The 3× table turns into a nephroid
300 points on a circle. Join each point to three times itself and a curve appears.
i → 3·i (mod 300)Put 300 points evenly around a circle and number them 0 to 299. From every point i, draw a straight line to point 3·i, wrapping past 299 back to 0. Point 10 joins point 30, point 150 joins point 450 − 300 = 150 (itself), point 200 joins point 600 − 600 = 0.
A curve made of straight lines
None of the lines is curved. But they crowd together along a smooth outline with two inward points, shown in gold in the video. That outline is a nephroid, from the Greek for kidney.
Each line touches the nephroid at exactly one point. The curve is the envelope of the lines: the edge they all lean against.
Rolling circles
A nephroid is also the path of a point on a small wheel rolling around the outside of a fixed wheel twice its size. Two trips of the small wheel's rim per lap give two cusps.
The rule ×2 gives a cardioid with one cusp; ×4 gives three cusps. In general the ×m table gives m − 1 cusps.
In a coffee cup
The bright curve you see on the surface of coffee when light shines in from the side is also a nephroid. It forms the same way: many straight light rays reflected off the round wall, all touching one curve.
