An Introduction to Theoretical Physics (Part 3 of 5)

Posted on 2011/10/19 by

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Feature Image - n-cube Resistor

Introduction to Part 3

Having solved the initial problem, calculating the resistance of a cube of identical resistors, what can be done to make the equation more general?  The first abstraction was to move from 2 dimensional circuit diagrams to trying to solve 3 dimensional circuit problems.

Could we investigate higher dimensions?  What about a 4 dimensional “cube”?  Let’s see if we can answer this:

What is the total resistance between two opposite vertices of a 4-dimensional cube of identical resistors?

As previously mentioned, questioning the purpose of the asking something like this is unhelpful for a theoretical physicist.  Of course, intuition will guide what is worth spending time on and which questions should be dismissed, but there are no strict rules.  If the questions are motivated purely by pragmatic considerations, it could stultify an innovative discovery.

Carl Sagan in his book “Demon Haunted World” wonderfully articulates the public advantage of funding to undirected research – inquiry that is motivated only by a physicist’s curiosity.

If Queen Victoria had ever called an urgent meeting of her counselors, and ordered them to invent the equivalent of radio and television, it is unlikely that any of them would have imagined the path to lead through the experiments of Ampere, Biot, Oersted and Faraday… They would, I think, have gotten nowhere. Meanwhile, on his own, driven only by curiosity, costing the government almost nothing, himself unaware that he was laying the ground for the [radio and the television], [Maxwell] was scribbling away. Its doubtful whether the self-effacing, unsociable Mr. Maxwell would even have been thought of to perform such a study. If he had, probably the government would have been telling him what to think about and what not, impeding rather than inducing his great discovery.

Carl Sagan, “Demon Haunted World“, Chapter 23: Maxwell and the Nerds. p. 390 [1995]

So, although our present work clearly isn’t dealing with the key question of the day, as Maxwell was investigating, working by our own curiosity mirrors a key aspect of being a theoretical physicist.  We can always look to see if there are any practical applications after we reach a solution!

To answer the above question, we will to know exactly what it means.  There are two things we need to learn:

  • What is a “4-dimensional cube”?
  • What are “two opposite vertices” on a 4-dimensional cube?

Let’s begin with the first question.

A Tesseract

A 4 dimensional cube, on first hearing, sound like an oxymoron.  Cubes are 3 dimensional, everybody knows that.  Well, we are using “cube” in a more abstract way.  Here is a drawing of a 4-dimensional cube (or a “tesseract”).

“I learned very early the difference between knowing the name of something and knowing something.”

Richard Feynman ”What is Science?”, presented at the fifteenth annual meeting of the National Science Teachers Association, in New York City (1966) published in The Physics Teacher Vol. 7, issue 6 (1969)

Knowing the name “tesseract” doesn’t really help us at all, its just a way of keeping track of what we are talking about. You don’t know anything about it from just its name.

To begin, we could notice how we compared the square and the cube by counting how many edges came from each vertex.  For the square, we had two vertices meeting at each vertex and noted it was a 2-dimensional object.  The cube has 3 edges joining to each vertex and is a 3-dimensional object.  Now, you might see where i’m going with this.  The tesseract, a 4 dimensional object, has 4 edges joining at each vertex.  Check the diagram to see if this is the case.

However, it is not just the number of edges from each vertex that determines the dimension.  It was noted that for the square and the cube, all the edges from one vertex were all at right angles to one another.  This is the difficult (or perhaps impossible) thing to picture about the tesseract.  From one vertex, there are 4 edges, all at right angle to each other!

As far as we know, we live in 3-dimensional space – at least we certainly all operate under that assumption.  So the tesseract is impossible to represent fully in our space.  Similarly, the cube was impossible to represent on our 2D computer screen.  The flat screen was only able to reproduce a projection of the cube, and the we interpret it as a 3-dimensional object.  The 3 edges from each vertex were no longer all at right angles to each other, but it was still all connect together the same way.

[Interpreting flat images as 3D is not something you only do with diagram, pictures or painting - all of your visual information from the world is received as 2-dimensional images.  The only difference is that with a real 3-dimensional object, you receive two slightly different flat images, one on each retina.  This defines your perspective and your eye-brain interprets this as one 3D picture of the world.]

So the above diagram of the tesseract is a projection – it’s connected together the same way, but the edges from each vertex aren’t all at right angles to each other any more.  It’s also a double projection.  The diagram is convincing you that you’re seeing a 3-dimensional projection of a 4-dimensional object.  But how can this be on a flat 2D screen?  You are actually looking at a 2D projection of a 3D projection of a 4D object!

Building A Tessearct

We could just as well call the tesseract the 4-cube, which is likely to be more beneficial.  That way, we can easily talk about 3-cubes (just normal cubes), 2-cubes (squares), 1-cubes (lines) and 0-cubes (points).  Choosing “cube” as the comparator is a natural choice – we live in 3 dimensions.  It is often the case that we choose the most familiar thing as a reference.  For example, having ten fingers is probably why we count in 10s, and not 2s as computers do or something else.

Talking about 2-cubes and 4-cubes is especially advantageous when we consider another way to think about the tesseract.  It is possible to construct it from the lower dimensional “cubes”.  Lets start with just a point:

This is a 0-cube.  It 0-dimensional object: 1 vertex, joined to zero edges.  If we take another 0-cube, we can join them with an edge:

We now have a 1-cube (a line).  It is  a 1-dimensional object: each vertex is joined to 1 edge.  Joining two of these together gives us a 2-cube (a square):

The 2-cube was made by taking each vertex of a 1-cube and joining it to one (and only one) vertex of the other 1-cube.  We carry this procedure on to get the 3-cube:

And lastly, for our purposes at least, we can construct a 4-cube by pairing up vertices from two 3-cubes:

I’ve draw this 4-cube in such a way that its projection in 2D is much more symmetric than our original picture.  Take time looking at it until you’re convinced it’s two cubes joined together.  Also,  to confirm that it is a tesseract, check each vertex has 4 edges joined to it.

