Introduction to Hyperbolic Geometry by Amber Case | Ignite Portland 8


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Traditional geometry is something we all learned in school. Some of us like it, and the rest of us didn't. In traditional geometry, parallel lines stay parallel, and triangles are always 180 degrees. Theses are the rules, and they cannot be broken.

Enter Hyperbolic Geometry. It's not new, but it is an AWESOME field of mathematics. Basically, it breaks all the rules.

In this presentation, I demonstrate some of the history and implications of this field, a field that has inspired Escher and sent Euclid rolling in his grave. I also explain the mysterious 5th Postulate and how all of this applies to the web.

Sound complicated? Don't panic! I'll use a lot of pictures and analogies. And with all of the beer you'll all be consuming, the ideas should enter your brain smoothly and enjoyable. This won't hurt a bit.


Amber Case is a Cyborg Anthropologist studying the interaction between humans and computers and how technology affects culture. She consults with a number of large and small companies on extending online presence. In her free time, she does independent research and has exceedingly long conversations.

Case gave her first lecture on Hyperbolic Geometry to her mathematics class at age 14. She hasn't given one since, but uses the mathematics all the time.

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  • I’ll give a disclaimer here. I used to be a big math nerd when I was a teenager. I haven’t given this speech since I was 14. One major advance have affected the world of hyperbolic geometry since then! That goes at the end of the presentation.
  • You’ve seen Hyperbolic Geometry before. If you’ve seen this famous piece by Escher. He uses the Poincare projection. It’s a lot like Biodome!
  • A guy named Euclid published the best textbook ever! It was called the Elements, and contained 5 axioms that were to both set geometry in stone as well as constrain it for the next thousand years.Elements is one of the oldest extant Greek mathematical treatises. It has proven instrumental in the development of logic and modern science. Euclid’s Elements was the most successful and influential textbook ever written.
  • 1. A straight line can be drawn from any point to any point. 2. A finite straight line can be produced continuously in a straight line. 3. A circle may be described with any point as center and any distance as a radius. 4. All right angles are equal to one another. Read more: Non-Euclidean Geometry - The History Of Non-euclidean Geometry, The Founders Of Non-euclidean Geometry, Elliptic Non-euclidean Geometry http://science. jrank. org/pages/4705/Non-Euclidean-Geometry. html#ixzz0gzsVepqa
  • The 5th postulate was more like a rule. That these lines would never touch. Mathematicians and societies in the coming years would not have the flexibility of truth and assumption that Euclid had. They would be bound by Euclid’s axioms for centuries, doomed to consider the world in terms of completely flat spaces. ---Hyperbolic geometry is based on changing Euclid's parallel postulate, which is also referred to as Euclid's fifth postulate, the last of the five postulates of Euclidian Geometry. Euclid's parallel postulate may also be stated as one and only one parallel to a given line goes through a given point not on the line. Read more: Non-Euclidean Geometry - The History Of Non-Euclidean Geometry, The Founders Of Non-Euclidean Geometry, Elliptic Non-Euclidean Geometry
  • He was a very precocious youth. At 13 he knew calculus, and his father was friends with the famous mathematician gauss. In 1816 Farkas wrote to his friend Gauss asking him if he would let János live with him and take him on as a pupil so that he might receive the best possible mathematical education. So he went to school in Vienna and completed a 7 year course in 4 years. He was 21 years old when he began to do his work on hyperbolic geometry. He wrote to his father: Between 1820 and 1824, he developed his new non-Euclidean geometry coming from the solution of the problem of parallels. He was only 21 years old when he reported his finding to his father: "I have discovered things so wonderful that I was astounded . . . I have created a new, another world out of nothing. . . ”But it turned out that Gauss had understood a lot of his work beforehand, which upset him greatly. He died at age 57 of pneumonia, poor and discouraged. The picture of him is taken from a stamp issued by the Hungarian Post Office to celebrate the centenary of his death. It is not believed to be authentic and no authentic picture exists.
  • In the nineteenth century, geometry, like most academic disciplines, went through a period of growth verging on cataclysm. Gauss basically thought of the axiom-oriented world of Euclidean geometry as “imaginary”, so he set out to prove that the 5th axiom could be worked around, but Lobachevsky published first. But the name was really awkward, so Felix Klein got to name it. He came up with the name Hyperbolic geometry, else we’d be stuck with the name Lobachevskian geometry and I wouldn’t be giving the presentation!The writings of Gauss showed that he too, first considered the usual attempts at trying to prove the parallel postulatetrying to compose himself. However, a few decades later, in his unpublished reports in his correspondence with fellow mathematicians such as W. Bolyai, Olbers, Schumacker, Gerling, Tartinus, and Bessel showed that Gauss was working on the rudiments of non-Euclidian geometry, the name he attributed to his mathematics of parallels. Gauss shared his thoughts on this topic and asked them not to disclose this information but Gauss never published them. It has been proposed by historians that Gauss was concerned that these concepts were too radical for acceptance by mathematicians at that time. And if this was the case, it probably was correct since the two founders of non-Euclidean geometry, Bolyai and Lobechevsky, received very little acceptance until after their deaths. Read more: Non-Euclidean Geometry - The Founders Of Non-euclidean Geometry http://science. jrank. org/pages/4703/Non-Euclidean-Geometry-founders-non-Euclidean-geometry.
