Hi my name is Teri, I am a student and this question is 6-9th grade.. I wanted to know why a three legged stool is always steady, and why a four legged stool is not. I am wanting to know the mathematical reasoning behind this. Thank you!

Hi Teri,

There is some nice geometry in this (and related) questions. I once had a 3rd year undergraduate geometry student do a major project on this and related questions!

First imagine trying to place a two legged stool on a floor. For convenience, we will assume the ends of the legs are rounded, so that only one point at a time touched the floor, for each leg. To make it simpler, imagine the ends are very fine points. The stool will, essentially, pivot about the line joining those two points of contact. It will not be stable.

However, any third leg (of a reasonable length) will contact the floor at exactly one of the positions as it pivots about the line. When it does, that will be what you call the 'steady' position.

IF you make a small change in the length of this third leg, there will still be a nearby position in terms of the pivoting, at which this new length will contact the floor. The top of the stool may not be parallel to the floor but the stool will be in a stable position. The stable position is robust in terms of small errors in the lengths of the legs or the flatness of the floor.

Now imagine it is sitting on the three legs. Try to add a fourth leg on some particular line out from the top of the stool to the floor. There is a single exact length which will now just reach the floor. If the fourth leg is a little to short (or the floor sags a little) then this fourth leg will not contact - but the entire stool can pivot about two of the legs, lifting one of the others and putting the fourth one in contact.

Similarly if the fourth leg is a little too long (or the floor has a raised bump) then placing in the fourth leg will have to raise one of the other legs off the floor, making things wobble again.

It is not obvious, but IF you had a stool with four exactly equal legs forming a square on the bottom (at the floor) AND the floor is bumpy - THEN there will be some angle you can TURN the whole stool so that all four legs touch the floor. You could experiment with that - and even talk with your teacher about how to convince yourself it is true. [The corresponding theorem about functions is the intermediate value theorem. If a function is -1 at point a, and +1 at point b, and the functions has no breaks in it, then it is 0 at some point between a and b! For example if the temperature was -1 degrees at midnight and +1 at 10 am then at some time between midnight and 10 am the temperature was 0 degrees.]

Of course to call this three legged stool 'steady' the weight has to be placed, on average, over a point above the interior of the triangle made by the bottoms of the three feet.

The same principle explains why a photographer uses a tripod (with the camera over the center of the triangle formed by the feet). Sometimes, the photographer does use a monopod (one leg) and then the photogrphers feet and arms complete the basic tripod - three legs support for stability.

You COULD consider the simpler plane version of this problem. For this the floor becomes a line (or curve). The three legged stool becomes ... a two legged flat stool. One leg lets the stool pivot till the second leg makes contact. Once you have contact, any third leg is likely (because of small errors in the lengths) to push one of the legs off the line. Unless the 'weight' is balanced between the two legs now in contact the plane (flatland) stool will wobble over till the right two legs are in contact.

This approach by analogy (with other dimensions) is a common strategy of mathematicians in problem solving.

Walter Whiteley
York University

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