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Jeff Grimes
7,752 followers -
Father of 2, car lover, and space travel dreamer.
Father of 2, car lover, and space travel dreamer.

7,752 followers
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I'm sure the math here is interesting, but my initial reaction was "Mmmmm, donut."
As the world turns, but the map stays still

The animation below was suggested by Michael Hardy, whose question on MathOverflow about the properties of a 2-to-1, almost-everywhere-conformal map from the torus to the sphere got me interested in the subject.

In a previous post, I showed a fixed mapping of two copies of the surface of the Earth onto a torus, and then rotated the resulting torus to reveal more of its surface than can be seen in any one fixed view.

But in this animation, while the torus itself stays fixed, the Earth rotates around its axis relative to the coordinate system used for the mapping.

The result makes it clear just how strange things are at some points on this map (in the previous view, the strangeness was deliberately hidden in the ocean). There are four branch points on the torus; if you walk around these points, the version of you mapped to the Earth will complete two circles around the corresponding point. You can only see two of the branch points in this view, but the positions of the other two are easy to imagine from the symmetry.

[Edited to add] Another version of this image (with a grid of longitude and latitude marked, and twice as many frames to give a smoother rotation – the file is about 17 Mb):

http://www.gregegan.net/images/tw4.gif

My previous post on the torus map:

https://plus.google.com/113086553300459368002/posts/8XXBvq9hjoK

Michael Hardy's question on MathOverflow:

http://mathoverflow.net/questions/260202/jacobis-elliptic-functions-and-plane-sections-of-a-torus

+Henry Segerman's interactive 3D model (which uses essentially the same map, but orients the Earth differently relative to the torus):

https://skfb.ly/MYpC


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I'm pretty excited about this. 

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When the Subaru dies, I think I know what I want to replace it. Not going back to snow chains ever again.

#practicalporsches
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2/19/17
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New Lupe Fiasco

#drogas

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Thrilled with Yale's decision to rename Calhoun College and honor Grace Murray Hopper.

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Giving thanks for sugar and spice and everything nice. And flour. And butter. Definitely butter.
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You might live in Silicon Valley if the guy next to you in the Costco parking lot uses a Noble as a daily driver.

#noble #dailydriven

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Bentley wants to make sure you know that when properly equipped with a roof rack and trailer rack, the Bentley Bentayga can carry just as much gear as, well, any other vehicle with a roof and tow rack.
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The scariest Halloween decoration ever.
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