  • A triangle in a two dimensional plane always has angles that add up to 180 degrees. But if you curve the space, you can have triangles with less than 180 degrees!
  • While the measures of the internal angles in planar triangles always sum to 180°, a hyperbolic triangle has measures of angles that sum to less than 180°, and a spherical triangle has measures of angles that sum to more than 180°.This one has angles that add up to 270°! What a strange and fantastic object this is! How absurd! No wonder mathematicians were afraid to publish!
  • Reimann created the mathematics that made it possible to calculate spherical space. Riemann's idea was to introduce a collection of numbers at every point in space (i. e. , a tensor) which would describe how much it was bent or curved. Riemann found that in four spatial dimensions, one needs a collection of ten numbers at each point to describe the properties of a manifold, no matter how distorted it is. This is the famous construction central to his geometry, known now as a Riemannian metric.
  • Now armed with Hyperbolic geometry, Einsterin was able to create the general theory of relativity! What a perfect way to describe planets and the shape of space! That’s how important Hyperbolic Geometry is! BUT THAT’S NOT ALL! There’s much, much MORE!
  • For the longest time, one was unable to even consider pacing 4th dimensional manifold in 3space. As in, one could not place a 4 dimensional object in a 3 dimensional plane. And then a professor from Cornell showed up and began knitting them. The material that the yarn provided made it completely possible to create 4d manifolds in 3 space! She was like, what…you can just knit them! In 1997 Cornell University mathematician DainaTaimina finally worked out how to make a physical model of hyperbolic space that allows us to feel, and to tactilely explore, the properties of this unique geometry. The method she used was crochet. This image shows a normal hyperbolic Octagon.
  • Now we have all sorts of cool objects, like hyperbolic coral reefs and hyperbolic strange looking things, and even hyperbolic pants! There are math and fiber shows all around the world, and blogs have popped up all of the the place with patterns like, pick one number N, and for any other N stitch a bunch of material. I stumbled upon an interesting project where, through crochet, activists are physically simulating hyperbolic space and organisms that embody hyperbolic patterns, such as coral, kelp, sponges, and many other underwater organisms. Check out their fantastic creations and their work against protecting the oceans. I think this one is about 6 feet wide. Tons of people worked on these projects.
  • The patterns spread! You can even knit Hyperbolic Pants!
  • On social networks, everyone is at the center of a hyperbolic disk. The self is surrounded by family, with friends and colleagues forming the outer sections and the public. This matches the shape of a hyperbolic disk.
  • 12,000 computers around the world sent out probes to map the Internet, and this is what came out. An image very hyperbolic in nature. One of the only ways to accurately map the Internet. Following these probes, the scientists counted how many routes connected each subnetwork to the others. This widespread method revealed three layers within the Internet. At the core of the virtual jellyfish are about 100 of the most tightly connected subnetworks. These include some subnetworks you've probably heard of, such as Google. Surrounding this core is a much larger group of subnetworks that have lots of connections to each other and to the core. Finally, about one fifth of the Internet's subnetworks can communicate to the rest of the world only by sending information through the core. The scientists compare these fringe subnetworks to the tentacles of a jellyfish. Because 80 percent of subnetworks can reach each other without going through the core, the new map suggests that the Internet is less vulnerable than scientists previously thought. Attacks or outages to any one part of the Internet probably wouldn't take out the whole system.
  • CodeRed is the name of a worm (and variants) that spread with alarming rapidity on July 19, 2001. Beginning on the morning of July 19th, UTC, a random seed variant infected over 359,000 machines in a 14 hour period. Hyperbolic geometry is one of the best ways to map the spread of the infection through the Internet.This visualization depicts the number of hosts infected by CodeRed (and variants) during the 24 hour period starting on July 19, 2001, UTC.
  • What you’re seeing here is the Internet mapped in Hyperbolic space by the Cooperative Association for Internet Data Analysis. 500,000 nodes and 600,000 links mapped together in a hyperbolic sphere. On the right is an animal cell, but you’ll have to make that analogy, because we’re out of time!
  • Thanks so much everyone. If you want to hear about more ideas like this, and contribute some of your own, come to CyborgCamp Portland in October at Webtrends!
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