Toothpick Construction of a Square: What is it?
A Mathematical Droodle

Explanation

Matchsticks and a rusty compass serve formidable geometric tools to play with. The applet below suggests a construction of a right angle and a square using only toothpicks. (Matchsticks would of course perform the same job as well.) The rules of the toothpick constructions allow placing toothpicks end-to-end or at certain marked points. The best the end-to-end construction can do is to produce an equilateral triangle or, provided there is a sufficient supply of the material, a triangular lattice, or grid, whose algebraic properties have been discussed elsewhere. To enable non-trivial constructions we mark a point somewhere on a stick and allow placing toothpicks at this point.

In the applet, AB is the starting toothpick, with point C marked on it. All other points are named alphabetically in the order of their construction.

Points D and E are obtained by two toothpicks joint at their endpoints with the other ends at points A and C. Let's denote equal angles CAD and CAE as α. By constructing successively three equalateral triangles AEF, AFG, and AGH, we get a straight line segment EAH. We add another equilateral triangle ADI and note the point J of intersection of toothpicks GH and DI. We show that angle BAI is right, independent of the position of point C on AB.

Indeed,

BAH = 180o - CAE   = 180o - α

Since, also BAD = α, the two toothpicks, AH and AD are symmetric in a line perpendicular to AB at A. (Both angles HAJ and DAJ equal 90^o - α.) Thus we may place a toothpick AK over point J. KAB is right. When two toothpicks, one with an end at B and the other at K, have their other ends touch, they touch at L such that BAKL is a square.

We got a funny construction of a square. But it must be taken seriously. Chapter 8 in [Martin] ends with a theorem that states that any point that can be constructed with a ruler and compass, can be constructed with a supply of toothpicks, provided a point is marked on one of them.

References

1. G. E. Martin, Geometric Constructions, Springer, 1998, Chapter 8.