Given a triangle there are many well known ways to construct a point in the interior of the triangle. Two well known examples of this include the intersection of the angle bisectors and the median. Using this point one can then connect vertices to opposite edges which splits the triangle into six new triangles. We can then repeat this process with each newly formed triangle and so forth and so on with each new triangle created.
What happens to these triangles as we continue this process? Obviously they get much smaller in their area so we will look up to similarity. In other words, we will only care about the angles of the triangles. Does the smallest angle of such triangles go to 0? If not what can be said about the distribution of the smallest angle? It turns out that there can be quite disparate behavior. For example triangles in the recursion with medians tend to go to "flat" triangles with two angles close to 0, on the other hand triangles in the recursion with bisectors tend to go to some "fractal-like" distribution with essentially no minimal angles of 0 in the limiting distribution.
To help explore some of the properties of what is going on we have written some basic JAVA applets related to this dynamic problem.
Applet |
Brief description of applet |
|---|---|
| stretch | This looks at pairs of triangles simultaneously. The most important feature is to use this to see how far the daughters spread compared to the original triangles. (The available recursion methods are: bisectors, medians, Lemoine point, Gergonne point, Spieker center, and the isogonal of the Spieker center; in addition you can look at splitting into 3 or 6 daughters.) |
| deep_new | [An improved version of recurse.] Given an initial triangle this produces the daughters down at some depth (<=6) and graphically plots them. You can then dynamically drag it around to see what happens to the daughters as you change the initial triangle. (The available recursion methods are: bisectors, medians, Lemoine point, Gergonne point, Spieker center, and the isogonal of the Spieker center; in addition you can look at splitting into 3 or 6 daughters.) |
| compare | Given an initial triangle this produces the six daughters and graphically plots them. You can then dynamically drag it around to see what happens to the daughters as you change the initial triangle. In addition you can compare two different recursion methods simultaneously. (The available recursion methods are: bisectors, medians, Lemoine point, Gergonne point, and Spieker center.) |
| recurse | Given an initial triangle this produces the daughters down at some depth (<=6) and graphically plots them. You can then dynamically drag it around to see what happens to the daughters as you change the initial triangle. (The available recursion methods are: bisectors, medians, Lemoine point, Gergonne point, and Spieker center.) |
| Gergonne | The Gergonne point exhibits some rather unusual behavior. This seems to be caused by the fact that there are many triangles which are not daughters. In this applet we look at what daughters show up at some given depth (<=20). |
Information about the triangle centers used above can be found on the triangle centers page maintained by Antonio Gutierrez. A more exhaustive (or perhaps exhausting?) list of triangle centers is at the"Encyclopedia of Triangle Centers" maintained by Clark Kimberling.
The JAVA code for these applets are stretch.java, deep_new.java, triangle.java, compare.java, recurse.java, and Gergonne.java. The code is largely based on the excellent JAVA tutorial about applets created by Michael McGuffin along with a fair amount of trial and error (mostly error). Please keep in mind that these are JAVA programs written by someone who is not a JAVA programmer, so if you look at the code and say "that seems terribly inefficient" you are correct. The programs are being posted here in hopes of inspiring others to improve upon it.
These applets were created by Steve Butler and any questions regarding these applications should be directed to him.
Last updated on 22 July 2008.