**The Shiekh (Fearless Blue)** | Thursday, March 10, 2005 - 05:37 pm Nice maths Nimz, (I think, I didn't bother to check it because its not particularly relevant) A pile of sand as we view it only exists in 3 dimensions. A pile of sand conforms most closely to a conical shape, and a pyramid is an approximation to a cone. The larger the number of sides to the base of a pyramid, the closer the pyramid resembles a cone. So a hexagonal pyramid, more closely resembles a cone than a square pyramid, and a square pyramid more closely resembles a pyramid that a triangular pyramid, etc. The pyramidal number, as I had envisaged it was a number that reflected the number of grains of sand (spheres) that would fit into a pyramid if the pyramid was turned upside down and the sand (spheres) poured in. This is problematic, because spheres only pack neatly in a few circumstances. It works nicely for a pyramid with a 3 sided base, 1 + 3 + 6 + 10 ...., and for a pyramid with a 4 sided base, 1 + 4 + 9 + 16 ..... A 5 sided base does not pack neatly into flat layers, (try it yourself), a 6 sided base packs nicely but what do you do for the top level, is it 1 sphere or 6? I would list 1, 4, 10, 20, ... and 1, 5, 14, 30 ... as all being pyramid numbers, but being of seperate series. There are obviously more series, but the computation of them gets ambiguous when the spheres do not neatly pack into flat levels. Anyone care to hazard a guess at the number of spheres on the 3rd level of a pentagonal pyramid? And of course the whole cone/pyramid solution is just a huge simplification for the pile of sand problem. Grains of sand are not spherical, a pile is pretty much never of a neat geometric shape, there is no requirement for the taper of the pile to be constant or to match a nice mathematical formula, a pile is often lumpy and can contain internal cavities. |