[katie]

katie asked the Naked Scientists:

Hi- when he was about 3, my son Calon was asking me about 2D and 3D shapes.

Then he asked what 4D and later what 5D is. I did some research but although Calon seems satisfied with the various answers I found, I could do with some more help to get my head round the ideas.

As you are all so good at explaining the difficult theories out there, could you explain the concepts of 4D and 5D for me?

Thanks so much for your excellent show,

katie


What do you think?

[JP]

The problem with understanding anything bigger than 3D is that we live and think in 3 dimensions.  Going to 4, 5, 6, ... dimensions is similar to trying to draw a 3D picture on a piece of paper.  You can get the idea of a 3D object across by shading and perspective tricks, but in the end, you're really just drawing a 2D object because you're working on a 2D piece of paper.  It's tricky to think about 4 dimensions because--well how would you represent a 4 dimensional object by using familiar 3 dimensional ideas?

However, if you take a step back and ask "what is the mathematical meaning of a dimension" you can go ahead and describe higher dimensions mathematically: It's basically just the number of coordinates you need to specify in order to describe an object.  I think an easy way to get your head around the idea is to think in terms of what use you might find for 4+ dimensions in real life.  How many dimensions do you need to specify:
How far the car has gone?  1 dimension for distance.
Where the car is on a map? 2 dimensions, one for latitude and one for longitude.
Where the car is on a topological map (i.e. where is it on a map and what is its altitude?)  3 dimensions, latitude, longitude and altitude.
Where the car is on a topological map and what its speed is?  4 dimensions, latitude, longitude, altitude, and speed.
Where the car is on 2D map, and what it's speed/direction is on that map?  4 dimensions, latitude, longitude, north/south speed and east/west speed. 
Where the car is on a 3D map and its total speed and direction?  6 dimensions, 3 for speed/direction + 3 for position/altitude.

There's other uses for 4+ dimensions in physics, depending on what you're trying to plot.  A really famous one is the 4 dimensions you get in relativity "space-time" which is 3 dimensions (space) + 1 dimension (time).  The physical meaning of the higher dimension depends on why you're using them.  In relativity, its thought that the fabric of the universe itself is 4 dimensional.  In my examples above, the 4 dimensions are a mathematical tool for describing the car. 

You can extend shapes to any dimension by extending their definition to higher dimensions.  For example, a sphere is "all points that are equally distant from a central point."  As long as you know how to measure distance in your space, you can plot a sphere.  Usually (but not always) distance is given by the "Pythagorean theorem" which says that the distance between two points can be calculated by adding the squares of the distance between them on each axis, and then taking the square root of that number. 

[LeeE]

Perhaps the easiest way to visualise n-dimensions is to think of them hierarchically.

If you start with zero dimensions, you can have a zero-sized point.  If you then add a dimension, you could arrange a number of points in a line.  Extending this to two dimensions means you could put a number of lines side by side to form a square.  The next step is... you guessed it - you stack squares one on top of another to form a cube.  That's the easy part that everyone is familiar with.  The important bit though, is seeing the relationship between the different number of dimensions - for each new dimension you add, you duplicate what you've already got and extend it in the direction of the new dimension.

In the next step, of going to a four dimensional object, we hit problems because we can't directly represent it in three-dimensional space.  However, what you can do is to think of a three-dimensional object over a period of time, so the size of the cube in the time dimension is the length of time that it exists.

Another way to represent a four dimensional object is to use several cubes placed side by side.  In this representation each cube represents 'the' cube at a different point in time, and at any instant you're looking 'the' cube at all points in time at once.

All you then need to do to view your collection of cubes as a five-dimensional object is to look at them for a period of time, or once again, duplicate them all again and find a football pitch to spread them all out.

Good lower-order everyday dimensional objects to think about are films & videos and music.  Films & videos at any one point in time are two dimensional, and sound is one dimensional (although we use two or more streams simultaneously, like laying two lines side by side), but it is only once we've seen or heard them over their duration in time that we can say we have seen or heard the entire object.

[chris]

These are superb answers. Thanks.

Another analogy to make this simple to think about is the concept of a road junction and how to avoid a crash.

On a straight piece of road - 1 dimension - cars can run up and down the road but not turn off.

At a cross roads cars can move in two directions; they can even collide because they can occupy the same X and Y coordinates.

Now add a third dimension - an underpass or a flyover - now two cars can be in the same X and Y coordinates - the crossroads, but they won't necessarily collide because one can be on the ground and another on the flyover - i.e. have a different Z coordinate.

Now add a fourth dimension - cars can arrive at the junction at different times - now they can have exactly the same X,Y and Z dimensions but still won't collide, because their t (time) dimensions can be different.

I can't think of a sensible fifth dimension to fit the road analogy, but hopefully you can get the picture!

Chris

[LeeE]

Hmm... perhaps if you then put your cars onto car transporters...   

There should be some way of working roundabouts into this scenario too