This is as simple as it gets. We take an broken E-string (the small one) off an electric guitar and make a hot wire foam cutter out of it.

The details are flexible. A hot wire cutter is simple. Take a wire and pump some electrons through it so that it heats up and bingo, it'll cut foam.

What we need:
A wire (the guitar string)
A frame (to hold the wire and put tension on it)
Moving electrons (the power supply)

Step 1: The Wire & Electrons

Before you build the frame, you should test your wire. It doesn't need to be taunt to test its heating capability. In this case, I hooked the wire up and attached to it a power supply.

Lets talk about the power source:
Ideally you might have a variable DC power supply. In my case pumping 11.1 Volts @ 1.7 Amps did the trick. Varying the voltage will change the current. (V=I*R) The power (P=I*V) dictates the temperature of the wire. The more current, the more electrons are moving and smashing into the wire (resistance) and thereby heat it up.

If you don't have a power supply you should get one, they are very useful. But really, if you don't have one you can get buy with other power sources. Here are a few ideas:

1. A wall wart.
These are cheap and prevalent, but it will be hard to find one that can put out a couple amps. That is the problem with any power supply you try to use. The power supply that I have can put out 5A @ 18V which is pretty good. Usually, wall warts can do 0.5A which probably isn't going to cut it.
2. A hacked computer power supply
This is a simple hack. PC power supplies are all over the place. You can make a nice 5V/12V supply for next to nothing. Just search for instructables related to that.

Fixed vs. Variable.
Because we are using some random wire the voltage needed to drive the right ammount of current through the wire will vary. This is because the resistance of the wires will vary. So, ideally, we want a variable power source. This will also allow us to adjust the voltage and therefore the temperature of the wire if we'd like.

If your power source is fixed, that is no problem. We have another option. We can still adjust the current draw of the wire by changing the resistance of the system.

Step 2: The Circuit

The circuit is a simple as it gets. Current through a resistor. This is the classic V=I*R. Given that my voltage is 11.1 V and my current is 1.7A I can find the resistance of the wire 11.1/1.7=6.53 Ohms.

A quick note. The resistance of the wire will increase as it heats up. For instance, when I first connect my wire at 11.1 V it pulls more than the 1.7A. Then, as the wire quickly heats up the current decreases to a lower value.

!!!!MATH!!!!
These are the important equations (only simple algebra here).
Ohms Law: V=I*R
Power Law: P=I*V
The reistance of the wire is dependant upon a factor r, (r=R/L) i.e a shorter wire of the same material has less resistance.
Temperature is proportional to power dissipation per length. T~P/L

Playing with the math get us a relation between the temperature and Voltage and Length and the properties of the wire.

T~P/L=I*V/L=V^2/(R*L)=V^2/(r*L^2)

T~V^2/(r*L^2)

Thus for a given voltage, decreasing the length of the wire, increases the temperature. This makes sense, as the shorter wire has less resistance and therefore with the set voltage will have a higher current draw, thereby heating it up more. Plus it is shorter, so the shorter wire has more power dissipation/length thus heating it even more. (hence the 1/L^2 factor)

Also, changing the wire type 'r' changes the temperature. Use a higher resistance material and that will decrease the temperature because less current is drawn, therefore less power.

See, this is simple. These relations then give us a way to adjust the size of the hot wire cutter if need and keep the temperature roughly the same.

Step 3: Adjustments and Variability

Adjustments & Variable Control

If you don't have a variable power supply, don't worry. Lets look at my case. I was running at 11.1 Volts @ 1.7 Amps. Now lets pretend that I only had a 12V supply that can put out the required amperage. If I used that 12 volts, it might just work fine. The wire will get hotter. But lets say that I want to have the same temperature. Now I need an 11.1 Volt drop across the heating wire with 1.7 Amps. This is going to require a resistor in series with the heating wire to drop that additional 0.9 Volts. Thus, @ 1.7 Amps and 0.9 Volts we get

R=V/I = 0.9/1.7 = 0.53 ohms.

This is pretty small and it is going to be hard to to find a resistor or potentiometer that can work accurately at such a small value.

An easy way around this is to just increase the length of the cutting wire.

T~Vo^2/(ro*Lo^2)=V^2/(ro*L^2)

Where, ro is the resistance per length of the wire, Lo is the original length, L is the new length and Vo is the original Voltage and V is the new voltage.

11.1^2/(ro*Lo^2)=12.0^2/(ro*L^2)

The resistance per length of the two wire are the same. The voltage has changed. Now we can find out what the change in length of the wire needs to be.

12.0^2/11.1^2 = L^2/Lo^2
12.0/11.1 = 1.081 = L/Lo

So we just need to make the wire 8% longer and everything will be fine. This is the power of mathematics. We can working around these problems easily. So, by making the wire a little longer we were able to work with the slightly larger power supply.

Step 4: The Wire

Most likely, your wire won't be just the same as mine. The resistance is going to vary. This is why you will have to play around with the voltage and test the heat of the wire on whatever foam you'd like to cut.

Besides this fact, really any typical metal wire will do. Steel has a good resitance to it. You can try different gauges of wire, but as you increase the diameter of the wire you will decrease the resistance.

T~V^2/(r*L^2)

Then, for a given voltage, the wire will heat up more. However, this is considering a constant voltage. And since I=V/R the current is going to increase substantially.

You can use the relations shown before to make changes in your design. Also, having a variable power supply lets you get around all this math. You can just hook it up, start at a low voltage, and crank it up until it heats up as much as you would like.

Step 5: The Frame

The frame puts tension on the wire so that it will cut well. The only restriction is that the frame should not conduct electricity. We want the current to pass through the wire, not the frame.

The frame that I have made here is made of conductive metal, however I've made sure that there is no conductive path through it.

This frame also has an adjustable tension. This is useful since I can tighten the wire to whatever tension I would like.

I also made it very big. I needed this to cut out wing sections. You certainly can make a small (or larger yet) one yourself. All that really matters is that you wire is properly heating up and has proper tension. (and remember, changing the wire length changes the heating) So the frame needs to be able to hold that tension consistently.

All of the parts shown here are available at any normal hardware store.

My frame consists of a aluminum cross bar and two 1/4" threaded rods. One the the rods is mounted tight. The other one, the one with the electrical tape on it preventing conduction, is mounted a little loose. The bolts keep it in place, but we don't want it too tight, otherwise the tensioning system won't work well. It should just pivot around the main bar.

The wire was attached by cutting too grooves with my old craftsman rotary tool into the threaded rods. One end of the guitar wire has a ball with a hole in it. I fed the wire through that hole and looped it through the groove. This gave a sold anchor at that point. The other end of the wire was simple wrapped around the threaded rod and through the groove. After a few wraps it was holding well.

The tensioning system is attached to the other end of the threaded rod. It is just a sting (avoiding conduction again) attached to a wire tensioner. Both ends are fixed with bolts. The tensioner is twisted until a good amount of tension is on the guitar string. Don't over do it, or you'll break the wire.

Step 6: Cutting

The power supply is connected to the wire. I attached the alligator clips to the threaded rod.

Cutting foam is a straight forward process. You want the wire to be hot enough cut easily, but not too too hot, otherwise it will melt away too much foam. As the wire is cutting, you'll want to cut at a smooth speed. If you sit at one place for a while you will melt the area around it leaving bumps.

Also, don't cut too quickly as you will bend the wire altering it's shape and therefore the shape of your cut. So, use a steady slow cut and all will be good.

And there we have it a simple hot wire cutter + theory. All is well.