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That was great, the only thing missing was why. I am sure there are some interesting applications of the principles they explored and I would have loved to have

That was great, the only thing missing was why. I am sure there are some interesting applications of the principles they explored and I would have loved to have heard a few at the end. Feel free to fill me in.

trafficlight 16y ago
Exactly. Was there a practical reason for the restrictions on tight bends and creases? Or was that just to make it interesting? Edit: Please don't downvote this, I legitimately want to know why.
- Devilboy 16y ago
  If you avoid sharp edges you can take differentials - if not you end up with discontinuities which makes is less useful mathematically.
- andreyf 16y ago
  Was there a practical reason for the restrictions on tight bends and creases? sigh You'll understand when you're building biological spaceships: you see, the inversion of n-spheres under those conditions is the simplest workable model for how evolutionary phenomena interact with each other. The inversion of an n-sphere of dimension 3 is representative of the evolution in just one layer of a system. After you discover the generalization of inversion across n-space in 2072, it will just take one ambitious biologists to apply it to genetic data, and generalize to most of the evolutionary phenomena around you: memes and cultures. By 2103, Human Directive 72 will be achieved, and all children under the age of four will be able to conceive of the basic geometries of infinite-dimensional topological models.
- leif 16y ago
  When you avoid tight bends and creases (singularities---places where the derivative vanishes), you have a nice structure on which you can do useful things like take derivatives, measure lengths and angles, find "straight" lines, and such. In short, the structure behaves, as long as you're small enough, just like the space we live in and have studied for centuries. Now you can extend this notion from a property of one surface to a property of a continuous family of surfaces, and then look at the beginning and the end (imagine it varying with respect to time, just like in the video). The niceness property above, if maintained throughout, gives you more things to say about what happens to substructures. For example, you can say something like "if I start with a triangle inside this surface, and morph it into another surface keeping it nice and smooth the whole time, I'll still get a triangle, and it won't do {big class of bad things}".
mikeknoop 16y ago
There is a large branch of mathematics which this is tied to called Knot Theory, I'll bet you could find some practical applications associated to it. (http://en.wikipedia.org/wiki/Knot_theory http://en.wikipedia.org/wiki/Knot_theory)
nitrogen 16y ago
I can imagine extensions to this concept being applied to shape tweening in computer graphics and to materials science.

trafficlight 16y ago

Exactly. Was there a practical reason for the restrictions on tight bends and creases? Or was that just to make it interesting? Edit: Please don't downvote this, I legitimately want to know why.

Devilboy 16y ago
If you avoid sharp edges you can take differentials - if not you end up with discontinuities which makes is less useful mathematically.
andreyf 16y ago
Was there a practical reason for the restrictions on tight bends and creases? sigh You'll understand when you're building biological spaceships: you see, the inversion of n-spheres under those conditions is the simplest workable model for how evolutionary phenomena interact with each other. The inversion of an n-sphere of dimension 3 is representative of the evolution in just one layer of a system. After you discover the generalization of inversion across n-space in 2072, it will just take one ambitious biologists to apply it to genetic data, and generalize to most of the evolutionary phenomena around you: memes and cultures. By 2103, Human Directive 72 will be achieved, and all children under the age of four will be able to conceive of the basic geometries of infinite-dimensional topological models.
leif 16y ago
When you avoid tight bends and creases (singularities---places where the derivative vanishes), you have a nice structure on which you can do useful things like take derivatives, measure lengths and angles, find "straight" lines, and such. In short, the structure behaves, as long as you're small enough, just like the space we live in and have studied for centuries. Now you can extend this notion from a property of one surface to a property of a continuous family of surfaces, and then look at the beginning and the end (imagine it varying with respect to time, just like in the video). The niceness property above, if maintained throughout, gives you more things to say about what happens to substructures. For example, you can say something like "if I start with a triangle inside this surface, and morph it into another surface keeping it nice and smooth the whole time, I'll still get a triangle, and it won't do {big class of bad things}".

mikeknoop 16y ago

There is a large branch of mathematics which this is tied to called Knot Theory, I'll bet you could find some practical applications associated to it. (http://en.wikipedia.org/wiki/Knot_theory http://en.wikipedia.org/wiki/Knot_theory)

nitrogen 16y ago

I can imagine extensions to this concept being applied to shape tweening in computer graphics and to materials science.