Is it right to express a plane as an infinite set of points?

>>8706276
A plane is determined by 3 non-collinear points and the only thing that can be inside of it are lines and points.

And lines are determined by two points, and the only thing that can be inside lines are points so yeah, a plane is made entirely of points.

>>8706287
Can you express a plane, what is made of points, and each point allows itself to be expressed by also non-numeral information?

For example, I want to have a plane made of dogs with no spatial extent. Can you put those dogs into an equation that expresses the plane?

>>8706276
A plane is topologically a sphere

>>8706335
Simply find 3 dogs that are not in the same dogline.

>>8706338
Nope, try again freshman.

>>8706340
Then how do you answer a question "is this dog in the dogplane defined by these 3 non-collinear dog-points"?

Remember you can't do basic arithmetic operations like + - * / on dogs!

File: dogmetry.png (23KB, 746x584px) Image search: [Google]

23KB, 746x584px

I made a picture representing the dog-point, dog-line, dog-plane.

>>8706383
Get this shit on ArXiv

>>8706362
>Remember you can't do basic arithmetic operations like + - * / on dogs!

No need. You don't need arithmetic to do geometry.

>"is this dog in the dogplane defined by these 3 non-collinear dog-points"

Simple, first connect the dog triangle and then try to find to points in that dog triangle such that their dogline contains the dog. If such a dogline exists then the dog is in the plane.

If such a dogline does not exist then the dog is not in the plane.

This would entail doing an infinite amount of work (if you wanted to prove that the dog is not on the plane) but in geometry you can usually get around this.

File: makets.png (7KB, 515x263px) Image search: [Google]

7KB, 515x263px

>>8706344
>this poor newfriend doesn't know spherical topology

top pleb desu senpai

>>8706397
A sphere is compact, a plane isn't.

That shit picture explains nothing.

How would a plane made of dog-points look like compared to a plane made of cat-points? They obviously can't be the same since everyone of the infinite objects carry different information!

How about a dog plane where each dog is moving their legs? How could the plane express this? Clearly a plane with non-moving dog-points cannot be the same as plane with moving dog-points.

>>8706393
Can't argue with this. I guess this solves it then. Geometry seems pretty cool.

>>8706400
>implying sin(2*pi*n)!=0 for all integer n

too bad you can't grok simple diagrams

>>8706400
a finite plane is topologically identical to a finite sphere. An infinite plane is topologically identical to an infinite sphere.

It is not that hard

>>8706438
Define topologically identical.

All this time I've been supposing you mean homeomorphic.

>>8706480
>not knowing what identical means

>>8706535
You're officially retarded.

>>8706826
>your retarded
brainlets often obscure their ignorance behind a veil of hostility

>>8706397
whatever definition of "topologically equivalent'" you're using is extremely non-standard, and it's understandable how someone else could get confused

>>8706438
>"An infinite plane is topologically identical to an infinite sphere."
>durr what the FUCK is the RIEMANN SPHERE

>>8706276
Generally no, but specifically Yes. A complete plane can be represented by RxR, which are just 2d coordinates.

>>8706335
If the dogs are a field.

>>8706338
No it's not.

>>8706383
I mean, if "dog" is a field then you can easily throw a vector space over it, or define an R-action if dogs form an abelian group and get some sort of module structure going. But if you want your "points" to be "dogs", you're not really doing anything but renaming things.

It also depends what properties you want. Rational numbers are closed and form a field, so it's easily possible to have rational dogs and finite sums of rational dogs but they aren't complete under infinite sums.

>>8706393
>No need. You don't need arithmetic to do geometry.
Uh, no, nothing drawn in that image is well-defined if you don't have any algebraic structure.

>>8706397
They're both 2d manifolds but a sphere, like anon said, is compact. The plane is homeomorphic to a sphere with a hole. If you know anything at all about topology then you know that means the structures are not homeomorphic, which is what "topologically equivalent" means since homeomorphic is an equivalence relation.

>>8706438
"A finite plane" is what is known as a disc, and no, you're fucking wrong.

>>8706383
Also if you want dogmetry to have nice properties like being a field and being complete, you're actually literally creating the real numbers because it's provable that the real numbers are unique, in the sense we care about in analysis.

>>8706276
>Is it right to express a plane as an infinite set of points?
>Or are points, lines, planes, volumes, etc, different entities?
In set theory terms, all of these entities embedded in R^3 are just collections of points. They're nothing more.