I did, and I'm sure there was a lot going on in Mr. Caveman's mind.
He might have started out making a walking bike, using old odd shoes that were laying around the cave. I imagine him thinking like this:
That shoe thing sort of works, but it's a bit tough getting all the materials together.
What if it could just be made of sticks?
And since simpler is better, I suppose he tried using triangles.
No doubt the first thing out of his mouth was, Won't roll.
Ok, what about squares?
That's better, it rolls on the right kind of surface. How about 5 sides?
Oh, good. Even smoother (though not great). How about 6, 7, 8?
Maybe if I make one with as many sides as possible, each side as small as possible, it will roll without any bumping at all?
And so (in my imagination) that's how the wheel was discovered.
NOTE: Unfortunately for Mr. Caveman, he had to make these wheels and try them, the hard way. He didn't have Wolfram Mathematica to help create computer simulations. Every math lover should take a look at Wolfram's math tools.
Anyway, back to our bike riding caveman. Maybe he found the wheel, but having a good idea is no guarantee of success. He could have gone overboard with the circle/wheel thing. Trying to build a bike to take all his friends along, he could have created this:
You know that experimentation can sometimes turn into a vicious circle/cycle (sorry!). Well, eventually my caveman may have gotten his round-wheeled bike together, like this beauty built by Marco Facciola.
Or perhaps he did not go to round wheels. He might have felt square was hard work, but okay for his bumpy ground. A square-wheeled bike is demonstrated here by Professor Stan Wagon of Macalester College. Stan was honored by Ripley's Believe It or Not for creating and riding this specially-constructed cycle and surface.