February 27 2001

Email: It seems I can receive email via andrea@spicynoodles.com at the moment, though I’m not sure it will stay this way. Apart from the problem in the US it seems that the German Telekom is having problems at the moment as well…

By the way, if anyone has got a large digital image of Circle Limit IV by M. C. Escher (at least 1,000 x 1,000 pixels), please let me know!

circle limit 3: Hyperbolic Geometry

Hyperbolic Geometry is a strange world. There is more than one parallel to a straight line through a certain point, and the three angles of a triangle add up to less than 180°. There’s no way to build a real model for this geometry in our Euklidian world, but there is a model: The whole hyperbolic world fits into a disk.

To me, hyperbolic geometry is one of the most interesting branches of mathematics. One of the most beautiful works of art visualizing the hyperbolic plane is Circle Limit III by Maurits Cornelis Escher.

Science News: Visions of Infinity – Tiling a hyperbolic floor inspires both mathematics and art. By Ivars Peterson.

Helaman Ferguson, sculptor and mathematician:

“I chose to do creative math in a liberal arts college rather than an engineering school. Our society tends to compartmentalize people and professions, maybe with good reasons. Overcoming this compartmentalization has been a continuing battle for me.I refuse to be diminished by being described as just a mathematician, by being described as just a sculptor–I persist in both. Fortunately for me our society is diverse enough to permit both. […]

Mathematics, its ideas, symbols, and equations are an essential part of my personal design language. Much of my sculptural body of work celebrates the remarkable achievements of mathematics as an abstract art form–a human activity spanning thousands of years. “

Look at his hyperbolic quilt!

Douglas Dunham: Transformation of Hyperbolic Escher Patterns. The author also presents two colourful hyperbolic sculptures.

Spherical Geometry

is sort of the “opposite” of hyperbolic geometry. No parallels exist in the spherical world, and the angles of a triangle add up to more than 180°. I just received email from Dave Fisher (Thanks!), who wrote a poem about a spherical triangle: My Favourite Triangle.

As you may have gathered from the poem, spherical geometry is what’s happening on a sphere. Try to draw two straight lines on a sphere and have them not meeting anywhere! See, no parallels.

Note: straight lines on spheres are great circle paths. (The great circle path is the intersection of a spherical surface with a plane passing through the two points on the surface and the center of the sphere.)


Hal, it’s great to read that the CPAP is working for you. Although I guess it’s really pretty strange to have air blown up your nose all the time. But if it helps… fine!

4 thoughts on “February 27 2001

  1. Max Smith


    If you still need a large Circle Limit IV, I may be able to help you.

    I have a postscript version that I can resize and export to any file format you need. However, it is a black and white image, without the detail of the image you posted.

    Let me know if you want it.


  2. Andrea Frick


    Sorry my reply is so late… I was out of town and offline for a couple of days. Thanks for your offer to send me “Circle Limit IV” by Escher. If it looks like the one you pointed to, it’s not really what I need. I need a picture of the original work by Escher, if at all possible. Postscript would be fine, though…

    I’m going to search on.

    But thanks anyway!


  3. Oliver


    we did a cool little QuickTime VR a while back: We took an image by M.C.E., scanned it in and let the QuickTime VR tool do its magic. Result: a perfect VR. Only very minor adjustments had to be made to correct for the scanning misalignments and I think we had to duplicate the image to make it broader.

    Anyways, here it is: http://static.userland.com/gems/farsideofmymind/Escher.mov

    Hope you like it.




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