
scalene triangle with mirrorlike edges
Posted:
Nov 29, 2012 7:36 AM


Say we have a triangle
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ABC with side AB of length 5, BC of length 6 and AC of length 7. A (5,6, 7) triangle.
Suppose AB is thought as the 1dimensional analog of a reflective, mirrorlike, surface.
Then if a point P is not on AB, it's 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 ...
Similarly, we have the lines l_{A, C} and l_{B, C}.
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.
Out to infinity through iterations of the three reflection affine transformations, we get the minimal set S_{P} of points such that: (a) P is an element of S_{P} (b) if x is in S_{P}, then the reflections of x through any of l_{A,B}, l_{A,C} and l_{B,C} are in every case elements of S_{P}.
(c) S_{P} is minimal under the orderrelation induced by the relation created by "[...] is a subset of [...]".
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Upon 'n' iterations of the three reflections, we have up to 3^n ways of composing three functions, 'n' at a time.
If a triangle doesn't or can't tesselate the plane through iterated reflections of the whole triangle (or one of its images) though its side (respectively, the through reflections of an image through sides of that same image), reflections of a generic point P through up to n compositioned reflections could produce at most 1 + 3 + 3^2 + ... + 3^n point images.
So, there's a potential for exponential growth in the number of ditinct images of P through 0 to 'n' compositioned reflections. [ meaning: I can't rule out exponential growth].
Every reflection is an affine transformation, so (x, y) > ( ax+by+c, fx+gy+h), which seem "simple".
So, I wondering what a typical minimal set S_{P} might look like, say for the (5, 6, 7) triangle and a generic point P within the interior of triangle ABC.
Intuitively, this seems related to "playing billiards" on a triangular table (studying the trajectory of a ball started from a point on the table going initially in a straight line).
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I sometimes wonder if the heat kernel for the infinite plane domain , when reflected repeatedely and counting multiplicities by multiple paths yielding the same image (case of an equilateral triangle) could produce a "theoretical formula" for the heat kernel in a homogeneous triangular region with reflective sides, which intuitively to me seems related to Neumann boundary conditons in heat equation PDE, or the heat equation for a homogeneous medium with a thermally insulated boundary (so that heat neither enters the triangle from outside, nor leaves the triangular region towards the outside), as in animated figure here: http://en.wikipedia.org/wiki/Heat_equation#General_description
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I was led to thinking about arbitrary reflections of a point P through the three sides of a triangle containing it from taking part (just a little wee bit) in the Polymath Project on the (fiendish) Hotspots Conjecture:
http://polymathprojects.org/2012/06/03/polymathproposalthehotspotsconjectureforacutetriangles/
One probabilist mentioned Brownian motion, and a variant of Brownian motion in case of Neumann boundary conditons, which I associate with heat diffusion with a thermally insulated boundary.
I don't know if I should have mentioned the standard heat kernel and its reflections with multiplicities at the infinitely many image points of a point P, where the standard heat kernel at P evolves over time. Honestly, I did not think it worth mentioning.
And from there my arose my interest in things like the minimal set S_{P} closed under the three reflection operations.
David Bernier

