Peter Liepa

@peterliepa

Visual math, mainly geometry -- euclidean, hyperbolic, projective, conformal, computational.

Toronto
Joined May 2009

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  1. 7 Dec 2019

    Live version and code at . (Doesn't work well on iPhone Safari, but what else is new.) 3/3

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  2. 7 Dec 2019

    The gradient used is the gradient of the angle ACB for two fixed points A and B. That's an unusual gradient, with a funky formula, but the circular trajectories follow from the fact that inscribed angles in a circle subtended by the same chord (AB here) are equal. 2/3

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  3. 7 Dec 2019

    A gradient flow where the trajectories form orthogonal sets of coaxial circles. Some liberties taken in this visualization, because particles flow in both directions along flow lines and level sets. (short thread)

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  4. 4 Dec 2019

    of the Hesse Transfer Principle in Richter-Gebert's Perspectives on Projective Geometry. 2/2

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  5. 4 Dec 2019

    Projective geometry is endlessly fascinating. The six points of a complete quadrilateral (blue) are centrally projected to a conic. The images of opposite points define 3 lines (green) that are concurrent (as indicated by the red arrow). Based on an illustration 1/2

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  6. 18 Sep 2019

    Conformal mappings for the given pictures are normalized Jacobi sn, inverse stereoscopic, cos+i sin, and a frequency modulated version of cos+i sin. These were rendered in online Mathematica and have some artifacts. 3/3

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  7. 18 Sep 2019

    To compute these neighborhoods, find a conformal mapping that takes a real interval (or the entire real axis) to the circle, and then map the image of a grid containing the domain. (This of course can be generalized with maps to any curve) 2/3

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  8. 18 Sep 2019

    Conformal neighborhoods of circles and subcircles. I.e. square grids that follow the curve. Square size depends on arc speed of underlying parameterization. Grid on one side of the curve is Schwarz reflection of grid on the other side, which here is inversion in the circle. 1/3

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  9. 9 Aug 2019

    I feel this way about math (especially geometry): "Writing songs is a great thing. It's like a jigsaw puzzle and a kaleidoscope put together" -

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  10. 1 Aug 2019

    [2/2]: A Riemann sphere tessellated by a packing of hyperbolic tessellations. Induced by a Kleinian reflection group whose limit set is a Sierpinski circle packing. Other visualizations by Chéritat are at and

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  11. 1 Aug 2019

    [1/2]: Planet Chéritat - . Webgl implementation of "a Kleinian reflection group with a Sierpinski limit set" devised by Arnaud Chéritat ( )

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  12. 24 Jun 2019

    Oddly satisfying contrail doodler. (requires a mouse).

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  13. 13 Jun 2019

    From the archives, vintage 2010. A hyperbolic tiling mapped to the exterior of a filled Julia set. Frames from a zoom sequence -- wide shot, mid-zoom, closeup.

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  14. Retweeted

    Elegant proof that the square root of 2 is irrational, by Stanley Tennenbaum & see the interesting comments & links

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  15. Retweeted
    7 Jun 2019
    Replying to

    This is a version showing particle trails. Perhaps not exactly what you were requesting, but it gives an idea of how particles move approximately along geodesics.

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  16. 5 Jun 2019

    Hyperbolic billiards. A close-up.

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  17. 5 Jun 2019

    Hyperbolic billiards. Elastic collisions with clumping. There are really only three balls - black, green, blue.

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  18. 23 May 2019

    I'm working on a hyperbolic doodler -- an online paint program for triangle group tessellations. Here are some test images. Wabi-sabi meets the hyperbolic plane meets webgl.

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  19. 8 May 2019

    Conformal morph of rectangle to the unit disk. Via Jacobi sn(). Created using and

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  20. 8 May 2019

    Conformal mapping of rectangles of varying aspect ratio (w/h) to the unit disk. Via Jacobi sn(). Created using and

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