This cutter heats a 16 gauge nichrome wire hot enough to glide through the densest foam.  It doesn’t seem to be effective at cutting anything else.  Not even paper.  Nichrome is the same wire used in electric baseboard heaters, toaster ovens, blow dryers etc.  Its only purpose is to get very hot when the correct wattage is put to it.  This wire is readily available online.  It is my understanding that all nichrome will eventually pop after a certain amount of use, so I went with a lower gauge for longer life.  Nichrome Popping Experiments

The design is super simple.  All you need to do is set up a transformer that will put enough wattage through the wire to get it red hot.  It is important that the wire is attached to a spring on one end.  The wire relaxes and becomes less taught as it gets hot.  The spring is necessary for keeping the wire taught as you cut.

The aluminum angles are there to keep the heat of the nichrome a superfluous distance from any of the wood.  As it turns out though, the nichrome does not seem to transfer any heat whatsoever beyond the wire itself.  When running it does not get the aluminum angles nor the spring hot. 

The switch is wired in before the transformer; directly in-line with one side of the mains supply.  The outputs of the transformer are wired to alligator clips connected on either side of the nichrome.  These can be clamped anywhere on the nichrome and will heat whatever wire is in between.  It is important to note that the distance between these connections makes a substantial difference on the temperature the nichrome reaches.  Tables can easily be found online which outline what temperatures are reached depending on the wattage applied to a one foot length of a specified gauge of nichrome.

I used scrap 1/2″ plywood to make a rudimentary fence for guiding the foam as it passes through the wire.  Getting a clean cut depends almost entirely on passing the foam through at an even speed.  The cut will get wider or narrower depending on how quickly the foam is passed through the wire.