Times Tables on a Circle

Place ten points evenly around a circle and label them 0 to 9. For each point, double its label, then draw a straight line segment to the resulting label. If the result is 10 or more, subtract 10 until it is back in the range 0 to 9.

For example, label 3 connects to 6. Label 6 doubles to 12, which wraps to 2, so it connects to label 2. The multiplier changes the destination label, not the length of the segment. A label that connects to itself produces no visible segment.

Repeat for every point. With more points, the straight segments trace an increasingly clear heart-shaped outline called a cardioid. The finite drawing contains straight segments, not a continuous curved stroke; the curve describes their limiting envelope.

Changing the multiplier to 3 uses three times each label and produces an outline called a nephroid, with two pointed cusps. The whole-number multiplier and cusp relationship is explained below; non-integer settings need their own interpretation.

With the optional overlay switched off, only the straight connections are drawn. Their apparent outline is an envelope, a curve tangent to members of the continuous family of lines. Tangency is a local relationship; it does not mean a line cannot cross another part of the curve.

An envelope is easy to make and you have probably made one. Take a piece of card, mark points along two edges, and join first-to-last, second-to-second-last, and so on with straight thread. A curve appears in the middle. No thread is curved. The curve is where the threads bunch up - where consecutive lines are so nearly parallel and so nearly touching that their crossing points trace a smooth path.

The same limiting idea applies here: where nearby members of the continuous line family intersect, those intersections approach the envelope as the starting angles approach one another. The finite drawing samples that family. Increasing the point count makes the outline easier to recognise, while the optional overlay shows the curve itself.

Drag the multiplier slowly from 2 to 3 and watch the endpoints move. With ten labelled points, multiplier 2.5 sends label 1 to position 2.5 between labels, while label 2 lands exactly on label 5. Fractional multipliers do not preserve all the numbered labels, but their endpoints still have well-defined positions around the circle.

The construction therefore extends beyond whole-number times tables. Its lines and envelope change continuously with the multiplier. For each whole-number multiplier $k\ge 2$, the continuous envelope has exactly $k-1$ cusps. That cusp-count formula is not a claim about fractional multipliers, and the finite chords are samples of the continuous family.

Work on the unit circle. For $\theta \in [0,2\pi )$ let $P(\theta )=(\cos \theta ,\sin \theta )$, and consider the family of chords

Claim. For $k>0$ with $k\ne 1$, the envelope of $\{L_{\theta }\}$ is

When $k$ is a whole number at least 2, this closed curve is an epicycloid with exactly $k-1$ cusps. At $k=1$ every chord has zero length.

Taking $k=2$ gives the cardioid, $k=3$ the nephroid.

The construction with $n$ points draws the members $L_{2\pi i/n}$ for $i=0,\dots ,n-1$. It is a finite sample of the family, and the envelope belongs to the family rather than to the sample.

Write the chord as a level set. The line through $P(\theta )$ and $P(k\theta )$ satisfies

Substituting either endpoint makes this expression zero, so it is the line through both points.

The envelope of a family $F(x,y,\theta )=0$ is the set of points satisfying both $F=0$ and $\partial F/\partial \theta =0$. Differentiating,

Solving the two linear equations for $(x,y)$ - the determinant is $(k+1)(1-\cos ((k-1)\theta ))$. It vanishes when the two chord endpoints coincide, at the smooth rim contacts; the formula below follows at the other values and extends to those contacts by continuity:

which is $E(\theta )$. ∎

The cusp count for whole $k\ge 2$. A cusp is where the traced point comes to rest - where the velocity vanishes, not where the curve happens to touch the rim. Differentiating $E$,

which is zero exactly when $e^{i\theta }=-e^{ik\theta }$, that is when $e^{i(k-1)\theta }=-1$. So $(k-1)\theta =\pi +2\pi j$, giving

• exactly $k-1$ values in one turn. ∎

A distinction worth making, because this page got it wrong. The curve also meets the rim $k-1$ times, at $(k-1)\theta =2\pi j$, and it is tempting to call those the cusps: there are the right number of them, and they are the points where something visibly happens. They are not the cusps. At $k=2$ the rim contact is at $\theta =0$, where $|E^{\prime }|=4/3$ and the curve is as smooth as it ever gets; the one real cusp is at $\theta =\pi$, which puts it at $(-\frac{1}{3},0)$, a third of the way out from the centre and nowhere near the edge. Two interleaved sets of $k-1$ points, and counting the wrong one gives the right answer for the wrong reason.

And the tangency is checked, not assumed. The test suite verifies that $E(\theta )$ lies on the corresponding chord and that $E^{\prime }(\theta )$ is parallel to it, then samples the rendered curve to guard against the page drawing a plausible shape that is not the one the algebra describes.

There is no open question here. That is unusual for this guide and it is the reason this page is worth reading. Everything above is settled, and the interesting thing is not the mathematics but what the picture does to a reader.

The frontier here is a habit, not a theorem. Every other exploration on this site trains the same reflex: is this pattern real, or an artefact of a finite search? That question has been the right one on six pages running. Here it is the wrong one - the pattern is completely real, it has a closed-form description and a three-line proof, and it still is not a fact about the two times table. It is a fact about a family of straight lines, of which the two times table picks out two hundred.

So there are two different ways a picture can mislead, and this guide has now shown both:

• A pattern that might not hold. Gilbreath's left edge, 196, every Collatz orbit - where the evidence is finite and the claim is not.
• A pattern that holds perfectly and is about something other than what you were looking at. This one.

The second is harder to catch, because nothing about it ever fails. Recamán's arcs are the same species: the drawing convention genuinely reveals structure, and the structure is partly in the convention. The only defence is the one this page is built around - being able to ask what exactly is drawn here, and getting an answer you can check.

What this laboratory computes. Up to two thousand chords, in exact index arithmetic, and the envelope from its formula. Nothing here is a search, so nothing here has a bound to report.