Just returned from a tour of California and came across this piece of engineering at a restaurant at SFO. Sadly, the table designers had chosen to have four points of contact between the table and the floor which is of course a disaster. Apparently, a section of pizza had the correct size and compliance to form a stable interface between the table and the floor. An adjacent table had what seemed like a better design in that a single pole came down from the table to a circular disk base but the base had five points of contact with the floor. A waitress noticed my interest and a more general discussion ensued, causing some hilarity to other patrons. It was explained that I was a nerd (we’d visited CalTech after all) and that seemed to take care of it.

Believe it or not, restaurant table stability is a common subject for discussion here, especially since one of our number is a mechanical engineer and also because we spend too much time at restaurants. So, time to develop a proper (although not very rigorous or perhaps even correct) theory…

If an infinitely smooth flat surface has some other infinitely inelastic solid resting on it, there will only be three points of contact that form a triangle (just using classical physics here – quantum effects don’t occur in respectable restaurants). How stable this is depends largely on how big the triangle is and how far away from the center of gravity is the nearest vertex or edge. If a moment is applied to the solid, it may be able to tip the solid to another triangle formed by two of the original points and a new third one. So, moments applied to the solid cause the three points in use to transition from one triangle to another.

What does this means for restaurant tables?

- Table with three points of contact with the floor. This could be a table with three separate legs or else a central pole with a wider circular base with three points of contact (which is much more compatible with likely seating arrangements). This is stable until a sufficiently large moment is applied, the magnitude depending on the size of the triangle of contact. Essentially, a wide circular base with three points of contact will work very well.
- Table with four points of contact with the floor. Total disaster. The four points of contact effectively form two triangles sharing one of the diagonals. The diagonal in question will be the one formed by the two longest non-adjacent legs. The center of gravity of the table is directly in line with the diagonal. So, almost any size moment will cause the table to change modes between the two triangles. Hence the use of pizza, wooden wedges, folded paper etc etc.
- Table with five (evenly spaced on the edge of a circle) points of contact with the floor. While better than four points of contact, there are still multiple triangle combinations that allow triangle mode switching and the center of gravity will be closer to the edges than it would if only three points of contact were used for the same diameter base. Consequently, there’s no point having five rather than three.

Clearly, the designers of the five point of contact table had snatched defeat from the jaws of victory. So why do restaurants continue to use tables with four or five points of contact and then have to waste time stopping them wobbling? It’s a mystery.