
Re: scalene triangle with mirrorlike edges
Posted:
Nov 29, 2012 1:26 PM


On Thu, 29 Nov 2012 08:11:43 0500, David Bernier wrote: > On 11/29/2012 07:36 AM, David Bernier wrote: >> Say we have a triangle [...] ABC with side AB of length 5, >> BC of length 6 and AC of length 7. A (5,6, 7) triangle. ... >> Then if a point P is not on AB, its reflection >> through AB is obtained by extending AB to the >> Euclidean line l_{A,B} that contains the side AB, >> the points of the side AB. P might be on the line, >> l_{A,B}, but that doesn't matter in defining the reflection >> of P through l_{A,B}: by definition, any point on >> l_{A,B} is equal to its reflection through l_{A,B}. >> >> The notion of reflection of the point P through the >> line l_{A, B} is the familiar one from the elementary >> study and classification of isometries of the >> Euclidean plane ... ... >> If P is interior to the triangle ABC, >> we can obtain its reflections (points) through >> l_{A, B}, l_{B, C} and l_{C, A}. >> Then, we can iterate one time, and get the >> reflections of the reflections, thus 3x3 = 9 ways. >> >> Iterating once more, the reflections of the reflections >> of the reflections of P: 3*3*3 = 27 ways.
I probably misunderstand, because it seems to me a reflection of a reflection of a point P is P. As a concrete example, suppose P=(0,0) and AB is a line from A=(0,10) to B=(5,0). The perpendicular from P to AB intersects AB at (4,2). Do you say the reflection of P through AB is at (8,4), or some other location?
>> [big snip] >> I sometimes wonder if the heat kernel for the infinite >> plane domain , when reflected repeatedly and counting >> multiplicities by multiple paths [snip] >> <http://en.wikipedia.org/wiki/Heat_equation#General_description> [snip] >> <http://polymathprojects.org/2012/06/03/polymathproposalthehotspotsconjectureforacutetriangles/> [snip] >> from there my arose my interest in things like the minimal >> set S_{P} closed under the three reflection operations.
Again I probably misunderstand; this time it isn't clear to me why (or whether) heat distribution on a triangular plate doesn't act like it does on a line and steadily approach a nearlyflat temperature distribution on the plate. If temperature becomes uniform, then saying that extreme temperatures are in the most acute corners of the triangle seems misleading.
 jiw

