dimension, and also give a taste of "real mathematics". The idea to model a set of physical constraints as a high dimensional "space" can be really captivating!
You could use basis vectors as a basis.
In 2D, because everyone talks about lower dimensions when talking about any dimensions, you can draw two lines at a right angle to each other.
Then, by only translating, but not rotating, that group of lines you can get to any other 2D point. You may also be able to describe the vectors as "rails" that the object slides along to get to its point.
Now, you can draw a third line coming from the same point. You can show that any two of the lines can get to the same point that the third gets to. This allows you to talk about linear independence.
With one line, though, there are many many points we can't get to. So, the paper is 2D because we need two lines to describe points.
If you make a 3D set of axis, you can show that in the same way, moving along those lines can get them anywhere they need to go.
A fourth line is redundant, and two lines obviously misses a lot of points, so we can perceive 3D.
If you want to boggle minds, then you could start with some hyperbolic crochet, which is only flat in a 4th dimension, and then get your 12-year-olds to understand that the world that they (believe they) live in is simpler than that.
Or you could play the scene from Woody Allen's The Purple Rose of Cairo, where a character from a 2-dimensional film comes out of the screen into a 3-dimensional world, and get them to discuss what it would mean to live in world with a different number of dimensions.
It might be helpful to explain that space is popularly thought as being three dimensional because it requires three components for a coordinate to locate any point from a mathematical perspective. Mathematics is a very popular tool for describing the universe in scientific terms, but while useful for study and application, our perception of the universe does not explain why the universe works the way it does. Sometimes the things that we believe must be revised to more closely coincide with reality, so we see that while we do not know precisely how reality works, over time we are coming up with mathematically better approximations to correspond with our observations and experiments.
Though not an answer to the original question, here's how you might explain that you live in a four dimensional world, the fourth dimension being time. Maybe this intuition is helpful.
If you ask someone to meet you, you would never say, "Meet me at the intersection of first and second street, on the third floor", without also specifying a time. Likewise, you would never specify a time without including a location to meet.
Yes, there is an experiment that shows that we live in a 3D world, and not in 2D or 4D world. The answer comes from Coulombs law. The force F follows the inverse square law. You can test that by doing an experiment. You can prove that if our world had more or less dimensions that force would be impossible to follow the inverse square law. For me this is the most straightforward way to convince someone that our world is 3D. It links a hypothesis (is it 3D) with an experiment. The same holds for gravity since it also follows the inverse square law.
I will not go into the maths. Im sure that if you search the internet with the words "how would coulombs law, or gravity be in 4D" you will find many things to find your answer. The main idea is that the concepts of flux and curl (basic in maxwells equations) are different in 2D 3D and 4D. Those concepts are not that hard to explain in children because you can always show them with basic hydrodynamic examples (water flowing from a hose, or curls in water etc)
Use an android or ipad tablet to represent a portion of an infinite 2 dimensional plane you can turn or incline that plane but any object you draw can only exist only on that plane. Then place a book on top of that plane and you have a third dimension.
Maybe 15 graphing papers on top of each other bottom graphing paper is 0 on z axis, next graphing paper is 1 on z axis etc.
For easy transition 3d to coordinate system - childrens blocks representing dimensionless points - a 2x2 grid (in case you don't have enough blocks) with labelled coordinates for 2 dimensions, another grid on top, for the third dimension. The students can label a block's coordinate for more interactivity. You can easily choose a 0,0,0 coordinate if you need to explain negative coordinates.