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MathPrinceps's profile
Laurens Gunnarsen
Laurens Gunnarsen
Laurens Gunnarsen
@MathPrinceps

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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

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    1. Ian Agol‏ @agolian 8 Apr 2019
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      The Indiana University Bloomington mathematics department has a rendition of the Seven Circle Theorem on their mugs. While visiting, I came up with a synthetic geometric proof. I'll give some hints in the replies. https://en.wikipedia.org/wiki/Seven_circles_theorem …https://www.youtube.com/watch?v=kPJURKUg13c …

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    2. Ian Agol‏ @agolian 8 Apr 2019
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      The first hint: three lines connecting the tangency points of opposite circles can be thought of as geodesics in the Klein model of the hyperbolic plane. https://en.wikipedia.org/wiki/Beltrami%E2%80%93Klein_model …

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    3. Ian Agol‏ @agolian 8 Apr 2019
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      Whereas the 6 tangent circles may be thought of as horocycles in the Poincaré disc model of the hyperbolic plane. https://en.wikipedia.org/wiki/Poincar%C3%A9_disk_model … https://en.wikipedia.org/wiki/Horocycle 

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    4. Ian Agol‏ @agolian 8 Apr 2019
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      Three geodesic lines will intersect in the Klein model if and only if three geodesic lines intersect in the Poincaré disc model. Hence we may replace the three lines with three circles orthogonal to the boundary of the outer circle, and try to show that these intersect.

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    5. Ian Agol‏ @agolian 8 Apr 2019
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      Now the configuration is invariant under linear fractional transformations preserving the Poincaré disc, i.e. isometries of the hyperbolic plane. So we may perform such a transformation sending two of the geodesics to intersect at the origin, in which case they are now straight.

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    6. Ian Agol‏ @agolian 8 Apr 2019
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      For the cycle of 6 horocycles, we may shrink every other one by hyperbolic distance r, and increase the other three by r, while maintaining the cycle of tangencies. Note that opposite horocycles will maintain their distance under this transformation.

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    7. Ian Agol‏ @agolian 8 Apr 2019
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      Recall that two of the three geodesics are straight lines meeting at the origin. Resize the horocycles so that one opposite pair has the same size. By symmetry (a π rotation in the origin), the other pair will as well.

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    8. Ian Agol‏ @agolian 8 Apr 2019
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      Now one also sees by symmetry that third pair of horocycles has equal size and are related by rotation through π in the origin. Hence the Poincaré geodesic connecting the tangency points is a line through the origin, confirming the theorem.

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      Laurens Gunnarsen‏ @MathPrinceps 8 Apr 2019
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      Replying to @agolian

      This is indeed a lovely argument. Thank you very much for sharing it here. And thank you, too, for making the theorem itself better known in this way; this was my introduction to it.

      11:24 PM - 8 Apr 2019
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