∂ part of node [[2021-10-15]]
- Mathematicians are people that do a lot of thinking.
- When you use a category, what other categories do you have to ignore to use that category?
- We come up with general rules to avoid case-by-case judgments.
You need rigid rules for shapes when you're building something with material that may not fit together if they are not put together in a particular way. This we call geometry.
- Such a rule is: two shapes are the same shape if you can change one into the other by squeezing or stretching instead of gluing or ripping.
In topology, we don't care about size. It's just if the shapes can be stretched into each other. Because of this, a triangle, a square, and a circle are all kinda the same shape.
- S-one is the shape of a rubber band or a necklace. So it can be stretched to be a circle or a square or rhombus, etc.
- A line is a shape in topology. You can bend a line into an almost-circle, but the ends won't join, so it stays a line.
- A figure-eight is also a shape in topology. You can stretch it however but the place where the line crosses on itself cannot be changed, in the same way that the ends of the line cannot cross.
- There are infinite shapes, given these rules. Since you can just add crossing-points and end-points to denote a new shape and you can keep adding those indefinitely, there are infinite shapes.
- If you want to show there is an infinite amount of something, show how there's a way to keep on making more of that thing. This is called the infinite family argument.
- How are proofs accepted as proofs?
- Since there are so many shapes, people who research topology focus on manifolds. A manifold is a smooth, simple, and uniform shape such as a circle, line, plane, or sphere.
- A shape is a manifold when it has no end, crossing, edge, or branching points. It also has to be the same everywhere.
- People want to know what all the kinds of manifolds are.
- You can have manifolds that are like sheets, or like dough. If the world does not suddenly stop somewhere, and if it doesn't cross over itself, it may be a manifold.
- Manifolds that you can make out of string are one-dimensional manifolds. Since manifolds can't have end points, there are only two kinds of one-dimensional manifold: an infinite string (R-one) or a circle (or S-one). So basically this is any closed loop or shape that goes on forever.
- In the second dimension, shapes look like 3d shapes but they're made of 2d material- sheets. Two dimensional shapes are sheet like all over- there are no edges and no cliffs (or places where they just stop).
- Through stretching and squeezing, you can turn a sphere into almost any other continuous 3d shape, such as a cube.
In math, a sphere is hollow, while a ball is filled.
- A sphere can be stretched or squeezed into a cube, cylinder, cone, etc, etc.
- A sphere in topology is called S-two. Recall that the circle is S-one. So we may find the equivalent of R-one (a line)in two dimensional space, too.
- The earth might be flat in terms of topology- a manifold has no end or cross point, so all points might look the same if you look at it from the point of view of one point. So you can't see the curves- which means that if you live on a sheet, it would look like a flat plane.
- "More dimensions means more freedom of movement."
- Process is what drives to the front.
- We come up with categories to limit our possibilities when the possibilities are too much to cut out a path from.
- What are the fewest categories you need to move?