I don't think it's quite that simple, Jeff - you've got an inconsistency there.
On the one hand, you say to constrain it while rotating, so that A, B and C remain in contact with the floor. You can do that. But you then say that after 90 degrees D will be contacting the floor and A will be off it. A can't be off it if the chair is constrained so that A remains in contact.
I would guess that making a block with a single flat side is in fact far easier and cheaper to mass produce than any type of tripod. The cheapest way to make a tripod with precision flat feet might even be to make a block with a flat side and them remove the extra material to turn it into a tripod.
Making a tripod with pointed feet so that you don't have to deal with the flatness issue ends up with something that will damage your surface plate so that's not much of an option. Making hard rounded feet on the tripod is again, only harder than just making a block with a flat surface and also risks damage or excess wear to the surface plate. So though it might seem like an interesting approach, I think the answer is that in practice, there is no type of tripod that makes more sense than a block with a flat side for use with a surface plate.
BTW, my new surface plate just showed up today and I just spent the last hour making a flat piece of aluminum for the fun of it.
Very cool. I like your logic much better than mine. :)
Even without the constraint of the rotation, it's (now) easy to see that if you keep three legs on a wavy floor and move it around randomly, you will always be able to find a spot where the the forth leg would hit the floor making all 4 touch at that point. The rotation idea is simple proof that there there must always be at least one location where that was true (assuming the chair legs were correctly aligned on a plane).
My curious mind got to wondering when I first got involved with this whole silly musing about four legged stools and wavy floors.
Looks like it was over 12 years ago, and this link provides a grandios explanation of why what I said really does work:
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Happy Independance Day to all.
Jeff (Who can't recall any other US holidays referred to by their dates instead of their given names. He doesn't think that the shipment of mayonaise on the Titantic is a US holiday yet.)
I did that, although I was making a small aluminium sanding block (I'm now sure my sanding block needed to quite THAT flat, but it was, as you say, fun).
I don't quite follow this. I thought the important feature was that the scriber should be at a consistent height - and it will be even if you rotate it.
Stool with legs A, B, C, D all the same length, sitting on a wavy surface. Surface is only mildly wavy with no steps or discontinuities.
Stool is unstable - it rocks on legs diagonally opposite. Let's say they are A and C. It appears that legs A and C are too long. Rotate the stool 90 degrees, so that legs B and D are where A and C were, and since they are all the same length B and D will now be solid and A and C will appear too short. They went in that 90 degree rotation from too long to too short. At one place at least in that rotation they were just right.
If it's a continuous floor surface with no steps, there's guaranteed to be at least one spot where they will all touch the floor at the same time.
Before the A or C leg can lift off the floor in the rotation, the B and D legs must first both touch the floor. It's at that point in the rotation where the B and D legs first touch the floor that they will all be touching the floor at the same time.
It's the same reason you can't draw a graph of two lines - one tiled up, and the other tilted down, that you can't get from the point on the x axis where the line A is above B, to the point where line A is below B, without going through the point where the lines intersect. It's at that point where both lines are the "same height" on the graph.
Doesn't work for me with a pub table, all that happens is I get the "what's he up to now?" look from the landlord. So back to the tried and tested folded beer mat that the bloody cleaners throw away each night. Grrrrrr!
So do the landlord a favour and install leveling feet on your table - different table every visit, the people will really wonder what's up with you. Gerry :-)} London, Canada
So you have a non standard wavy floor. It's what I believe the mathematicians would call 'ill conditioned' Get the landlord to replace it with a better one.
Any continuous-surface-floor (no jumps, steps, holes) would be ok, if the ends of the legs are coplanar. Most likely one of the pub table legs is shorter than the others, by about the thickness of a folded beer mat.
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