# Mechanisms and the shape of a polygon

If we have three coloured rods which we want to join together to make a triangle, and we fasten one end of the green rod to one end of the brown rod, the other end of the green rod must lie on the green circle. Similarly if we fasten one end of the magenta rod to the other end of the brown rod, the other end of the magenta rod must lie on the magenta circle.

The third point of the triangle must be where the two circles cross (intersect). Two circles can intersect at only two points, so we can only get two triangles, and they are congruent. There is therefore only one way of joining the three rods.

If however we add a fourth, dark grey, rod to make a quadrilateral there are three ways of joining the rods since any of the grey, magenta or green rods can be opposite the brown. In what follows we have shown the green rod opposite the brown.

If we fix the brown rod, and then one end of the magenta rod to one end of it and one end of the grey rod to the other end of it, and the green rod to the other ends of the magenta and grey rods, the free ends of the magenta and grey rods, and therefore the ends of the green rod, must lie on the magenta and grey circles.

Wherever the end of the green rod fixed to the magenta rod is, the other end must be on the green circle. If the end of the magenta rod moves the green circle moves. It will still intersect the dark grey circle, but at a different point, that is, there is more than one way of joining the four rods.

In fact four or more rods pinned together at their ends make what is called a mechanism. You can try this for yourself with Lego or Meccano or any other similar construction kit.

Mechanisms like this have many uses, from controlling the valves on a steam locomotive to forming the 180° opening hinges on a kitchen cabinet door. You can see some examples of mechanisms on the Mechanism Web Site.

Of course, there are many more times when you want to join a number of rods in a way which does not form a mechanism, assembling a climbing frame for a small child for example. You can lock a quadrilateral into a particular shape by putting in a diagonal: this converts it into two triangles, and for a triangle once the length of the sides are fixed so are the angles.

There are also other ways of using diagonals or cross-members to lock rods or poles into place.

Next time you pass a building with scaffolding round it look to see if you can spot the cross-members which keep it all in shape.

All of this leads to one conclusion: even if we know all the sides of a quadrilateral, or any polygon except a triangle, we do not know anything at all about its shape.