- 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.
collapsed:: true
- 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.
collapsed:: true
- 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?