Idran

jtotheizzoe:

harbard reblogged your quote: … when people thought the earth was flat, they…

HANG on the earth is spherical?

Actually it’s not. It’s an oblate spheroid. We’ve talked about it before.

It’s actually not quite regular enough to be an oblate spheroid (even ignoring mountains and oceans and whatnot) due to uneven distribution of mass.  The proper term, in a hilarious bit of circular definition (which is the main reason I bring up this bit of pedantry, because I love this term for how essentially useless it is), is geoid. :D

I say circular definition because if you really think about that definition, it essentially means “something shaped like the shape the Earth has”.

intothecontinuum:







Rotation around the symmetry axis of 4th order in the disk and band models

intothecontinuum:

An ideal rotation around the point at infinity in the disc and band model of the hyperbolic plane

Triangles?

Just a quick post here.  I mentioned in an earlier post that in spherical geometry, one could have triangles with total interior angles summing to more than 180 degrees.  How does this work?

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Heptakaidecagon!

intothecontinuum:


Back in the ancient Greek days, a lot of effort was put into doing geometry with nothing but a compass and straight-edge (not a ruler, mind; no markings, just something to draw a straight line).  No way of sliding a line segment, and no eyeballing.  Everything had to be exact or it didn’t count.  They managed a lot of impressive stuff with such simple devices, though there were three long-standing problems that lasted for literally millennia before they were finally proven impossible.  (A topic for a later post, perhaps.)

Still, it was very important, especially in the early days of mathematics, because to show something was correct, you had to prove it was correct.  It was just this sort of idea that the first concepts of rigour came out of; that it’s not enough to make a statement, you have to demonstrate exactly why it’s true.

But as for what you can do with just a compass and straight-edge: This is an animation showing how you can make a perfect 17-sided polygon using only those two devices. :D

M.C. Escher is a great example of math in art; though he had no head for the details of math, he had a great visual intuition for a number of mathematical concepts, and he loved to use his art in order to represent them in a clean and simple manner.  He had a special love for tessellations: a shape or set of shapes that can be tiled infinitely to cover a plane.  This tessellation of his in particular — Circle Limit III — demonstrates very simply an example of non-Euclidean geometry; specifically, he drew this tessellation as if it was in the hyperbolic plane rather than the traditional one.  Notice how they get more scrunched together as they get closer to the edge, this represents that the hyperbolic plane contains infinite area within a finite bound.  Here, the swooping white lines down the centers of the tiles are (essentially, though not quite) the equivalent of straight lines in the hyperbolic plane.
Hyperbolic geometry is one of two major non-Euclidean forms of geometry.  (There’s Euclid again!)  In normal Euclidean geometry, if you have a line and a point, there’s exactly one new line through that point parallel to the first.  This is also called “flat curvature”.  In spherical geometry, there are no parallel lines through that point; we also call that “positive curvature”.  While here, in hyperbolic geometry, there are an infinite number of parallel lines through the point, and we call this “negative curvature”.
We can also notice an interesting fact about triangles in the hyperbolic plane through this art.  Of course, in the normal plane, all triangles without question have a total interior angle sum of 180°.  Here, though, triangles have a total interior angle sum of ≤180°; closer to the center, it’s closer to “straight”, while closer to the edges the triangles are more skewed.  Similarly, in the spherical plane, triangles have a total interior angle sum of ≥180°
This all ties into Euclid’s Parallel Postulate, which I’ll cover in a later post, along with covering both these variants of geometry in more detail.

M.C. Escher is a great example of math in art; though he had no head for the details of math, he had a great visual intuition for a number of mathematical concepts, and he loved to use his art in order to represent them in a clean and simple manner.  He had a special love for tessellations: a shape or set of shapes that can be tiled infinitely to cover a plane.  This tessellation of his in particular — Circle Limit III — demonstrates very simply an example of non-Euclidean geometry; specifically, he drew this tessellation as if it was in the hyperbolic plane rather than the traditional one.  Notice how they get more scrunched together as they get closer to the edge, this represents that the hyperbolic plane contains infinite area within a finite bound.  Here, the swooping white lines down the centers of the tiles are (essentially, though not quite) the equivalent of straight lines in the hyperbolic plane.

Hyperbolic geometry is one of two major non-Euclidean forms of geometry.  (There’s Euclid again!)  In normal Euclidean geometry, if you have a line and a point, there’s exactly one new line through that point parallel to the first.  This is also called “flat curvature”.  In spherical geometry, there are no parallel lines through that point; we also call that “positive curvature”.  While here, in hyperbolic geometry, there are an infinite number of parallel lines through the point, and we call this “negative curvature”.

We can also notice an interesting fact about triangles in the hyperbolic plane through this art. Of course, in the normal plane, all triangles without question have a total interior angle sum of 180°. Here, though, triangles have a total interior angle sum of ≤180°; closer to the center, it’s closer to “straight”, while closer to the edges the triangles are more skewed. Similarly, in the spherical plane, triangles have a total interior angle sum of ≥180°

This all ties into Euclid’s Parallel Postulate, which I’ll cover in a later post, along with covering both these variants of geometry in more detail.