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scalene triangle with mirror-like edges
Posted:
Nov 29, 2012 7:36 AM
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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 1-dimensional analog
of a reflective, mirror-like, 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 order-relation
induced by the relation created by
"[...] is a subset of [...]".
===
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).
===
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
===
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/polymath-proposal-the-hot-spots-conjecture-for-acute-triangles/
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
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