By George Hart for the Museum of Mathematics

A complete graph is what mathematicians call a collection of items in which every pair is connected. If the items are spaced evenly around a circle and the connections are shown as straight lines, the lines form an attractive pattern of concentric circles.

This is a complete graph with eleven vertices. We made it at Mathcamp 2010 using plastic surveyor’s tape. There is a simple algorithm for constructing it in which people stand in a circle, and pass the roll from one to the next while counting aloud, wrapping it around their left wrist at the proper intervals.

A similar algorithm, executed by people standing in two straight lines, gives this large parabola. The construction steps for both of these figures can be seen in the additional photos here.

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## 12 thoughts on “Math Monday: Human-scale string constructions”

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http://david.rysdam.org/blog/says:“Above is the standard axes orientation. Can you prove these lines are tangent to a parabola?”

No, because they aren’t, I don’t think. The figure shown is a quarter circle. If you linearly distorted one axis, you’d have a section of an ellipse (because a circle is a special, undistorted ellipse). If NON-linearly distorted one axis in the right way, then you’d have (part of) a parabola.

I know this because just last week my officemates and I figured out the formula for the non-linear distortion.

George Hartsays:The “envelope of the lines” is a parabola. I am assuming people stand at integer points on the axes and there is a segment from (x,0) to (0,y), where x+y=c. (c is a constant, roughly half the number of people available.) For example, with c=10, there could be a segment from (5,0) to (0,5), and one from (6,0) to (0,4), and generally from (n,0) to (0, 10-n). For very large or small n, these tangent lines approach a 45 degree slope, so they become parallel, which is characteristic of a parabola.

http://david.rysdam.org/blog/says:(n,0) to (0, 10-n). For very large or small n, these tangent lines approach a 45 degree slope, so they become parallel, which is characteristic of a parabola.For large and small n, the slopes would approach 0 and 90 degrees, respectively. I.e. with c = 100 and n = 99 (large), a line from (99,0) to (0,1) would be almost horizontal while for small n = 1, a line from (1,0) to (0,99) would be almost vertical.

But even if you were right, I don’t see how that makes this figure a parabola. If you assume equal spacing on the x and y axes, then by symmetry it can’t be a parabola. Swapping the axes does nothing to the envelope, while swapping the axes of a parabola *does* do something.

http://david.rysdam.org/blog/says:That first part is me quoting you. My remarks begin with “For large and small n, the slopes would approach 0 and 90…”

Also, I always check the “Email me if someone responds” and never get emails. What’s up with that?

George Hartsays:You are thinking of the familiar y=x^2 parabola, which has its line of mirror symmetry vertical, because the mirror line is the Y axis. But if a parabola is moved, scaled, or rotated it is still a parabola. For the integer coordinates I used in explaining the string figure parabola above, the line of mirror symmetry is the line x=y of slope 1. The string parabola is then symmetric in X and Y. It is rotated 45 degrees clockwise from the y=x^2 parabola. Then, to understand how the slope of the tangent is changing as n gets very large or small, you must pick any constant c, keep it fixed and vary n.

http://david.rysdam.org/blog/says:I’m going to have to look at this closer to try to align my understanding with yours.