Times table circle · Cardioid generator
Times-Table Circle Studio
Space points around a circle, pick a times table, and join every point n to the point at multiplier × n. The 2 table draws a heart called a cardioid, 3 a kidney called a nephroid, and 9 points 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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How it works
Straight lines, curved shapes
1. Points and a rule
Number the positions from 0 at the top, clockwise, and join position n to position factor × n. When the answer runs past the last one, keep counting around the circle like a clock. With 200 positions and the factor 2, position 150 joins 300, which lands on 100.
2. The curve nobody drew
Only straight lines are drawn, yet every line of the ×2 rule touches one heart shaped curve, the cardioid. The ×3 rule touches a nephroid with 2 cusps, and ×k touches an epicycloid with k − 1 cusps, the path of a point on a small wheel rolling around a bigger one.
3. Why big multipliers turn strange
Only the remainder after dividing by the size of the circle matters, so ×201 on 200 marks is ×1 and ×199 is ×−1, a set of parallel lines. Near the top the cusps crowd closer than the marks, and the lines overlap into moiré.
Where the 3 6 9 vortex comes from
On 9 positions the ×2 rule sends 1 to 2, 2 to 4, 4 to 8, 8 to 7, 7 to 5 and 5 back to 1. 3 and 6 swap and 9 stays put. Choose "By root" and the loop glows copper while 3 and 6 glow violet. It is the same drawing as vortex math, ordinary arithmetic on a 9 hour clock.
Computed for this page
How many loops does each rule draw
Follow one point along its chord, then along the chord of the point it lands on, and keep going until you are back at the start. That round trip is a closed loop, and a point that lands on itself is a loop of one. The drawing hides how many separate loops it holds, so we counted them in Python for every factor from 2 to 30 against every size from 2 to 30.
The cardioid row is not one thing
On 29 positions the ×2 rule runs through 28 of them in a single grand loop and leaves 0 alone. On 31 it breaks into six loops of 5 plus the fixed point, even though both counts are prime and the two cardioids look alike. The multiplicative order of 2 decides it, which is 28 on 29 and only 5 on 31.
Dark cells, bright cells
The ×2 rule on 16 marks has exactly 1 loop. Every chain of chords runs into 0 and stops, so the picture is drawn but nothing closes. Any power of two in that column does the same. The brightest diagonal is the multiplier one above the dot count, where each dot joins itself, no chord is drawn, and the loop count equals the number of marks.
Bigger figures follow the same rules. On 200 points, ×2 gives 3 loops, ×3 gives 21 and ×34 gives 5. The mirror rule ×199 gives 101, which is 99 loops of two plus 0 and 100 standing still. On 1000 marks ×2 gives 4.
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?
Space marks evenly around a circle and number them 0, 1, 2 and so on. Pick a factor, then join every mark n to the mark at factor × n, counting around the circle again whenever you pass the last one. With 200 marks and the factor 2, 1 joins 2, 2 joins 4 and 150 joins 300, which is mark 100. The straight lines outline a curve.
Why does the factor 2 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 appears on its own. The main body of the Mandelbrot set is a cardioid too.
What shapes do other factors make?
The factor 3 makes a nephroid, a kidney shape with 2 cusps. Factor 4 has 3 cusps, 5 has 4, and in general ×k has k − 1. These curves are epicycloids. When the factor gets close to the total, the lines overlap into moiré patterns instead.
Why does ×201 on 200 marks look like ×1?
Only the remainder after dividing by the total matters. 201 × n and 1 × n land on the same spot, because the extra 200 × n is whole trips around the circle. ×199 on 200 marks is the same as ×−1, which draws parallel lines.
How is this the 3 6 9 vortex?
Use 9 points and the factor 2. 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, ordinary arithmetic on a 9 hour clock.
How many closed loops does a times table circle have?
Follow one point from chord to chord until it comes back to itself and you have traced a closed loop. On 9 points the ×2 rule gives three, which are 1 2 4 8 7 5, then 3 6, then the fixed point 9. On 200 points, ×2 gives 3 loops, ×3 gives 21 and ×199 gives 101. The heat table on this page lists every factor and point count up to 30.
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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Sources
- Cardioid: en.wikipedia.org
- Wolfram MathWorld, Cardioid: mathworld.wolfram.com
- Modular arithmetic: en.wikipedia.org
- Multiplicative order: en.wikipedia.org
- Wolfram Demonstrations, Modular Multiplication on a Circle: demonstrations.wolfram.com
- Loop counts: computed for this page with
times-table_data.py, brute force over every factor and point count from 2 to 30