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Re: scalene triangle with mirror-like edges
Posted:
Nov 29, 2012 1:26 PM
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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/polymath-proposal-the-hot-spots-conjecture-for-acute-triangles/>
[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 nearly-flat
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
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