Strategies for arranging coins in a circle

Question

Suppose you have $n$ identical circular coins and you would like to arrange them on the table so that their edges touch and their centers lie on a circle.

Mathematically, there is no trouble. "Just" put the center of each coin at $re^{2ik\pi/n}$ for $k$ in $\{0, 1\ldots, n-1\}$ and some suitable $r$. But in practice, one can't easily calculate $e^{2ik\pi/n}$ and one wouldn't be able to position the coins even if the coordinates of their centers were given.

What I want are heuristics that allow one to position the coins approximately correctly, which can be executed by someone with an ordinarily good eye and ordinarily good hands, without any special measuring devices.

Good solutions for $n\le 3$ are trivial. There is also a good solution for the special case of $n=6$, which is to arrange the six coins around a seventh. In practice it does not seem too hard to arrange four coins into a square, by first estimating the right angles and then looking to see if the resulting quadrilateral is visibly rhombic. But I would be glad to see a more methodical approach.

This is a soft question. I expect the solution to be informed by mathematics, but not purely mathematical.

Answer 1

For a large number of coins, I would apply an iterative scheme inspired by mean curvature flow. In other words, just keep nudging in coins that are jutting out, until the curvature of the whole arrangement is uniform.

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We can consider the arrangement of coins as a polygon whose vertices are the centers of the coins, and whose edges connect adjacent coins and are all of equal length. The polygon is regular if and only if all its internal angles are equal, which holds if and only if each angle is equal to (i) the average of all the angles, and/or (ii) the average of the angles at adjacent vertices. If you can compare adjacent angles by eye, or tell which vertices have the largest and smallest angles of all, then you can nudge them in the right direction.

In practice, I found that this has a couple of limitations. First, it's easy to push coins inwards because the rest of them will automatically move out of the way, but if you push a coin outward, you have to carefully reconnect the arrangement. Second, the procedure converges quite slowly. You quickly arrive at a nice smooth oval, after which moving coins one at a time is a frustratingly constrained way to round out the arrangement.

Answer 2

If you are allowed to use surplus coins, pack them together to form a large hexagonal grid in the (approximate) shape of a circle with centre $C$ and radius $r$. It is then algebraically straightforward to find which of these coins are closest to $re^{2 i k \pi / n}$.

Now remove all the other coins and use them to buy some string. Place a string from $C$ to the centres of each of the remaining coins; contract along the strings.

Here's an example with a few hundred coins and $n = 11$:

Answer 3

The main problem at hand is to compute the radius of the big circle of coins, since if we know this we can just measure out 2 adjacent sides from a central point and go from there.

The plan is to arrange the coins "like a regular n-gon", by which I mean associate the diameter, $D$, of a coin to the unit length of the n-gon.

Assuming that for N coins we can "easily calculate" the roots of unity, we can compute the distance in the complex plane between two adjacent roots of unity, $L$, and scale the whole n-gon formed by the roots of unity so that it each side has length $D$. In other words, the radius we are looking for is $\frac{D}{L}$.

Answer 4

As our eyes easily detect (absence of) mirror symmetry I propose the following: Start with a more or less circular layout; then symmetrize iteratively with respect to randomly selected axes.