Timeline for Convergence of sequences formed by orthocenters, incenters, and centroids in repeated triangle constructions
Current License: CC BY-SA 4.0
44 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 11 at 6:17 | history | edited | pie | CC BY-SA 4.0 |
added 2913 characters in body
|
| S Jun 5 at 3:29 | history | bounty ended | pie | ||
| S Jun 5 at 3:29 | history | notice removed | pie | ||
| Jun 4 at 1:04 | history | edited | pie | CC BY-SA 4.0 |
added 105 characters in body
|
| Jun 1 at 23:27 | answer | added | Saúl RM | timeline score: 5 | |
| Jun 1 at 23:26 | comment | added | pie | BTW I don't know if this is useful or not but the triangle with sides $1, \ 0.7239, \ 0.365808$ the point of convergence is very close to the obtuse angle | |
| Jun 1 at 22:50 | comment | added | Saúl RM | I will leave an answer shortly explaining a bit more | |
| Jun 1 at 22:25 | comment | added | pie | @SaúlRM What is the thing special about $1,0.05730946,0.959413912$ that make the sequence diverge (How did you discover it?)? and what are these numbers ? Graphing this triangle seems to diverge but I have my doubts | |
| Jun 1 at 21:08 | comment | added | Saúl RM | Also, it seems like in most cases (likely except in some measure $0$ subset of the set of triangles, if you give some reasonable measure to the set of triangles), the ratio between the sides of the triangles will converge to something close to $(1:0.31720:0.85873)$. And then your Scenario 1 will be satisfied, and the triangles will converge to a point. But convergence to the ratios I mentioned above is very slow | |
| Jun 1 at 20:28 | comment | added | Saúl RM | It looks like for a triangle whose side lengths are some values close to $1,0.05730946,0.959413912$, the $9^{\text{th}}$ iteration is an isometric copy of the original triangle but around $5$ times bigger. This means that this triangle would satisfy your scenario $3$: "The points will completely diverge". That said, 1) I am worried about the precision of my geogebra, especially since the 4$^{\text{th}}$ iteration of the triangle I mention is almost degenerate. 2) I'm not even going to try to prove it rigorously | |
| May 31 at 2:47 | comment | added | Gerry Myerson | 19th version of this question. | |
| May 31 at 0:43 | history | edited | pie | CC BY-SA 4.0 |
added 4 characters in body
|
| May 30 at 20:38 | comment | added | pie | @DarrenLi The area is not strictly decreasing | |
| May 30 at 14:24 | comment | added | GChromodynamics | Is it possible to prove that the area is strictly decreasing? | |
| May 30 at 10:59 | history | edited | pie | CC BY-SA 4.0 |
deleted 1 character in body
|
| May 30 at 10:25 | history | edited | pie | CC BY-SA 4.0 |
deleted 15 characters in body
|
| May 29 at 15:17 | history | edited | pie | CC BY-SA 4.0 |
added 120 characters in body
|
| May 29 at 11:24 | history | edited | pie | CC BY-SA 4.0 |
added 25 characters in body
|
| May 29 at 1:48 | comment | added | pie | @crow It seems that the limit point is not any of the centers listed in ETC. | |
| May 29 at 1:34 | history | edited | pie | CC BY-SA 4.0 |
added 479 characters in body
|
| May 28 at 19:35 | history | edited | pie | CC BY-SA 4.0 |
added 144 characters in body
|
| May 28 at 14:50 | history | edited | pie | CC BY-SA 4.0 |
deleted 5 characters in body
|
| May 28 at 14:44 | history | edited | pie | CC BY-SA 4.0 |
deleted 5 characters in body
|
| May 28 at 14:34 | history | edited | pie | CC BY-SA 4.0 |
edited title
|
| S May 28 at 14:32 | history | bounty started | pie | ||
| S May 28 at 14:32 | history | notice added | pie | Draw attention | |
| May 28 at 13:49 | history | edited | pie | CC BY-SA 4.0 |
added 18 characters in body
|
| May 27 at 17:33 | comment | added | Gro-Tsen | This is reminiscent of a kind of “iterated mean” process (see this question on this subject), but in dimension 2 and invariant under Euclidean similitudes rather than in dimension 1. To prove convergence you probably need to rephrase it as a map in the modulus space of triangles, but it is unlikely that the point will have interesting properties, as already the arithmetic-quadratic mean of two real numbers is largely unstudied (as per previously linked question). | |
| May 27 at 17:30 | comment | added | Pietro Majer | @crow i’d say a bit less ${40000 \choose 3}+40000^2$; still enough, for a while (then we may iterate) | |
| May 27 at 16:56 | history | edited | pie | CC BY-SA 4.0 |
added 76 characters in body
|
| May 27 at 13:57 | comment | added | pie | @PietroMajer It is not defined, in fact if the sequence reached a degenerate triangles starting from a non-degenerate one the sequence will stop as I sated in the point 4 | |
| May 27 at 12:36 | comment | added | Pietro Majer | How is the map defined for degenerate triangles, e.g. if A, B, C are collinear? (or maybe this never happens starting from a non-degenerate triangle?) | |
| May 27 at 12:27 | history | edited | pie | CC BY-SA 4.0 |
added 4 characters in body
|
| May 27 at 12:08 | history | edited | pie | CC BY-SA 4.0 |
added 9 characters in body
|
| May 27 at 7:15 | comment | added | crow | @Daniel Asimov It appears to ocillate between approximations to two specific triangles. Again this is based on a few crude simulations so pure speculation. But the dynamics of the iteration proposed by the OP seem to be quite fascinating. | |
| May 26 at 20:05 | comment | added | Daniel Asimov | If the sequence of points converges to a point, it is possible that (considering angles alone) it converges to a specific triangle. (Or not.) | |
| May 26 at 19:09 | history | edited | pie | CC BY-SA 4.0 |
deleted 141 characters in body
|
| May 26 at 16:06 | history | edited | pie | CC BY-SA 4.0 |
added 280 characters in body
|
| May 26 at 16:04 | comment | added | pie | @crow If the triangles converge to a point this point is unique to the triangle, I wonder what extra properties this point have. The only point (which is very unique to a triangle) that I didn't include is the circumcenter but this sequence doesn't converge to it. | |
| May 26 at 15:31 | comment | added | crow | Well, if my simulations are anything to go by, the iterates of $\Phi$ tend to $(0,0,0)$ for triangles of very different shapes (i.e., the triangles converge to a point), in rather interesting ways, and so I will stick my neck out and conjecture that this is the case for any shape. | |
| May 26 at 14:11 | comment | added | crow | If we denote the sides lengths of $A_1B_1C_1$ by $a_1,b_1 ,c _1$ and those of $A_2B_2C_2$ by $a _2,b_2 ,c _2$, then this defines in a natural way a mapping $\Phi$ on three space and your question is asking for the dynamics of its iterates. It is possible to compute $\Phi$ explicitly (I have done so) but the expression is rather complicated and I have no idea about its dynamics. However, one can use it for some simulations. The few attempts that I have made look as if they are going to a point but they are hardly enough to use even as the basis for a conjecture. | |
| May 26 at 14:05 | history | edited | pie | CC BY-SA 4.0 |
deleted 2 characters in body
|
| May 26 at 14:02 | history | edited | LSpice | CC BY-SA 4.0 |
tingle -> triangle
|
| May 26 at 12:26 | history | asked | pie | CC BY-SA 4.0 |