Laurens Gunnarsen

@MathPrinceps

Mathematical physicist and mentor to mathematically talented youth. Talent is that which bridges the gap between what can be taught and what must be learned.

Joined June 2012

Media

  1. Jan 11
    Replying to
  2. No woman aiming to compel the attention of the male should contort herself into a simpering push-over, nor should she suppose that a carefully cultivated saccharine neoteny will do the trick. She should rather apprentice herself to this incomparable master.

  3. 9 Oct 2019
    Replying to

    It's possible to do better, though -- by which I mean, to make the correspondence clearer between the terms of the series and the areas in the figure. For example, here's how I'd say it should go for r = 2/3. Note the ease of counting the thirds, the ninths, the twenty-sevenths.

  4. 8 Oct 2019
    Replying to

    Here's a starting point, showing how things go for r = 1/2:

  5. Replying to
  6. Replying to

    Perhaps by juxtaposing the previous picture with this one I can show what I meant (for surely it is clear that the pink equilateral triangles have side lengths equal to one-half the unit length.)

  7. Replying to

    I think it might be simpler to focus on the pink equilateral triangles in this picture, which evidently may be shifted slightly so as to stellate the small central square whose area is the fraction required.

  8. Here's the best way to picture all the (interesting) Pythagorean triples. Also great fun to try to figure out how to extend this indefinitely.

  9. "You do know how to whistle, don't you, Steve? You just put your lips together, and blow."

  10. 6 May 2019

    Here are some pictures that may be worth sharing with those who are first encountering geometric series. The last one, in particular, is likely, I think, to provoke some interesting conversations.

  11. Replying to

    This, I think, may be the most efficient means of solution. But the real question is: where does it come from?

  12. I wonder what your son will think of this picture:

  13. Replying to
  14. Replying to

    It is perhaps worth remarking, as Edward Gibbon once did, that things may go spectacularly awry, even when every conceivable pain has been taken to educate and enlighten one's offspring. Here is Gibbon on Aurelius and his irredeemably vicious son Commodus.

  15. Right: this picture should do the trick. The green triangle is now naturally divided into pieces whose area can be readily expressed in terms of the area of the smallest triangles in the picture. Since 24 of these make up the blue hexagon, this area is 1. Thus, 13 = 3*(6/2) + 4.

  16. Sorry if the link accompanying my previous remark appeared suspicious; this is what it points to:

  17. Replying to

    Perhaps not so much happy as enthralled? I wonder if that counts.

  18. A postscript, for those who feel I made the above picture rather more symmetrical than I should have: nothing essential changes in the more general case, as this picture shows.

  19. Replying to

    Again, I maintain: always symmetrize! I think this picture makes the whole thing clear.

  20. 5 Feb 2019

    Pity there's no hint he'll mention what Abel Prize winner Jacques Tits seems to consider the most wonderful thing about 6, namely, that S_6 alone among all the permutation groups has a non-trivial outer automorphism -- which can be visualized using this wonderful figure:

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