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Math Help - senior moments, equilateral triangle, infinite geometric series and fixation

  1. #1
    Member jonah's Avatar
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    senior moments, equilateral triangle, infinite geometric series and fixation

    Ever notice that mathhelpforum equilateral triangle logo? You know how some things are there but never really paid much attention. Got fixated about that logo for a few days and then after some hard drinking, came to realize that it figured in some problem that I once ran into. Senior moment solved.

    The length of the side of an equilateral triangle is 4 inches. A 2nd equilateral triangle is inscribed by connecting the midpoints of the sides of the 1st equilateral triangle, a 3rd by connecting the midpoints of the sides of the 2nd, and so on. Find the sum of the areas of the infinitely many equilateral triangles thus formed, including the 1st.

    There are of course other problems tuat can be constructed from that logo but that's for another drinking bout.
    Speaking of fixation, been catching up on Breaking Bad episodes and ran into the following line by
    Jane Margolis: Why not a door? Sometimes you get fixated on something, and you might not even get why. You open yourself up and go with the flow, wherever the universe takes you.

    Ever wondered what that defining moment was when you became fixated with math?
    Last edited by jonah; July 14th 2014 at 02:24 AM.
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  2. #2
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    Re: senior moments, equilateral triangle, infinite geometric series and fixation

    I think it's called a Sierpinski triangle...
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  3. #3
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    Re: senior moments, equilateral triangle, infinite geometric series and fixation

    Hi Jonah,
    If your question is not rhetorical, here's a solution.

    Clearly by elementary geometry the area of your inscribed triangle is 1/4 the area of the outer triangle. The series then sums to 4/3 the area of the first triangle. In your case this is:

    $$ {4\over3}\cdot{\sqrt3\cdot 4^2\over4}={16\sqrt3\over 3} $$
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