A Fun Geometric Proof

I learned something new today from visitor Sophos. He has an interesting geometric proof of the series 1/4 + 1/4^2 + … = 1/3. You can read it here. I think it is brilliant, and it reminds me of Tetris, an 80s arcade game where you try to score as high as possible by twisting and turning different edged geometrical objects to get a line without gaps. Think of a rotated L-shape that fits to the left of the L shape that is shaded, and think of another one where we “linearise” it and then use it as a base. This completes a square, of which the shaded L-shape makes up 1/3. The shaded L-shape can be thought of as a series of decreasing L-shapes by a factor of 1/4, so it equals to 1/4 + 1/4^2 + 1/4^3 + … . This completes the proof.

About Tsung Fei

A teacher, researcher in the bioinformatics division at the University of Malaya
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3 Responses to A Fun Geometric Proof

  1. Sophos says:

    I’ve given a long thought at your proof and I think it is not easily comprehensible. It requires some visualization to actually realize that the first “L” (The one with darkest gray) is 1/4 of the entire geometry.

    But it is interesting how you used that L-shape in this way to give you 1/3. It made me think how I can use this technique for other geometric series proof.

  2. Tsung Fei says:

    Yes, you are right, it’s not easy at first inspection.

    Following the principle of “simple is beautiful”, the best proof is still the one using the triangle in my opinion :-D.

  3. Sophos says:

    Personal preference I think. Haha. Still like my L proof lols..

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