If you still want to think more about the picture in my previous post, stop reading now!
Here’s a simple way to think about how the picture is made, as noted by Fergal Daly. The th circle (starting with ) has evenly spaced radial spokes, which we think of as being numbered clockwise from through , with spoke always lying on the positive x-axis. Then spoke has a blue dot at its end if and only if is relatively prime to , that is, . So, for example, the tenth circle has dots on spokes , , , and , since every other number shares a factor with . Note in particular that , and in general (this is not a special case, it’s just a natural consequence of the definition of $). This is why no circle except the first has a dot on the positive -axis. (Though as noted by Macbi, we can also think of the blue dots on circle as highlighting the generators of the cyclic group —this is almost the same as the definition in terms of , but means that possibly we should not put any dot on the first circle.)
It’s worth pointing out a few consequences of this definition in terms of :
As noted by Naren Sundar, prime-numbered circles always have blue dots: one on every spoke except the th.
The dots always have reflection symmetry across the -axis, since if is relatively prime to , then so is .
As noted by Dan Kearney, once a blue dot has appeared in a specific location on a circle, no blue dot will ever appear there again. For suppose a blue dot appeared on circle , spoke . Then if a spoke ever appears in exactly the same spot later, then it must be spoke on circle for some multiple —but then . For example, note the blue dots on spokes and on the third circle. Those same spokes show up on circle —but now they are spokes and , so they have no dots. Likewise, the same spokes show up again on circle , as spokes and .
We can use this last fact to make some cool pictures: for example, we can give a different color to each and then superimpose all the circles on top of each other. No dot will ever overlap with another dot (well, at least they wouldn’t if the dots were infinitely small). To make it more visually obvious what’s going on, I’ve also scaled circle to have radius :
This kind of reminds me of Paul Salomon’s Stars of the Mind’s Sky, except where we have deleted any star that can’t “see” the origin when it is exactly blocked by another star.
Now, Naren Sundar also made another conjecture: he observed that in addition to the reflection symmetry, the blue dots seem to have rotational symmetry whenever is divisible by some with , that is, whenever it is divisibe by the square of a prime. This turns out to be true, but unlike the other properties mentioned above, it is not at all obvious just from the definition! I hope to talk more about this in some future posts.