sdi

Very good question. It really is a mix! Some constructions, like Moss's Egg and the dodecagon, were found entirely by hand (though I used a computer afterwards to convince myself that there were no better solutions). Others, like that inscribed regular pentagon, were found entirely by computer. Most of the constructions I've made, though, are a mix: either I used a computer to guide my own search by hand (usually by asking it to find a circle with a given radius, and then studying it to see how I might do so in general), or I found something interesting by hand which I then explored using the computer (e.g. to prove that something that looks right actually is right).

The computer program I wrote to search for constructions can be found here (though it's not very user-friendly, and you'd have to be a computer programmer to get anything out of it). I've also been making pretty heavy use of Wolfram Alpha to verify constructions. (I really need to spend more time with Euclid; I'm not very good at geometric proofs, and my arithmetical ones are unsatisfying.)

My program is handy, but it's pretty limited; it can tell you that something is possible, and it can tell you how to do it, but it never tells you why. I'd say 80% of the time I've spent on this has been spent going back over things to try and make sense of them, and often I fail at it. So, the computer is an idea generator, but you have to go back over all those ideas with a lot of contemplation. Much of the reason I've been posting these has been so others can explore the constructions and maybe get a sense for how to construct, say, √2 without needing a program to search for it.

I've also destroyed most of a great, big sketchbook of newsprint and will need to get more soon :)