“Opposite” Vertices

In the original diagram of the tesseract above, it was represented as one large cube connected to a smaller cube.  This representation made it more difficult to see where the “opposite vertices” are.

For the square and the cube, the answer is intuitive - the pairs of opposite vertices are found by drawing diagonal lines through the shapes.  There aren’t any for the 0-cube, and the two vertices of the 1-cube are, by definition, opposite.  Mathematicians like to call this a ‘trivial’ case.

For the 4-cube, we’ll have to think a little more carefully and analyze our intuition.  What constitutes “opposite” in the 2-cube and the 3-cube?

For the square, if we move from one vertex along the edges, it take at least 2 moves to get to the opposite vertex.  For the cube, it takes at least 3 edges to get to the opposite vertex.  Of course, it is possible for it to take more than 3 edges – you could draw a path that twists and turns, double-backs and it could take as many edges as you like.  But opposite vertices are joined by at least as many edges as there are dimensions.

So, we have a very general insight that isn’t restircted to the 4-cube.  It would be fitting to condense the above considerations into a statement about n-cubes, which are n-dimensional objects.  Substitute any number in for “n” and you have a specific type of n-dimensional cube.  It’s just one more abstraction.

Opposite vertices on an n-cube are any two vertices where the minimum path connecting them requires moving through n edges.

So for our first diagram, with the big and small cube, the opposite vertices could be the front-top-left vertex of the big cube and the back-bottom-right vertex of the small cube.  It takes a minimum of 4 edges as it is a 4-dimensional object.  Check this for yourself – it’s not intuitive!

The diagram above highlights one example of a minimum path connecting those two vertices.  There are several others.  All the other vertices have opposites as well – they all pair up.

For the symmetric diagram, it is slightly easier to see.  On the outer ring of the 2-dimensional projection, the opposite vertices are simply directly opposite on the diagram.  The same is true for the inner ring.  For example:

Once again, we find the benefit of presenting the same thing from a different point of view.  In this case, the symmetry enables us to see something that was one obscure, now as intuitively as we saw the lower dimensional examples.  Now we have a better understanding of this mathematical object, let’s see if we can tackle the physical problem.

The Resistor Tesseract

Let’s start talking about resistors instead of edges, and look at the physics of solving the our latest question.  The first think to check is whether our analysis with equipotentials will work for the tesseract, as it did for the cube.  The first two steps are as before: connect the vertices that are one resistor away from the start, and one resistor away from the end.

Nothing new so far.  What about the remaining 6 vertices?  Well, we can consider that if we are successful with this method, we will end up with a diagram that has joined the equipotential vertices together.  How many would there be?  We might expect no more than 3, as this would give 4 edges between the start and end vertices.  These were chosen as ‘opposites’, so we would expect there to be 4 edges between them.

With this analysis, it would seem that there is only one more equipotential to find, and so all the remaining 6 vertices can collapse into one.  We are left with this:

You can now see why the resistor symbols have been left out of the diagrams!

We’ve now finished collecting the vertices into equipotentials.  But how many resistors are on each side of that central equipotential?  We can see how many resistors are unaccounted for by counting how many there are on a tesseract.  Using the first diagram turns out to be easier.  Each cube has 12 edges, we joined two cubes by pairing verticies, and each cube has 8 vertices.  So that means a tesseract has 12+12+8=32 edges.  4 resistors at the start and 4 at the end have been accounted for with our equipotential argument, so that leaves 32-4-4=24 resistors.

The solution is not as difficult as you might think.  All we need to do is consider symmetry, which is why this diagram is so much more useful than the initial representation of the tesseract.

The total resistance of the network has to be exactly the same no matter which vertex we choose as the start, as long as we pick opposite vertices.  So, the resistor network must be symmetric in this respect.  If we flipped the diagram around, top to bottom, we’d get the same answer.

This means we needn’t worry about studying our diagram too carefully to find out where these resistors are, the problem has been solved with a symmetry argument alone.  There must be 12 on each side.  However, for completeness, here’s the finished diagram:

Solution for the Identical Resistor 4-cube

The above diagram can now be convered into more traditional notation, putting the resistor symbols back in:

I’ve kept the presentation the same as the solution to the identical resistor cube from Part 2.  Again, the blue numbers keep track of how many vertices were squashed into one by noticing they were equipotentials, the red counts how many resistors are in parallel between equipotentials.

Using the same equations as before, the solution is as follows:

R_{T}=\frac{R}{4}+\frac{R}{12}+\frac{R}{12}+\frac{R}{4}

R_{T}=\frac{2}{3}R

Summary

The use of mathematics in physics is often spoken about with a sense of mystery.  We are asked why it is that the physical world can be described with abstract mathematics.  One fairly straightforward answer is to notice that mathematics is a systematic analysis of the symmetry, quantity and similarity we see in objects.  Mathematics is an abstraction of this process that considers ideas that don’t need any corresponding physical observation.

Every so often we find that there are things we require from mathematics that can be applied to physics.  This can be because we didn’t notice a symmetry in the real world that a mathematical idea suddenly clarified, or an experiment revealed something that wasn’t obvious about the world, but the symmetry is clear.  Or an abstraction has been made from physical observation to abstract variables, such as energy.  There are many other examples too.

In our case, we have tried to use real world physics and apply it to a mathematical object that has no physical analogue.  And yet, in Part 4, we will see how abstracting still further will result in a synthesis of familiar ideas, resulting in a greater understanding in both mathematics and physics.