Subdivision by bisectors is dense in the space of all triangles
Starting with any nondegenerate triangle we can use a well defined interior point of the triangle to subdivide it into six smaller triangles. We can repeat this process with each new triangle, and continue doing so over and over. We show that starting with any arbitrary triangle, the resulting set of triangles formed by this process contains triangles arbitrarily close (up to similarity) any given triangle when the point that we use to subdivide is the incenter. We also show that the smallest angle in a “typical” triangle after repeated subdivision for many generations does not have the smallest angle going to .
Given a triangle and an interior point of the triangle we can divide the triangle up into six smaller triangles, also called daughters, by drawing line segments (or Cevians) from the vertex through the interior point to the opposite side (see Figure 1). The process can then be repeated with each new triangle with its own corresponding interior point and again repeated over and over. When the interior point is the centroid this corresponds to barycentric subdivision.
Stakhovskii asked whether repeated barycentric subdivision for a starting triangle is dense is the space of all triangles, i.e., the space of triangles up to similarity where two triangles are -close if the maximum difference between their corresponding angles is less than . This question was answered in the affirmative .
Theorem 1 (Bárány-Beardon-Carne).
Successive subdivisions of a non-degenerate triangle using the centroid point contains triangles which approximate arbitrarily closely (up to similarity) any given triangle.
Moreover, it was shown that almost all triangles became “flat” (in the sense that the largest angle approaches for most triangles as the number of subdivisions increase). Ordin  extended their results and showed that this also holds if we choose the interior point of the triangle to be where are the vertices with and (the centroid corresponds to ).
Our main result is to show that a similar statement holds if we choose the interior point to be the incenter, which can be found by taking the intersection of the angle bisectors.
Succesive subdivisions of a non-degenerate triangle using the incenter point contains triangles which approximate arbitrarily closely (up to similarity) any given triangle.
We will see however that there is a difference in the behavior in that not almost all triangles become flat as in the centroid case.
2 Subdividing using the centroid
In this section we give a quick sketch of the ideas behind Theorem 1. The method of Bárány et al.  was to first associate triangles with points in the hyperbolic half plane, namely each triangle is associated with (up to) six points in the hyperbolic upper half plane as shown in Figure 2. This is done by placing some edge of with vertices at and and the third vertex is located at the complex coordinate with positive imaginary part. Observe that reflecting across the three circles , , and induces a natural action of on the hyperbolic half plane in which all six orientations of occur.
Now, the centroid point of a triangle with vertices at , and is the point , and so one of the corresponding daughters becomes when normalized. The argument reduces to showing that the group of automorphisms of generated by the map and the above action is dense in , in particular for any starting (i.e., any intitial triangle ) the set of all resulting points is dense in (i.e., dense in the space of triangles).
Further, using results of Furstenberg , it follows that almost all random walks formed from products of and elements of tend to infinity (in the hyperbolic plane) as the length of the product increases. This then implies that almost all of the th generation daughters have smallest angle tending to as increases. By different techniques, Robert Hough  was able to show that the largest angle approaches and moreover was able to give asymptotic bounds for the proportion of triangles with angles near .
3 Subdividing using the incenter
The important step in the proof of Theorem 1 was to find a way to associate triangles with points where the action of finding a daughter triangle was natural. The first step in proving Theorem 2 is to do the same. However, we will find it more convenient to associate each triangle with a point(s) in where the coordinates are the angles. The set of all possible triangles (including degenerate cases), denoted , is the intersection of the hyperplane with the first octant (see Figure 3). Note that also corresponds to a two dimensional equilateral triangle (see [1, 5, 10, 11] for previous applications involving ).
As noted in the introduction the incenter is found by the intersection of the angle bisectors. So the angles of the new triangles created by subdivision are linear combinations of the angles of the original triangles (hence the reason it is more convenient to work with ). In particular, if we let denote a triangle then the six new triangles are found by where
The union of the covers .
This can be seen by examining Figure 4. Alternatively, this says that every point in has a preimage in under some . For the case that with we have that is a preimage of in . Other possible arrangements for the ordering of can be handled with the remaining .
If we let denote Euclidean distance, then
To see this we can put into by ; note that this will preserve distance. Under this map we would also have that . If we now let then a calculation shows
The result now follows for and similar calculations establish it for the remaining .
We now prove Theorem 2. Let be the initial triangle we apply subdivision to. We need to show that for any triangle and there is some sequence of so that
Choose sufficiently large so that . By Observation 1 we can successively find a th generation preimage of in , which corresponds to multiplying by an appropriate at each step. Denote this preimage by
(where the are chosen according to how we construct the preimage). Repeatedly using Observation 2 we have
In the last step we used that points in are at most distance apart. This finishes the proof of Theorem 2.
