Wednesday, September 23, 2009

envelope of the Wallace line

I was looking at  Heinrich Dorrie's 100 Great Problems of Elementary Mathematics, and problem 53, which involves a surprising hypocycloid construction, caught my attention. The problem is "to determine the envelope of the Wallace line of a triangle" and the solution is "Steiner's three-pointed hypocycloid."

The construction of Stiener's hypocycloid  lends itself well to GSP, and also shows how naming coincidences lead to strange juxtapositions.

If you google "Wallace Line" you'll find that the Internet knows not about a mathematical construction, but rather about a line that divides Indonesia into two ecological regions whose fauna are generally described as Australian on one side and Asian on the other. This Wallace line is named after Alfred Wallace, the naturalist who is known for prompting Charles Darwin to publish his Origin of Species.

Searching a little more, you will find that the line that we are concerned with is more frequently called the Simson-Wallace line - that name  might remind you of Wallace Simpson, famous for her marriage to Prince (formerly King) Edward in 1936.

Wallace Simpson, not Simson-Wallace

Names aside, the Wallace line construction extends just a little from the construction of the circumcircle, and then the Steiner hypocycloid emerges when we look at the family of all Wallace lines.

The triangle and circumcircle
0. Start with a triangle whose sides have been extended
1. Construct the perpendicular bisectors of the sides of the triangle
2. Construct the circumcenter as the point of intersection of the bisectors
3. Construct the circumcircle, C, whose center is the circumcenter and whose perimeter crosses the verticies of the original triangle.

The Simson-Wallace (or just plain Wallace) Line
4. Choose a point P on the circumcircle C
5. Form the three lines through P perpendicular to each side.
6. Form the line, w,  that passes through the points of intersection of each perpendicular with its respective side - this is the Wallace Line.

The envelope and "Steiner hypocycloid"
We want to explore the family of Wallace lines as the point P moves around the circle. In GSP we can do this using the Locus construction (I think that families of lines are generally called a "pencil" rather than "locus", and that the latter term is usually reserved for families of points, but this distinction might be antiquated).
7. Select the Wallace line, w, the point P, and the circle C
8. Construct the locus generated by w as P moves about C

A GSP file for the construction is here.
Some good descriptions of the Simson-Wallace line and this construction can be found on Cut the Knot and Wolfram Mathworld. There is an interactive activity for constructing the Simson-Wallace Line on the NCTM Illuminations site here.