Wikipedia:Reference desk/Archives/Mathematics/2022 March 5

= March 5 =

Geometry.
Is there a structure that you can flip, that can land in 3 possible sides? A coin flip is 2, dice is 6, and tetrahedron (pyramid) is 4. If the answer is there is none, can there be a proof for that? 67.165.185.178 (talk) 15:38, 5 March 2022 (UTC).


 * This video might interest you. --DB1729 (talk) 15:51, 5 March 2022 (UTC)
 * So the doughnut-shape is the coin? Contributions/67.165.185.178 (talk) 16:30, 5 March 2022 (UTC).
 * 67.165.185.178. In the video, they are talking about an optimal cylindrical or disk-shape abject that would result in equal probability of landing on each of three sides when flipped...and as explained below by CiaPan. But yes, they did use a sort of doughnut-shaped roll of tape to demonstrate. So that can work too. --DB1729 (talk) 17:53, 5 March 2022 (UTC)


 * Another thing you can do is take a tall, thin triangular prism, and then slant the ends so it cannot balance on them. Danstronger (talk) 16:21, 5 March 2022 (UTC)
 * Oh, take a long-triangular prism, long like a pencil, and that can flip on 3 sides. I see. 67.165.185.178 (talk) 16:30, 5 March 2022 (UTC).


 * Actually, a coin can land in 3 positions, not 2: heads, tails and on the edge... So if you take very short cylinder (a thin coin), the probability of it landing on its side (edge) is very small. If you increase the length (approaching the shape of a pencil, or a rod) then the probablility of it landing on its side grows to 1, while probability of landing on any of its bases (ends) drops towards 0. Somwhere in between there is such length-to-radius ratio, that all the three probabilities of landing at either base and on a side are equal. --CiaPan (talk) 16:46, 5 March 2022 (UTC)
 * Regarding the coin landing on edge: see Mr. Smith Goes to Washington. --184.144.97.125 (talk) 22:51, 5 March 2022 (UTC)


 * There are shapes, which have only one position of stable equilibrium, so you could say whatever way you drop/throw/roll/toss them, they always land (finally) in one position. Two examples are Gömböc and Monostatic polytope. So I suppose there also exist convex bodies with arbitrary number of stable positions. Alas, I don't know any specific examples for 3. --CiaPan (talk) 16:54, 5 March 2022 (UTC)
 * Take an American football as a starting shape File:American Football 1.svg. Modify it so that the longitudinal profile of any 'meridian' remains, say, a semi-ellipse, while the equatorial crossection becomes a regular triangle. Such solid will have three stable positions on a plane, IMHO. Possibly you'll have to make the axial length (point-to-point) big enough relative to the width in the central ('equatorial') part. :) CiaPan (talk) 18:12, 7 March 2022 (UTC)


 * If all sides are flat, the structure is a polyhedron. Following the usual terminology, I'll call the sides "faces", and the corners "vertices". Faces are bounded by "edges", each of which connects two vertices. For example, the familiar cube has 8 vertices, 12 edges and 6 faces. Using symbols, we can say that for the cube V = 8, E = 12 and F = 6. For the standard pyramid on a square base we find that V = 5, E = 8 and F = 5 (counting the base as a face). Each edge is shared by two faces, and each face has at least three edges. Combining these two facts gives us that 2E ≥ 3F. Each vertex is shared by at least three edges, and each edge has two vertices. Combining these two facts gives us that, moreover, 2E ≥ 3V. (The similarity is not a coincidence. Either inequality follows from the other by polyhedral duality.) Next to these two inequalities we have the well-known fact that a convex polyhedral surface has Euler characteristic χ = 2, which means that V − E + F = 2, or, equivalently, V = E − F + 2. If we additionally require that F = 3, we can sum this all up as the system
 * 2E ≥ 3F&thinsp;;
 * 2E ≥ 3V&thinsp;;
 * V = E − F + 2;
 * F = 3.
 * Substituting the known value of F in the other (in)equalities results in the simplified set
 * 2E ≥ 9;
 * 2E ≥ 3V&thinsp;;
 * V = E − 1.
 * We can use the last equality to eliminate V from the second inequality:
 * 2E ≥ 9;
 * 2E ≥ 3(E − 1).
 * Since E is a whole number, the first inequality is equivalent to E ≥ 5. The last inequality is equivalent to E ≤ 3. Clearly, this has no solution. --Lambiam 17:28, 5 March 2022 (UTC)
 * Another way to reach the conclusion. We start with a system that holds for all convex polyhedra:
 * 2E ≥ 3F&thinsp;;
 * 2E ≥ 3V&thinsp;;
 * V = E − F + 2.
 * Eliminating V and simplifying, we obtain:
 * 2E ≥ 3F&thinsp;;
 * E ≤ 3F − 6.
 * So 3F ≤ 2E ≤ 2(3F − 6), which implies F ≥ 4. --Lambiam 22:25, 5 March 2022 (UTC)

Tapering down a coin's edge to a single sharp edge reduces the number of its stable sides from 3 to 2. Likewise, as suggested above, tapering a triangular bar's ends to sharp stake-like points yields 3, which works for any larger n-sided bar. In addition to the Gömböc, an odd-number sided equilateral polygonal bar (5 or larger) can be sufficiently weighted along one side to give only that one stable side if and only if its edges are rounded (so that when the weighted side is at the unstable upper most positions it still produces a torque at the base). -Modocc (talk) 21:25, 5 March 2022 (UTC)