# Tetrahendon from balls maximum height (static friction)

I could not find an answer by web search. From Why is rolling friction less than the maximum static friction? I understand static balls "experience" static friction, not rolling one.

I'm trying to build a tetrahedron from tennis balls, like here https://en.wikipedia.org/wiki/Tetrahedral_number.

I noted large pyramids are unstable, of about 10 balls edge is very easily breaking/falling/rolling apart during construction (8 balls are ok, 9 is doable but difficult).

My floor is rather smooth, so the structure can be approximated by placing ideal balls with some friction due to the rough surface on a no-friction floor.

The weight of the pyramid grows by the power of 3 whereas the surface of the basement by the power of 2, but static friction is proportional to weight, so still I do not immediately see why larger tetrahedrons should be more unstable. But building real ones shows they are. Why? can some formula be derived for maximum height with friction coefficients, size and weight of the balls?

• Tennis balls are not ideal balls. The deform, their surface is rough and irregular, and they are not exactly round. Commented May 10, 2021 at 12:59

Andria Rogava, Professor of Physics at Ilia State University in Georgia, Eurasia, has constructed some remarkable towers using tennis balls. For example, a 9 storey tower made of 25 balls and a frustrum of a pyramid made of 46 balls. In both structures some balls overhang so that they are supported from below by only 2 balls instead of 3.

Friction between the balls is essential to provide inward torques which counteract the radial outward torques arising from the balls pushing against each other. Which is why tennis balls are ideal. For the tower the top-most single ball (the 'keystone') is essential - remove it, even with extreme care, and the tower collapses. (See video in link.)

These tall structures have been achieved by careful placement of balls, and the use of temporary 'scaffolding' during construction. This suggests that the limit which you found is far from what can be achieved.

Each layer of balls pushes both outwards and inwards on the layers below and above. If the lowest layer is resting on a horizontal surface, there must be friction between the balls and the that surface in order to generate an inward force. Alternatively the outermost balls in the lowest layer could be welded together, or held in place by a frame; then friction with the floor is not required.

Mathematical Analysis

TO BE COMPLETED LATER.

Except for the simplest cases, such structures of rigid spheres are statically indeterminate and cannot be analysed without introducing other properties (or assumptions) such as elasticity and deformation.

• Thank you for your input. Could you please add some info if we do introduce elasticity etc., what would cause larger pyramid to roll apart? Commented May 6, 2021 at 2:50
• "There must be some friction between the floor and the balls, to balance the outward components of force within the structure." - why? single ball does not fall apart on ice - internal forces are enough to hold it together. Commented May 6, 2021 at 2:54
• @Martian2020 A single ball has internal forces which pull inwards. In a pile of balls there are no pulling forces which keep the balls together. If balls at the edge of the base layer were glued or welded together then the structure would not fall apart even on ice. Commented May 7, 2021 at 4:03
• Thank you for the link, towers are interesting and I think I would try to build one. Suggesting gluing is kind of strange, obviously it would, my guess it is not necessary for that. Might be interesting to find out what additional height rough surface adds compared to the ice. Commented May 16, 2021 at 3:15
• Some math would be nice cause nice example had not answered original issue. Tower in the link has fixed surface of base and growing weight, but pyramid has growing base, so might be no limit to height under "ideal" conditions. Commented May 16, 2021 at 3:19