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(Inspired by Why are four legged chairs so common?)
I've been wondering for a while... Why do most wheeled office chairs have 5 wheels?
My guess would be that while stability vs. simplicity results in 4 legs, adding mobility to the equation may result in the need for 5 wheels.
Edit: This question is about a mobile chair
Consolidating some of the points made in the answers to the question you linked, and comments:
When constructing a chair, 4 legs is easy when you use traditional (wooden) construction - 90 degree angles, and easy to make stackable. A little bit harder than three legs because you have to make sure they are all the same length (or the chair will wobble).
Once you have a "office chair" with a hydraulic center post, the construction argument goes away. That leaves us with greater freedom to pick number of legs. The considerations are:
All engineering design is a question of tradeoffs; in this case, I think that the first point argues for fewer legs, and the second / third point for more legs. The question then becomes: what is the additional value, and the additional cost, of one more leg? Below, I calculate the cost of adding more legs for the same stability and cost - this makes some assumptions but concludes five is indeed optimal.
But there is another important factor (tip of the hat to my daughter for this concept): when the floor is uneven, a chair will be not be supported by all its legs - it will "wobble". Now if you have four legs, this wobble will happen along one of the diagonals of the square, and this line will be below (or very close to) the center of gravity. That makes the energy needed to go from one side to the other very small. When you have five legs, the center of gravity is always displaced relative to the line of support. So five legs provide greater stability on an uneven floor. As you add more legs, the "diagonal of support" gets close to the center. Even numbered regular polygons always have the potential of having the line of support going through the center, making them the worst choice (incidentally this shows that a trapezoidal arrangement of four legs is slightly better than a square... you will sometimes see that, and now you know why).
All of that makes five the optimal number of legs - good stability in all directions. Note that from a construction perspective, it only makes sense to give a chair five legs when you start with a (metal or plastic) center post - the older (square wooden legs) construction makes four a more sensible number as the other answer stated. Once you want the chair to have vertical adjustment, a single center post makes sense - and then you have the flexibility to choose the number of legs.
Finally, a reference from a large supplier of office furniture:
The National Institutes of Health recommend a five–point chair base for maximum stability and minimal chance of the chair tipping.
In fact, Tom Reardon, executive director of the Business and Institutional Furniture Manufacturer's Association, says furniture manufacturers stopped making chairs with four–point bases in the 1980s because they weren't considered as safe as five–point chair bases.
UPDATE
I thought more about the problem of optimization, and think I can explain that five legs is best.
Assume that the chair has to support a constant weight $W$, and that we want a constant stability. Stability is determined by the shortest “tipping distance” $D$. For a radial distance $R$, a chair with $n$ legs has
$$D = R \cos\frac{\pi}{n}$$
So we can define a “stability factor” $S=\frac{1}{R\cos\frac{\pi}{n}}$
Thus, for constant $S$ we get $$R\propto \frac{1}{\cos\frac{\pi}{n}} \tag1$$
Next, we look at the stress on each leg. The stress will be greatest when the tipping torque $\Gamma$ is directly in line with just one leg. At that point,
$$\Gamma = W\cdot R$$
Now we want to calculate the shape (section) of the leg that can support this torque. The maximum stress $\sigma$ for a rectangular beam of width $w$ and height $h$ is proportional to $wh^2$, and the mass of the leg of length $R$ is $whR\rho$; if we assume a constant aspect ratio $\frac{w}{h}$, then mass is proportional to area times length:
$$m \propto h^2 R \tag2$$
where the first term is a function of the strength, and the second term a function of the stability.
Similarly, for given torque $W\cdot R$ we can write the bending stress as
$$\sigma = \frac{My}{I}$$
where $M$ is the bending stress, $y$ is the perpendicular distance to the neutral axis, and $I_x$ is the second moment of area about the neutral axis $x$. For a rectangular section, $y \propto h^4$.
For constant $\sigma$, the maximum will occur at the outer edge of the beam where $y=\frac{h}{2}$, leading to
$$h^3 \propto W\cdot R$$
For given weight $W$, it follows that
$$h\propto R^{1/3} \tag3$$
Substituting $(3)$ into $(2)$ we get
$$m \propto R^{5/3}$$
For constant breaking strength, we get the total mass of $n$ legs:
$$M = n\cdot m \propto n R^{5/3}$$
For constant stability, we use $(1)$ to obtain
$$M \propto \frac{n}{\cos^{\frac53}\frac{\pi}{n}}$$
We can evaluate this for n between 3 and 7, and obtain $M$ as a function of the number of legs:
n=3: 9.524
n=4: 7.127
n=5: 7.118 <--- lowest value
n=6: 7.625
n=7: 8.329
This shows that indeed the structure with five legs needs the lowest mass to support a certain torque - if we can equate "mass" with "cost", and stability is indeed the main driver, this proves that a chair with five legs is optimal.
The difference between an office chair's 5 wheels/supports and a regular chair's 4 legs is that the latter has all of its load going straight down. The legs only need to be strong enough not to shatter. In fact, a chair could easily get away with 3 legs but for the stability. In contrast the office chairs legs support load perpendicular to their orientation. They need to be strong enough not to snap. Therefore they need more supports to support the load from above.
Another reason to have 5 legs as opposed to 4 is that the wheels aren't in a fixed position relative to the chair. That means if there were only 4 wheels the tipping axis could be anywhere so a sitter could lean in a direction and tip without realizing it. In a 4 legged chair a sitter can lean diagonally and not worry about tipping. Since office chairs can't confer the certainty of where the tipping axis is, it makes sense to make it less able to tip.
i'm pretty sure that if you check the OSHA (the U.S. Occupational Safety and Health Administration) regulations for chairs found at https://healthfully.com/osha-regulations-workplace-chairs-5916239.html you will find the ultimate answer to this question.
in addition to specifications pertaining to the back, seat and armrests, they also have regulations about the...
Base
Your workplace chair must have a strong, five-legged base with casters that are suitable for the flooring type of your workstation. Workplace chairs with four or fewer legs may give insufficient support. Chairs that have no casters can make it hard for you to position your chair close to your desk. This could increase bending or reaching to access your workstation components, leading to fatigue and muscle strain.
I don't know the history, I am just speculating. The chair in the picture has features that you don't find on a typical 4 point chair: It has casters, it has a swivel seat, and it has a springy reclining back. All of these make the chair more likely to tip over without warning. Lets assume that you roll the chair forward so all the casters are rotated to the back, then lean back. At the moment that it tips onto the two back most casters, the base will swivel to the least stable orientation, and the casters will rotate to form a smaller circle. In a 4 point chair, the base can rotate from diamond to square, and the casters rotate to make the square smaller, with no warning. A rigid 4 point chair is very predictable, and yet we have all tipped them over. Add the swivel and casters and the chair becomes an unpredictable killer. To compensate, the base circle must be increased and/or the number of casters must be increased.
