I’ll be damned! I never noticed that before.
What if the surface is a sphere with a smaller diameter than the distance between any two of the four points?
Well, one suspects the proof covers that possibility 
I like the point in the above posts about the stability issues. Good insights here.
But with 5 legs, you have 25% more caster wheels that you have to push around. The net effect is the same.
Inflate it further until the desired size is met.
This. No more, no less. A three-legged chair would present large variations in the minimum required tipping moment, with the lowest moment occurring when trying to tip the chair toward a point midway between two of the legs (just 50% of the moment required to tip the chair over in the direction of any one leg). In order to have acceptably high resistance to tipping in this direction, the span of the three legs has to be very large, to the point of tripping passers-by.
To get around that, you add more legs. The ideal would be a continuous ring of support, which would provide equal tipping resistance in all directions, but would also be impractical. Four legs is still too much variability (between-leg tipping moment is 71% of the toward-leg tipping moment, but five seems to do it (between-leg = 81% of toward-leg).
The only time I’ve seen six wheeled legs was on a bariatric chair.
I’d like to add an observation I’ve made being an office worker for the past 15 years.
When pushing an office chair, typically the legs will rotate until there are two legs extending ahead. If you start to push an office chair so that a single leg is extended ahead (like the point of a pentagram) it will usually rotate so that one leg is pointing back from the chair and two ahead.
The advantage of a 5-leg chair is that there is less of a gap between legs and less chance it will tip when you hit an obstacle (like the edge of a rug or mat). Also, with one leg pointed straight back it makes it very unlikely to tip back the other way. It seems a 4-legged chair would be much more likely to tip over either way when being rolled.
IV stands in hospitals tend to have four wheels. This is probably cheaper to produce but is it practical? The five wheel units are double the price of the four wheel units.
Not at all. I’m talking about a chair with a human sitting in it. Imagine if you had just one wheel - it would dig deeply into the carpet because there’s so much weight above it. Your point would make sense if the legs/wheels weighed 100 pounds a piece or something, but since they are light, the more the weight of the chair and its occupant can be distributed among several legs/wheels, the less digging in there is.
The five wheel stands are probably more stable but theoretically could take up more floor space around other equipment.
BTW I used to work at Hill-Rom. Why would one offer the typical rich doc (or hospital manager) a Chevy when they can afford a Mercedes? Yeah, it’s all about patient outcomes… :rolleyes:
Often attributed to Abraham Lincoln, but I’ve never seen it attributed it to Mark Twain.
[Quote Investigator]
(Suppose You Call a Sheep’s Tail a Leg, How Many Legs Will the Sheep Have? – Quote Investigator®) says there is evidence that Lincoln did in fact use this, or some variation, but that it wasn’t original with him. Several earlier usages were cited. No mention of Mark Twain.
With fewer wheels, you’ll get more drag per wheel. Multiply the drag per wheel by the total number of wheels, and you’ll get the total drag.
As YamatoTwinkie pointed out, the stability criterion is cos(pi/n). We can use this to determine the most mass efficient leg system.
To achieve a given stability, which I’ll normalize to 1.0, we have the equation cos(pi/n) = 1/l, where l is the length of a leg. Let’s suppose for the moment that the mass needed is just the leg length times the number of legs: nl. Then, we need to minimize n/cos(pi/n).
This gives an optimal leg count of 3.65, which we have to round up to 4 legs. However, it’s unlikely that the mass is just nl; longer legs represent a longer lever arm, which means they need to be thicker at the base and smaller at the ends. You can see this on your chair legs. More likely, the mass formula is proportional to roughly nl[sup]2[/sup], which means we need to minimize n/(cos(pi/n))[sup]2[/sup].
The minimum value here is 4.80, which corresponds nicely to the standard 5 legs.
You are assuming that the amount of drag per wheel is proportional to the amount of weight per wheel.
For something like carpet I doubt that is true.
What if the chair is on a treadmill?
For thick carpet, probably not. But then, who uses rolling chairs on thick carpet? On thick carpet, even five wheels won’t roll well. On dense carpet, or tile, or one of those plastic floor mats that such chairs are usually used on, though, it probably is close to true.
That’ll never fly. The average office chair is too wide for a consumer grade treadmill. What do you propose, some specially built treadmill just for office chairs? 
Just don’t try standing on a four wheeled chair. I saw a colleague do that once and she ended up on the floor. No real injury, except to her dignity and a few bruises, but a lesson learned.
I’ll give you tile for all practical purposes (but IMO there probably still is a measurable effect, though its probably at instrument level, not worker drone leg level)
Plastic floor mats still sink into carpet where the legs are. And yes, while deep carpet (like the avacodo green 3 inch shag in my man cave) is probably worse than dense carpet, my gut feel is still that its going to be non linear for any carpet that is worthy of being called carpet.
Even if there are nonlinearities, though, they don’t necessarily work in your favor. Sink far enough into the carpet, and you’re effectively resting on the firm substrate underneath the carpet, and surrounded by strands that are still standing. I’d expect that once you get to that point, the drag is probably pretty constant, independent of further force. And 1/5 of the weight of person+chair is enough to get to that point.