I was googling something completely unrelated. Then an image stopped me cold.
A circle. Dozens of lines inside it. And somehow, impossibly, the shape those lines made together looked exactly like a heart.
I stared at it for a while. Then I went back and checked what made it.
Times tables.
The ones you memorized in class. The ones everyone memorizes and forgets the moment the exam is over. The ones nobody ever told you were drawing things.
Watch This First
Before I explain anything — just watch this for 30 seconds.
That is the ×2, ×3, ×4, and ×5 times tables. Drawing.
Take a moment with that. Because what you just watched isn’t a trick, isn’t a coincidence, and isn’t some advanced university mathematics.
It’s the same times tables you memorized in school.
So What Is Actually Happening?
Here’s the idea — and it’s simpler than it looks.
Take a circle. Place 50 evenly spaced points around it, like the dots on a clock face, but more of them. Number them 0 to 49.
Now apply the ×2 times table. Take each point’s number and multiply it by 2. Then draw a straight line connecting that point to the result.
Point 1 connects to point 2. Point 2 connects to point 4. Point 3 connects to point 6. Point 7 connects to point 14. Point 24 connects to point 48.
That’s it. That’s the whole rule.
One thing to know: the points wrap around. We only have points 0 to 49 — that’s 50 points total. So if multiplying gives you a number bigger than 49, you subtract 50 and use that instead.
Point 25 × 2 = 50. Subtract 50. Connect to point 0. Point 26 × 2 = 52. Subtract 50. Connect to point 2. Point 30 × 2 = 60. Subtract 50. Connect to point 10.
It’s the same idea as a clock — once you pass 12, you start again from 1. Here, once you pass 49, you start again from 0.
Do that for all 50 points and something extraordinary happens. The lines don’t scatter randomly. They start to curve. They lean toward each other in the same direction. And together — without anyone drawing it directly — they outline a shape.
That shape is called a cardioid. From the Greek word kardia, which means heart.
Your times table just drew a heart.
Why a Heart Though?
The honest answer is: the shape sneaks up on you through repetition.
Imagine you’re drawing the first few lines. Point 1 to point 2 — short line, top of the circle. Point 2 to point 4 — slightly longer, still near the top. Point 10 to point 20 — longer, crossing the middle. As you keep going, the lines start piling up near one side of the circle. Not because you aimed them there. Just because of where the ×2 rule sends them.
It’s the same thing that happens when you drop a handful of matchsticks on a table. No single matchstick is trying to point anywhere — but after enough of them, a pattern appears. The cardioid is that pattern for the ×2 rule.
Why Does Each Times Table Make a Different Shape?
This is the part that really got me. It’s not random. There’s a rule underneath the rule.
The ×2 table makes a cardioid — one loop. The ×3 table makes a nephroid — two loops. The ×4 makes three loops. The ×5 makes four.
The pattern is: the ×n table always makes n−1 loops.
But why?
Think about what the ×2 rule actually does to the circle. It takes every point and sends it to a location that is exactly twice as far around the circle. Points near the top stay near the top. Points near the bottom get sent to the opposite side. The whole circle gets “stretched” by a factor of 2, but because it wraps around, some of that stretch folds back on itself — and the fold is exactly where the cusp of the heart shape sits.
The ×3 rule stretches by a factor of 3. It wraps around twice before it closes. That creates two folds, two cusps, two loops — a nephroid.
The ×4 rule stretches by 4, wraps three times, makes three cusps.
Each times table is essentially a different way of folding the circle back on itself. The number of times it folds is exactly n−1. And each fold produces one loop in the final shape.
Here’s another way to see it. Pick any point on the cardioid — the ×2 shape. Draw a tangent line at that point (a line that just touches the curve without crossing it). That tangent line is exactly one of the chords you drew. Every single line in your diagram is tangent to the cardioid. The shape isn’t something you drew — it’s the boundary that all your lines agree on.
Mathematicians call this an envelope — a curve defined not by drawing it directly, but by the family of lines that surround it. The cardioid is the envelope of the ×2 chord family. The nephroid is the envelope of the ×3 family. And so on, forever.
You can go as far as you want. The ×10 table makes 9 loops. The ×100 table makes 99. The rule never breaks.
The Bigger Picture
Here’s what actually got me.
We spend years treating times tables like the most boring part of school. Pure memorization — no meaning, just repetition until it sticks. But the cardioid was hiding in there the whole time, waiting for someone to draw it out.
And it’s not just art.
The cardioid turns up in places you wouldn’t expect. That bright curved patch of light inside a coffee cup when the sun hits it? Mathematicians call it a caustic — a shape formed by light bouncing off curved walls, and a close cousin of the cardioid. Microphone designers use a mathematically perfect cardioid-shaped pickup pattern so the mic hears sound from in front and ignores noise from behind — that one is exact to the geometry. In F1, the study of how these curves describe the curl of air coming off a wing is fundamental to how engineers model the turbulence between cars. And in cardiac medicine, the shape of the pressure wave inside a healthy human heart ventricle traces — yes — a cardioid. The word was never just a coincidence.
None of those people were thinking about times tables. But the same family of shapes runs underneath all of it.
Simple rules. Extraordinary outcomes. That’s not a coincidence. That’s what mathematics actually is.
Try It Yourself
You don’t need software. You need a printed circle, a pencil, and about 20 minutes.
Here’s exactly what the process looks like — step by step:
Draw a circle. Mark 36 evenly spaced points around it — one every 10 degrees works perfectly with a standard protractor. Number them 0 to 35.
Pick the ×2 table. For each point, multiply its number by 2. If the result is more than 35, subtract 36 — that’s the wrapping. Point 20 × 2 = 40, subtract 36, connect to point 4. Point 19 × 2 = 38, subtract 36, connect to point 2.
Draw a line from each original point to its result. Do that for all 36 points.
Step back and look at what you drew.
Then try the ×3 table on a fresh circle. Count the loops.
I still think about the moment I found this. Not because of the mathematics — though the mathematics is beautiful. But because of what it means that it was there the whole time.
Every times table you ever wrote down in a notebook was quietly drawing something. Nobody told you. The curriculum moved on. The shapes just waited.
That’s what this blog is about. The things that were always there, just waiting for someone to look.
— Keep noticing.
Tagged: mathematics · art · patterns