Why do most still cameras have only one tripod mounting hole?

Previous answers have already suggested acquiring an Arca Swiss style dovetail adapter plate, typically made from aluminum, that is formed with a ridge or lip that mates with the body of the camera for a distance of typically around 2 inches (50-60mm).

This width of engagement of the ridge or lip against the body of the camera, combined with the screw point mounting hole that the adapter plate is affixed to the camera with, serves to eliminate rotation between the camera and the adapter.

Naturally, the adapter must be formed to fit the camera body, so these adapters are specific to the model of camera that you have.

To solve rotation or wiggling between the adapter and the tripod, an Arca Swiss B style dovetail head must be used on the tripod. Here again, the physics that stops wiggling and rotation is that 2 or so inch length of engagement that the machined dovetail plate (mounted on the camera) and receiver (mounted on the tripod) affords.

As you pointed out, a single point is a pivot point. But multiple points in a rigid line, and in the case of this dovetail plate engagement, two rigid lines, cannot rotate. This is why the Arca swiss B style dovetail plate arrangement is so popular among serious photographers.

Arca Swiss is a brand, but like the Xerox brand is to photocopying, or the Google brand is to searching... there are many brands and manufacturers to choose from when it comes to selecting a dovetail plate formed for your particular camera. I personally use the Really Right Stuff brand, but I'm not mentioning this to promote that brand... I merely mention this as a way for you to find a catalog online to visualize the variety of plates and tripod heads that just one US manufacturer makes. There are plenty of other, often cheaper, typically imported brands to choose from as well.

Previous respondents also mentioned L plates in passing, and this deserves more detailed discussion as L plates offer a significant step up advantage over just an adapter plate. A formed L plate, combined with a quick release head on the tripod, can make changing from portrait to landscape mode a snap, while maintaining close to the same framing.

If a traditional screw mount multi axis tripod head is used, and you want to switch from portrait to landscape, a lot more additional adjustments might be necessary to compensate for the change in framing when the camera is "swung down and to the side".

An L plate, combined with a quick release head, can with some cameras be designed and positioned such that the center axis of the lens remains unchanged when the camera is quick released from its' horizontal position and remounted in its' vertical position.

Again, this can be camera and L plate dependent, but it is one of the design goals of quality L plate manufacturers, that is sometimes undermined by data access ports for tethered shooting... which explains why it all depends on the specific model of camera.

Be that as it may, having an L plate adapter is still vastly more utilitarian than just a bottom plate adapter, because the L plate also affords additional attachment points for accessories.

For example, if you want to mount an on camera light, that is off axis from the lens, the hot shoe for the flash generally isn't strong enough to cantilever a light source off axis while remaining hand held... but an L plate, combined with various brackets attached to the L plate, can hold a light 3 feet up in the air above the camera, or 18 inches off to the side of the camera, if you wanted. The hot shoe would rip out, but the dove tail plates formed as part and parcel to the L plate can tolerate significantly more clamp load and leverage.

So while the foregoing response mainly served to clarify and augment the ideas that you've already been given (adapter plate and L plate), I'd like to introduce to you another way to support your camera, especially when using long lenses.

Long focal length lenses often have their own lens foot, and there are dovetail adapter plates that mount to these lens feet as well. As with camera mounting plates, lens mounting plates have an anti rotational feature incorporated as well. For the really long lenses, whose feet have more than one hole, the adapter plate takes advantage of the secondary hole in the lens foot.

But for shorter or more compactly designed (diffractive) long lenses with only one mounting hole in the lens foot, the adapter plate will have a ridge or lip that will engage along the fore or aft width of the lens foot. Some lens feet have curved edges, and in those cases, the adapter plate might have a couple of machined holes with allen head cap screws placed in them, and the round sides of these two cap screws are what engage with the radius edges of the lens foot.

Because there are two screws, there is no pivoting, because two points make a line, and we actually have three points of engagement when we remember that there is still the mounting screw that holds the adapter plate to the lens foot.

So why would you use a lens foot to mount the camera on a tripod rather than screw hole under the camera? For two reasons:

1. Better weight balance. Good glass is heavy, and the weight of a long lens can easily outweigh a still camera. The lens foot is positioned such that center of gravity can be optimized with the camera clicked into the lens, which provides the optimum balance for not only tripod mounting (reducing inadvertent lens droops and tilts), but also for hand holding.

2. Instantaneous portrait to landscape orientation changes while maintaining the same lens axis. Since the lens is rotating within the lens collar that incorporates the foot, the camera body is just along for the 90 degree rotational ride, and nothing changes in the central area of the frame. Because the entire rig is supported by the lens foot rather than the camera body.

Lens feet, and their adapter plates, tend to be longer, which provides more opportunity to fine tune weight balance in a dove tail grip system, because you can slide the plate within the receiver fore and aft until the balance is optimal.

All of the foregoing applies to both still photography, and videography using DSLR form factor camera bodies.

When I worked as a photographer (now retired), I mounted the longest dovetail lens feet I could find on my lenses, and mounted more stuff to the unused length, like radio transmitters for remote cameras, light triggers, pistol grips, reflectors, etc.

I think you asked a GREAT question, which is why I took this time to respond. Why don't manufacturers include anti rotation holes in traditional body style SLR camera bodies. There is no reason why they couldn't, so any answer to that specific question of "why" would amount to speculation and opinion.

So I endeavored to answer with ways you could solve the problem of "wiggling", as you described it. And believe me, when mounting remote cameras suspended up in the rafters of an arena with thousands of people below, we don't want any "wiggling". I found that the dove tail mounts were secure and confidence inspiring in my set ups, and I think you will too.