10
$\begingroup$

This problem is a restatement of this question, first announced in MathStackExchange.

We start with a triangle $T$ in the Euclidean plane and we define $A_n$ as the set of angles of the $6^n$ triangles obtained from the $n$-th barycentric subdivision of $T$. (Angles are geometric angles, i.e., the angles of $T$ and its children lie in $[0, \pi]$).

What we can say about set $A=\bigcup_{n=0}^{\infty}A_n$?

Is there a specific predefined triangle for which $A$ is dense on $(0,\pi)$?

If no, does there exist another simple alternative to this iterative procedure, i.e., barycentric subdivision, using other concurrent lines in triangles, which achieves our goal?

Improve this question
$\endgroup$
4
  • 2
    $\begingroup$ The sentence starting with "In $(n+1)$-th step" sounds obscure, even grammatically. Is $A_n$ a set of angles or a set of triangles? What is the meaning of "$\lim_n A_n$"? What is the meaning of "$A$ is dense on $(0,\pi)$"? What is "our goal"? Thank you $\endgroup$
    – Pietro Majer
    Commented May 29, 2018 at 19:36
  • 2
    $\begingroup$ Is perhaps $A_n$ the set of $6^n$ triangles of the $n$-th barycentric subdivision of $T$? $\endgroup$
    – Pietro Majer
    Commented May 29, 2018 at 19:37
  • 1
    $\begingroup$ @PietroMajer $A_n$ is set of angles of all triangles till $n$-th step (you're true: $6^n$ triangles in $n$-th step +$6^{n-1}$ in $n-1$-th step+...+1=$\sum_{i=0}^{n} 6^i$ tiangles) ,leads $A_n$ has $3×\sum_{i=0}^{n} 6^i$ angles. Our goal is to find a dense $A$ on $(0,\pi)$ $\endgroup$
    – Solar Galaxy
    Commented May 29, 2018 at 20:18
  • 1
    $\begingroup$ OK, so $A=\cup_n A_n$, right? $\endgroup$
    – Pietro Majer
    Commented May 29, 2018 at 20:51

1 Answer 1

Reset to default
15
$\begingroup$

The answer to the second question is yes, for any non-flat triangle $T$, the set of angles $A$ is dense in $(0, \pi)$. This follows from a stronger result of Barany et al. [Theorem 1, 1]:

Theorem. Successive barycentric subdivisions of a non-flat triangle contain triangles which, to within a similarity, approximate arbitrarily closely any given triangle.

Here is a nice divulgative article on the dynamics of iterated barycentric subdivisions. It also has a list of further readings on the topic. (The joy of barycentric subdivision, by Bill Casselman)-

[1] I. Barany, A. Beardon and T. Carne, "Barycentric subdivision of triangles and semigroups of Möbius maps", 1996. MR1401715.

Improve this answer
$\endgroup$
0

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .