Closed reflective paths in a polygon.

It is permissible for two nonadjacent segments in a closed reflective path to cross.

The perimeter of a polygon is a closed path. A square inscribed in a circle is a closed reflective path in the circle. We are interested in which figures are sure to have closed reflective paths and (when possible) describing them. For the general case this is a difficult problem. However there are cases which are readlily, completely, understood and others in which interesting, partial results can be obtained.

Here we investigate the closed reflective paths in (convex) polygons. We will concern ourselves primarily with triangles, describing a quite general technique for deducing interesting results from simple euclidean geometry and Maple's ability to solve systems of linear equations. We will illustrate the general method by applying it to the following question:

Question : Which triangles have closed reflective paths?

Easier Question: Which triangles have triangular closed, reflective paths?

The easier question is about a general triangle and whether it has closed trangular reflective path. We draw a general triangle and a prospective closed triangular reflective path.

rpathq1

We have six unknown quantities: A,B,C,a,b,c and five equations implied by the five labeled triangles. There are also three inequalities imposed by the situation.

> restart;

> eq0:=A+B+C=Pi;
eq1:=A+b+c=Pi;
eq2:=a+B+c=Pi;
eq3:=a+b+C=Pi;
eq4:=Pi-2*b + Pi-2*c + Pi-2*a = Pi;
ineq5:=a<Pi/2;
ineq6:=b<Pi/2;
ineq7:=c<Pi/2;

> sln1:=solve({eq0,eq1,eq2,eq3,eq4,ineq5,ineq6,ineq7},{a,b,c,A,B,C});

>

>

What do we conclude from this computation? Notice that and . Hence it follows that . and we conclude that the three corner triangles are all similar to the original triangle. Also, we see that all three angles of the orginal triangle must be acute. This proves the following theorem.

Theorem : Suppose a has a triangular closed reflective path. Then each angle of is acute and each angle of the path is the supplement of the angle of the triangle which is opposite it. Furthermore, each of the three corner triangles cut off by the path is similar to

Two natural questions occur here.

Question : Does every triangle with all angles acute have a closed triangular path?

Question : Can a triangle have more than one closed triangular path?

Exercise : Show that every non-obtuse, isosceles triangle has a closed reflective path and describe a simple way to construct it (even with straight-edge and compass).

Exercise : Create a Maple word which inputs the three vertices of a triangle and draws the triangle with its unique closed reflective path if it has one.

Look in the next worksheet if you want to see the answers to some of these questions and exercises.

QUESTION : Can triangles have a closed quadrilateral reflective paths?

One investigates this question just as we did for triangular paths. There are more sub-figures, hence more equations.

rpathq2

Exercise: Does the equilateral triangle have a closed quadrilateral reflective path?

Exercise: Does the 3-4-5 right triangle have a closed quadrilateral reflective path?

Exercise: Is there a triangle with a pentagonal closed reflective path of length 5?

DEFINITION : A figure F has a self-tracing reflective path if it contains a polygonal arc made of segments , ... such that (1) each has its end points on the boundary of F, (2) for each i from 1 to n-1, and have a common vertex which is their only point of intersection and (3) for each i from 1 to n-1 the line extending is the reflection of the line extending through the tangent to F at their common vertex, and (4) and are normal to the boundary of F.

The idea of a self-tracing reflective path is one that is reflected back upon itself. Note that a self-tracing reflective path in a triangle

Exercise: Find all triangles that have self-tracing reflective paths of length 2. Within those triangles find all self-tracing reflective paths.

rpathq3

Hint: The rectangle gives the equation

Exercise: Show that a triangle cannot have a triangular self-tracing reflective path.