James with Ben's first stone 1999
I have created an unintentional puzzle. I have challenged my great-nephews and nieces to find me the roundest stone. (It must be found as-is without any man-made modifications and be as like a ball as possible). I now have the problem of how to judge the winner.
Obviously a larger one would be better than a small one. If it has a small diameter hole in it this will be better than having an attached small diameter cylinder protruding from the surface.
I think I need a way to measure the ease with which it rolls. Any suggestions for a set of rules understandable by children from 2 to 100 years old (including myself?)
Your metagrobologically, James Dalgety
Date: 31-Oct-1999 From: Nick Baxter
Measure the smallest and largest diameters of the stone. The difference is the measure of roundness, the lower the better. Respectively, these diameters would be the smallest circular hole that the stone CAN pass through in SOME orientation, and the largest hole that the stone CANNOT pass through in SOME orientation (or simply the greatest distance between any two points on the surface). These aren't precise, but for generally convex surfaces this will work. This tends to ignore small holes in the surface and penalizes outcroppings, which was your desire.
Date: 31-Oct-1999 From: Ed Pegg Jr
Put all stones on a flat board. Lift the board by tiny amounts, perhaps by putting coins under one end. Whichever stone rolls off first, wins. But a cylinder might win. To defeat cylinders, add an array of nails to the board. The roundest stone will not be hampered by the nails. A cylinder will get trapped. Basically, you'll be finding the 'topple angle' of each stone. I used topple angles to figure out the odds of a normal 6-sided die with drilled holes. Loosely speaking, the topple angle is the amount of energy required to move an object from one stable face to another stable face. As a die bounces around, it loses energy, and eventually there isn't enough left to cause a face change.
Date: 31-Oct-1999 From: Allan Boardman
James - Yes, I think you are on to something with the rolling idea. Set up a slightly inclined ramp with a short 'shoot' at the top - a channel, as it were, to start the ball (stone) rolling in the right direction - and with a similar, but perhaps wider, slot at the bottom of the ramp to catch the perfectly round ball. This ramp can be rather accurately set up using a large (say, one inch), ball bearing. The ball bearing should go into the bottom slot most of the time if the ramp is quite flat and set up properly. A not-so-round stone, if it rolls at all, will not likely roll into the slot and the distance by which it misses the slot would be some sort of measure of non-roundness. For drama, you could set up a row of pegs on either side of the slot, one of which the test article would knock over. All of these stones should be checked for moss.
Date: 31-Oct-1999 From Adrian Fisher
Float the stone in a bath of mercury. Presumably it will float with at least part (if not most) of its surface above the level of the mercury. Set the stone in motion by stroking it until it starts to spin. Shine a light on this stone and record the extent to which it reflects the light within a narrow cone of space towards a light sensor. The rounder the stone, the greater the amount of light detected and reported. You might install a sound device to emit a sound signal, with the louder and more consistent the sound, the more perfectly the stone is round.
Date: 01-Nov-1999 From Professor Yoshiyuki KOTANI, Tokyo
My solution is: Measure the diameter, which is its longest length. Then Measure the volume, for example, dipping it in water like Archimedes. It is the rounder , the value, the third power of the diameter divided by the volume, is the lower. The length of the surrounding can be used instead of the diameter. This seems too normal.....
Date: 08-Nov-1999 From: DAVID SINGMASTER Professor of Mathematics and Metagrobologist
There have been a number of definitions of roundness used in the literature on convex sets which led to a famous limerick whose authorship has not been determined though I suspect it was Leo Moser.
Whether each circle is round
Is a question both deep and profound.
In a paper of Erdo's, Written in Kurdish,
A counterexample is found.
Supposedly Erdo made inquiries as to whether there was a Kurdish mathematical society in the hopes of writing such a paper! Practically, the easiest way would be to use a calipers or micrometer and take the largest and smallest diameters and use the ratio of the smaller to the larger as a measure of roundness which would range from zero to one. I would assume that non-convexity would rule out the stone immediately, or at least would detract from the score. Size could be given bonus points, or one could combine roundness score with size score. However, since these are quite different types of numbers, it's not obvous how one could combine them. If r is the ratio of the smallest diameter to the largest, I thought of using size * 1/(1-r) but if r = 1, then the score is infinite and the size doesn't affect it. I think some well worn beach stones will be so close to spherical that r = 1 might occur, at least within the accuracy of simple measurement. If this is thought likely, then one can adjust the factor to something like 1/(1.1-r). But these are rolling stones. So moss must be an immediate disqualification.
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