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A Three-Legged Stool

Date: 06/26/2001 at 11:49:02
From: Teri Brown
Subject: Why is a 3-legged stool always steady?

I am interested in finding out why a 3-legged stool is always steady, 
versus a 4-legged stool, which can be wobbly. 

Thank you for your help!

Date: 06/26/2001 at 14:05:54
From: Doctor Ian
Subject: Re: Why is a 3-legged stool always steady?

Hi Teri,

That's an excellent question! The answer has to do with something 
called 'degrees of freedom'.  

Think of it this way:  

(1) If you hold a cane in the air, you can move it in any direction, 
    twirl it, and so on. Its motion isn't constrained at all. That is, 
    the top of the cane can move freely in three dimensions. 

(2) If you put (and keep) one end on the ground, now its motion is
    constrained: you can't lift it, or rotate it... although you can 
    swing the top around in a variety of different arcs. That is, the 
    top of the cane can move freely in two dimensions. 

(3) If you connect the tops of two canes together and place the other
    ends on the ground, you can still move the tops, but only along a 
    single (straight) arc, back and forth. That is, the tops of the 
    canes can move freely in one dimension. 

(4) If you try the same trick with three canes, now you can't move the 
    tops at all. This is basically what's happening with a three-
    legged stool. The tops of the cans can move in zero dimensions... 
    which is to say, they can't.

Each time you add a cane, you remove one dimension in which the top 
can move freely - that is, each new cane removes one 'degree of 

Now, what happens when you add a fourth cane? Well, now you have too 
many constraints. This means that there are multiple ways that the 
stool can 'solve' the problem of which legs to use for support.  
Wobbling occurs when the stool can't 'decide' which solution to use, 
or, more precisely, when it's changing its mind about which solution 
to use. 

In effect, during the time that the stool is actually wobbling, it's 
really a two-legged stool, with one degree of freedom - which is the 
direction of the wobble.  

In math, we see this kind of behavior in systems of equations. If I 
have two variables, one equation will give me a whole range of 

   y = 2x + 3           Solutions include (1,5), (2,7), (3,9)

In fact, the range of solutions is just the graph of the equation. If 
I add another equation, I can have at most one solution:

   y = 2x + 3                  
                        Solution is (2,7)
   y = 4x - 1

That is, each new equation removes a 'degree of freedom' from the 
system. With two variables, I need two equations to lock things down.  
With three variables, I need three equations. And so on. 

Now what if I add a third equation? 

   y = 2x + 3                  

   y = 4x - 1           No solution

   y = 4x + 1

There is no pair of (x,y) values that will satisfy all three of these
equations simultaneously. Of course, I can _choose_ any two of the
equations and get a solution for that pair, and then I could switch to
another choice, which would be quite a lot like what happens when a
four-legged stool wobbles.  

So, the 'mathematical' answer to your question is: A 3-legged stool
'solves' a system of three equations in three variables, while a four-
legged stool changes its mind about which three of four equations to 
solve for the same three variables.   

Does this help?  Write back if you'd like to talk about this some 
more, or if you have any other questions. 

- Doctor Ian, The Math Forum   

Date: 09/02/2003 at 16:54:02
From: John
Subject: Three-legged stool

I found your answer very interesting.  If I understand you correctly, 
you feel that a three-legged stool is more stable than a four-legged 
stool.  Most furniture designers feel that a five-legged stool 
(especially one on casters) is more stable than a four-legged stool 
and, I believe, a three-legged stool.  For a given leg radius, the 
consensus seems to be the more legs, the more stability.  This concept 
seems at odds with your analysis.  Can you explain this for me?

Date: 09/02/2003 at 23:35:00
From: Doctor Peterson
Subject: Re: Three-legged stool

Hi, John.

There are different kinds of stability! A three-legged stool is 
guaranteed not to wobble, because the ends of its legs always form 
a plane. But a little wobble is only an inconvenience. More important 
for practical purposes, such a stool is LESS stable than one with more 
legs in the sense that its center of gravity is further inside its 
base: the more sides a regular polygon has, the greater its apothem 
(the distance from the center to the middle of an edge). That greater 
distance means that the sitter can lean farther out in any direction 
without tipping over. So if you don't mind a little tipping but don't 
want to fall on your face, or if you have a reasonably even floor, 
more legs are better. In a barn, I'll take the three-legged stool 
over the upholstered desk chair any time!

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum   
Associated Topics:
High School Basic Algebra
High School Euclidean/Plane Geometry
High School Geometry
High School Higher-Dimensional Geometry
Middle School Algebra
Middle School Geometry
Middle School Higher-Dimensional Geometry
Middle School Two-Dimensional Geometry

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