What's a hyperbolic octagon?

Question

And why/how do they tesselate?

Answer 1

Not all hyperbolic octagons tessellate, but before I can give you an answer that you'll understand I'll need to lay some groundwork.

Imagine standing on a ball. Look down at the surface, you can notice it curving. Now imagine that the ball gets bigger (its radius increases), you notice the curving less. Once the ball gets to be the size of the Earth (the radius of ball is very large), you can't notice any curving by looking at it. If the radius of the ball is infinite, there is absolutely no curve there at all that anyone can see. When drawing on the surface of this ball of infinite radius, you are working in Euclidean Geometry (the stuff you learn in high school).

If we draw a regular (all angles equal) octagon in Euclidean Geometry (I'll just call it EG for short), it doesn't matter how big you make it. The angles will always be the same. But is this always true for other geometries?

Let's go back to when the ball we were standing on was still ball-like and we could notice the curve. Triangles in EG have angles that sum up to 180 degrees, but you'll notice that if you draw a triangle on the ball the angles will be more than 180. For extremely small triangles they will have angles summing to approximately 180, but the sum will increase the bigger you make the triangle (try it out on a balloon or something). If you make a biggest "triangle" on the sphere, it will go all the way around the ball and look like a circle (all of it's angles tend to 180 degrees). This shows that angles aren't ALWAYS independent of size.

So when we decreased the radius of the ball from infinite (working in EG) to a radius that was finite (working in Spherical Geometry, SG) our angles increased.

But what if we made the radius of the ball BIGGER THAN INFINITE! Now instead of seeing the ball we were standing on curving down, we'll see it curving up! If you drew on the surface of this "ball" you'd see that the angles would decrease instead of increase. This is the world of Hyperbolic Geometry, HG. (As a technical note, you can mathematically express this as a sphere with a purely imaginary radius, that is (real number)*(square root of -1)).

You can't actually picture what shapes look like all at once anymore because what they look like depends on where you are looking at it from. The same was true of working in SG, but we could get around that problem by looking at the shapes from outside the sphere. In HG, we can't see things from the outside because our "sphere" has grown too big.

Just as the angles in SG grew to 180 degrees, the angles in HG shrink down to 0 the bigger we make our shapes.

Now we're at a point where I can answer your question.

Q) What is a hyperbolic octagon?
A) It is an octagon in HG.

Q) How/why do they tessellate?
A) Regular polygons tessellate when some multiple of their interior angles = 360 degrees. In EG, this is why squares and triangles and hexagons tessallate. Since in EG, an octagon has interior angles of 135 degrees, this doesn't fit the bill. But in HG, since small octagons start out with angles near 135 degrees, and the bigger we make the octagon, the smaller the interior angles get (tending to 0 when we make a "biggest octagon") we know that there is some octagon between small and biggest that will satisfy the condition for tessellation. Once you find an octagon such that there exists a multiple of the angles that will = 360, then all you need to do is just make copies of it and adjoin their sides. (there will be infinitely many such octagons, but still not ALL hyperbolic octagons will have this property).

Answer 2

Tessellation is a property of polygons meaning whether or not they tile. Pentagons tessellate into a dodecahedron. Hexagons tessellate in a plane. I am not sure how hyperbolic octagons tessellate, if indeed they do, but there would be no "why" involved.

Answer 3

Infinite forms can be extended to tessellate a hyperbolic space. Hyperbolic space is like normal space at a small scale, but parallel lines diverge at a distance. This allows vertex figures to have negative angle defects, like making a vertex with 7 equilateral triangles and allowing it to lie flat. It can't be done in a regular plane, but can be at the right scale of a hyperbolic plane.

Answer 4

I read about this in a Roger Penrose book. Basically, it's an octagon where Euclidean geometry does not work. You have to use hyperbolic geometry. I don't know why they tesselate.

Answer 5

hyperbloic:1) of or relating to hyerbola denoting trigonometrical functions defined with a reference to hyperbola rather than a circle. 2)deliberatly exaggerated. tesselate: cover by repeated use of a single shape, without gaps or overlaying. thats all i could find

Answer 6

It's an eight sided shape with "bolics" hanging from it.
They keep their tesselations private.

Answer 7

Your bolic's are hyper & shaped like a octagan because your sex patner is to demanding. They tesselate because tesseing early will give your bollocks another bashing . They tesselate because you are not attracted to your patner any more.

P.S My advice. "Give your balls a rest & start doing "Algebra!!!

"I hope i was "HELPFUL!"

They are not "Polygon's property" they are yours!

Answer 8

http://www.mkheydt.com/hyperquilt/

http://www.discover.com/issues/mar-06/features/knit-theory/

http://mathworld.wolfram.com/UniversalCover.html

http://www1.kcn.ne.jp/~iittoo/usDraft_A2.htm

Here are a couple sites that talk about the Hyperbolic Octagon, the last one is the best, and it is about half way down the page.

Enjoy!

Answer 9

If there is such a thing..go use Google..NOW!

Answer 10

Don't know but it sounds painful!