Times table circle · Cardioid generator
Times-Table Circle Studio
Put points around a circle, pick a times table, and join every point n to the point multiplier × n: with the 2 times table the straight lines draw a heart called a cardioid, with 3 a kidney called a nephroid, and with 9 points they draw the 1 2 4 8 7 5 vortex. Slide the multiplier or press play to watch one shape grow into the next.
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SVG and video
Clean vector lines for cutters, plotters and worksheets. The video is a 12 second loop at 1080 × 1920 that breathes from this times table to the next and back.
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The link reopens this exact figure.
How it works
Straight lines, curved shapes
1. Points and a rule
Space points evenly around a circle and number them from 0 at the top, going clockwise. Choose a multiplier. Join point 1 to point 1 × multiplier, point 2 to 2 × multiplier, and so on for every point. When an answer runs past the last point, keep counting around the circle, the way a clock goes from 12 back to 1. With 200 points and the 2 times table, 150 joins 300, which lands on point 100.
2. The curve nobody drew
No curve is drawn anywhere, only straight lines. Yet every line in the 2 times table touches the same heart shaped curve, the cardioid, so the curve appears where the lines crowd together. The 3 times table touches a nephroid with 2 points. Times k touches a curve with k − 1 points, called cusps. These curves are epicycloids, the path traced by a point on a small wheel rolling around a bigger one.
3. Why big multipliers turn strange
Only the remainder after dividing by the number of points matters, so times 201 on 200 points draws the same figure as times 1. Once the multiplier gets near the number of points, the cusps are closer together than the points themselves, and the lines overlap into moiré patterns. Times 199 on 200 points is the same as times −1, which gives parallel lines.
Where the 3 6 9 vortex comes from
Take 9 points and the 2 times table. 1 joins 2, 2 joins 4, 4 joins 8, 8 joins 16, which is 7, 7 joins 14, which is 5, and 5 joins 10, which is 1 again. 3 and 6 join each other and 9 joins 18, which is 9. Choose "By root" and the loop 1 2 4 8 7 5 glows copper while 3 and 6 glow violet. It is the same drawing as vortex math, and it is the 2 times table on a 9 hour clock.
Real vs legend
What is math and what is story
| Claim | Status |
|---|---|
| The 2 times table on a circle outlines a cardioid, and times k outlines a curve with k − 1 cusps. | Real |
| The main body of the Mandelbrot set is a cardioid. | Real |
| The bright curve of sunlight on the bottom of a cup is a nephroid, the shape of the 3 times table. | Real |
| 9 points and the 2 times table give the loop 1 2 4 8 7 5, with 3 and 6 swapping and 9 fixed. | Real |
| Only the remainder after dividing by the number of points matters. | Real |
| The vortex diagram is a map of energy, or a blueprint for free energy coils. | Legend |
| Tesla drew or described this diagram. | No source |
Questions
Times table circle FAQ
What is a times table circle?
Put points evenly around a circle and number them 0, 1, 2 and so on. Pick a multiplier, then join every point n to the point multiplier × n, counting around the circle again whenever you pass the last point. With 200 points and the 2 times table, 1 joins 2, 2 joins 4, 150 joins 300, which is point 100. The straight lines outline a curve.
Why does the 2 times table make a cardioid?
Each line runs from an angle to twice that angle, and every one of those lines touches the same heart shaped curve, called a cardioid. With enough points the lines crowd so close together that the curve they touch appears on its own. The main body of the Mandelbrot set is a cardioid too.
What shapes do other times tables make?
Times 3 makes a nephroid, a kidney shape with 2 points. Times 4 has 3 points, times 5 has 4, and in general times k has k − 1 points, called cusps. These curves are epicycloids. When the multiplier gets close to the number of points, the lines overlap into moiré patterns instead.
Why does times 201 on 200 points look like times 1?
Only the remainder after dividing by the number of points matters. 201 × n and 1 × n land on the same point, because the extra 200 × n is whole trips around the circle. Times 199 on 200 points is the same as times −1, which draws parallel lines.
How is this the 3 6 9 vortex?
Use 9 points and the 2 times table. 1 joins 2, 2 joins 4, 4 joins 8, 8 joins 16, which is 7, then 7 joins 5 and 5 joins 1. That is the loop 1 2 4 8 7 5. 3 and 6 join each other and 9 stays put. It is the same diagram as vortex math, and it is ordinary arithmetic on a 9 hour clock.
Does it work with decimals?
Yes. The end of each line is placed at its exact spot on the circle even when it falls between two points, which is why the animation morphs smoothly from one shape to the next.
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