The limiting distribution
In fact more can be said about the iterated subdivision of triangles using the incenter. Namely, since the maps are contracting with Lipschitz constant then it follows (see ) that there is a fixed limiting distribution on that the process converges to. Further it converges exponentially.
To get some sense of what this limiting distribution looks like we can simply start with any triangle (in our case we will use an equilateral triangle) and plot all of the th generation daughters for some in . This is done for in Figure a.
Examining Figure a we see that the daughters seem to fill in most of (agreeing with Theorem 2). However, a patient count will reveal that there are far fewer than triangles in Figure a. This is because some points have been mapped onto several times (a consequence of starting with such a symmetric triangle). So to get a better sense of the limiting distribution instead of plotting the individual triangles in it is better to look at a histogram. We will divide into a large number of small regions and then shade each region according to the number of triangles that fall into that region, the darker a region is the more triangles fall into that region. In Figure b we give the histogram for generations starting with the equilateral triangle.
Very little is known about the limiting distribution. Experimentally, it appears that the densest point on the limiting distribution (i.e., the darkest region in Figure b) corresponds to the triangle . This is likely because this triangle corresponds to an eigenvector of eigenvalue of two of the . In other words under subdivision using the incenter this triangle has two of its daughters which are similar to it (see Figure 6). No other triangle has this property, and the triangle is the only other triangle with one of its daughters similar to itself, but this triangle does not appear to play a significant role in the distribution.
If we were to draw the histogram for or and compare it to Figure b we would see almost no perceptible difference between them. This is because, as we noted above, the convergence to the limiting distribution is exponential. Or put another way, if we look at what happens when we map a triangle under applications of the in then knowing what the last few steps that we applied gives us a good handle on where we are in . This is essentially the heart of the proof of Theorem 2.
For example, the region corresponds to a triangle with vertices in at , and . In particular, looking at all the daughters for large at least of the daughters must lie in this subregion of (i.e., of the possible products of the will have as the leading term). Since points inside this subregion of must have minimum angle at least then we have that at least of the daughters in the th generation must have minimum angle at least .
Of course, by looking at larger products and looking over more prefixes we can say a lot more about what happens with the minimum angle. For example, in Figure a, b and c we have plotted all of the triangles that are the boundary of under all possible second (), third () and fourth () maps of into itself. (The number of such triangles is large so it is difficult to pick out the individual triangular regions. It also is interesting to note the similarity with the histogram in Figure b.)
We can now bound the limiting cumulative distribution function (CDF) for the smallest angle in the limiting distribution of triangles. This is done by considering all resulting images of in the th generation. Then for any angle , a lower bound for the number of triangles with minimum degree (or less) is found by counting the number of the images of which have largest minimum degree at most . Similarly, an upper bound for the number of triangles with minimum degree at most (or less) is found by counting the number of the images of which contain a triangle with minimum degree at most . Doing this for gives Figure 8. (By comparison the limiting CDF under subdivision using centroids is the constant function , showing that these two methods of subdividing are fundamentally different.)
We have seen that like the centroid, when doing subdivision using the incenter the resulting triangles are dense in the space of all triangles. However, unlike the centroid the smallest angle in a typical triangle does not tend to . This is important since certain methods can fail when the subdivision creates a large number of triangles with minimal angles going to as gets large (see [3, 13, 14]).
One interesting question is to understand the limiting distribution for repeated subdivision using the incenter, an approximation of which is shown in Figure b.
One can also consider what happens for subdivision using other interior points. For example, the Gergonne point is found by taking the inscribed circle in the triangle and connecting a vertex to the point of tangency on the opposite edge; these three lines intersect at the Gergonne point. When using the Gergonne point to subdivide it is known  that the triangles are not dense in the space of all triangles. In Figure a we have given a histogram of for the tenth generation of subdividing using the Gergonne point (notice the large white spaces where there are no triangles).
An interesting point for which little is known about what happens after repeated subdivision is the Lemoine point, which is found by taking the lines from a vertex through the median of the opposite edge and then flipping them across the angle bisectors; these three lines intersect at the Lemoine point. In Figure b we have given a histogram of for the eleventh generation of subdividing using the Lemoine point. It is currently unknown if this method of subdivision is dense in the space of all triangles and what the limiting behavior is (there is some experimental evidence that the triangles become flat, but the convergence seems to be relatively slow).
More information about the Gergonne and Lemoine points, as well as a large number of other interesting points available to investigate, can be found online (see ). More information about what happens under repeated subdivision using a central point can be found in .
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