You do realize that what you’re claiming to do is impossible, right?
Yes, but it depends on what you think I am claiming to do. I have an MS in math, and I spent many hours in grad school correcting the angle trisections that were often submitted to the department. There are many pitfalls, but this is not one of them.
The webpage does have an angle trisection on it. It’s very simple, and easily proved correct. I think Archimedes came up with it.
All right, I’m curiuous. What’s the difference between what you’re doing and what’s impossible?
To trisect an angle with an unmarked straightedge and a compass is impossible, what ultrafilter is doing is essentially marking his straightedge. The trick here is that in standard compass and straightedge constructions, the compass and the straightedge work independently, but that is not the case in ultrafilter’s demonstration.
So while possible to do in the real world, it doesn’t work in the realm of classical construction.
Also, it is possible to trisect angles and take cube roots and all sorts of cool stuff if you allow yourself extra tools beyond a straightedge and compass. Appropriate curves can even be “sort of” constructed, that is to say on a point by point basis, and then you fill in the blanks. This is exactly how you can plot a parabola with only a compass and straightedge, but only point by point, until you get the picture and connect the dots with nice curvy lines.
Tenebras
(Sorry if I jumped you ultrafilter… so rarely does a math question come up when I can see it. 
)
Sorry I switched that… replace throughout ultrafilter by RM Mentock and vice versa…
Tenebras
BTW, are you the same Mentock from the Bad Astronomy board?
The same.
But, notice. I do not mark the straightedge. There is still a difference between that trisection, and what was allowed classically.
How about using a bit of poetic license for the benefit of simplicity and divide the wheel into 16 segments.
Will some pedant come up and say ‘Hey that civil war cannon wheel has two extra spokes, whats going on here!’?.
Maybe not, but I will. Does that make me a pedant? I always thought I was just a careless anal-retentive. Is this a promotion?
Umm…I love clever geometric constructions as much as the next guy, but I’m not sure that this level of analysis makes sense.
SG, are you actually going to paint this wheel as if the axle were perfectly perpendicular (okay, normal) to the plane of the painting? If not then you have to divide an elipse into equal segments, not a circle.
Furthermore, unless you are doing this painting on a grand scale or you’re planning to give the wheel impossibly thin spokes, you have a substantial “fudge factor” to work with. As long as you don’t put all the spokes on one side of the wheel it’ll probably look fine.
Also, wouldn’t said wheel probably be standing in something like grass or mud, thereby obscuring the lower spokes anyway?
And lastly, I doubt that the original builders of circa Civil War cannon wheels went to all this trouble. They probably had a simple, practical solution that got it good enough for government work (as it were). Hyperacuracy might actually detract from the overall realism of the picture.
Anyway, just my 2 cents. YMMV
Yes, you did not mark the straightedge, but you have done something which is equivalent. When you put the compass up against the straightedge and then slide the straightedge up and down, you are measuring a fixed distance on the ruler, which is a no-no. The compass, strictly speaking, is to be used only to produce circles with a designated center and radius.
That’s what I meant when I said that you were “essentially marking the straightedge.”
Tenebras
Are you saying that it would be an acceptable classical method if, in step two, marked, with the compass where the left leg is pictured, and then used the straightedge to draw a line from that point to where line B intersects the circle?
The left leg of the compass slides along the line, from step two to step three.
Classically, the compass could not be used to transfer measurements. It collapsed after it was used.