Multiplication Circles
A multiplication circle puts numbers around a rim and draws what happens when you multiply by one factor. Products mode joins each point k to (k × m) mod n. Times table mode walks the multiples in order, the drawing from a Waldorf lesson: a pentagon for 2× on ten points, a star for 4×. With a few hundred points the 2× products settle against a cardioid. Join each point to its multiple, or walk one times table around the circle. Press play to move the factor. Click a point to read the product.
How it works
- Open this page. The 2× lines on 200 points are already a cardioid.
- Choose 2× table for ten numbered points, or 4× star for the five-pointed star.
- Press play to move the factor, click a point to read its product, then save a PNG.
Each point times the factor
Number the points 0, 1, 2, and so on, clockwise from the top. In Products mode a straight line runs from k to the remainder of k × m after division by the number of points. On a circle of 10, 7 × 2 = 14, and 14 leaves remainder 4, so the line runs from 7 to 4. Adding the point count to the factor draws the same lines: 12× on 10 points is the same figure as 2×.
A times table, or a cardioid
Times table mode walks the multiples. Start at 0 and keep adding the factor. On 10 points, 2× is 0–2–4–6–8–0, a pentagon, and 4× is 0–4–8–2–6–0, a five-pointed star. All cycles draws the other loops, such as the odd pentagon of the 2× table. Products mode with 200 points is a different picture: the 2× lines crowd against a cardioid, one cusp, and the 3× lines crowd against a nephroid, two cusps. An integer factor m hugs a curve with m − 1 cusps. The lines stay straight. The curve is the edge they will not cross.
Paper, chalk, and a board of nails
Pick the surface the figure is usually made on. Paper uses a colored pencil. Chalk uses a green board. Wood is a ring of nails with yarn between them. Ten or twelve points are for reading a table, and the numbers can sit outside the rim. A hundred points or more are for the curve. Save a PNG, or open the canvas in the PictorX editor.
Frequently asked questions
What is a multiplication circle?
Points numbered around a circle, and a drawing of multiplication by one factor. Products joins each point k to (k × m) mod n. Times table joins the multiples in order.
Why does times 2 look like a heart?
With a few hundred points, the straight 2× products sit against a cardioid, a curve with one cusp. Times 3 sits against a nephroid, which has two cusps.
What is the difference between Products and Times table?
Products draws every line from a point to its multiple. Times table draws one walk: 0, then the factor, then twice the factor, and so on. On 10 points that walk is a pentagon for 2× and a star for 4×.
What does mod n mean?
The remainder after division by the number of points. On 10 points, 14 mod 10 is 4, so 7 × 2 ends at point 4.
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