If you have an infinite amount of zero dimensional points, you have a one dimensional line. If you have an infinite amount of one dimensional lines, you have a two dimensional square. If you have an infinite amount of two dimensional squares, you have a three dimensional cube. In both Math and Physics, if you have an infinite amount of something that does not occupy a certain dimension, that will actually occupy that certain dimension.
Just.... why? Is there a reason for this? Or is it just one of the things you have to lower your head to and say "oh ok" in mathematics and science?
You don't have good concepts for "infinite" or "dimension". I understand the intuition that motivates the statement, "if you have an infinite amount of something that does not occupy a certain dimension, that will actually occupy that certain dimension," but it actually isn't sensical.
I honestly mean this in the most friendly way--you're probably in high school, and if you continue studying math you'll come to have a better notion of what's going on here.
Ayden Bell
Wouldn't it be just easier to answer the question instead of typing two paragraphs of passive aggressiveness? Again: points don't occupy space; yet an infinite amount of points will form a 1D line, which occupies space. The same happens with Physics, the most fundamental particles don't occupy space; yet here we are. Why? What is going on, exactly?
Gabriel Garcia
I'm not being passive aggressive at all, I'm giving you an honest assessment. But since you're insistent, I'll say a little about the mathematics question, and not physics, because I don't know anything about physics.
>if you have an infinite amount of something that does not occupy a certain dimension, that will actually occupy that certain dimension
No such mathematical fact exists--an infinite collection of "one dimensional" objects is not by necessity "two dimensional". What you seem to be getting at is that if we're restricted to considering points in R^n, there are infinitely many n-1 dimensional cross sections of the space. But this is obvious and intuitive from the definition of R^n--because there are infinitely many objects in R.
Easton Turner
>an infinite amount of points will form a 1D line It won't though. Put points at all the rationals from 0 to 1. Infinite number of points, gaps everywhere in your 'line'.
Colton Allen
If you are being honest and not trolling me then mathematicians have to put their shit together because on every other forum I have lurked it was said a line is an infinite collection of points. We could expand on this. Just like there are infinite points in one line, a square can be said to be made of infinite 1D lines, a cube of infinite 2D squares, and the Universe of infinite particles that do not occupy space. If you have MS Paint, do the following: open it; draw a line, use the selection tool to select the line, copy it, delete it; now paste the thing and, while holding ctrl+v, also hold shift and drag your mouse, you'll see what I mean. So, once again, the question would be how exactly a n dimensional thing can be made of infinite n-1 things (or, in better words, just why and how exactly does an infinite amount of n-dimensional things form a n+1 dimensional thing). Given this is Veeky Forums, I will say you're either trolling me or are just wrong until this is explained. And like you I hope it doesn't come off as rude or arrogant, I'm just curious.
Kayden Roberts
>If you are being honest and not trolling me then mathematicians have to put their shit together because on every other forum I have lurked it was said a line is an infinite collection of points.
An infinite number points do lie on a line. But there are lots of infinite collections of points that aren't lines, nor curves, nor well defined "surfaces" of any kind, even in 3D space. This fellow gave you an obvious example.
>how exactly a n dimensional thing can be made of infinite n-1 things
This follows from the definition of R^n, as I described above. If you knew any formal definitions, you'd understand what I'm talking about, which is I said you need to learn more and have clearly concepts.
>(or, in better words, just why and how exactly does an infinite amount of n-dimensional things form a n+1 dimensional thing)
This isn't true in general, as we've already said.
Jackson Collins
>If you have an infinite amount of zero dimensional points, you have a one dimensional line Not true. Not true for [math] {\aleph}_0 [/math] many points, for example.
Jackson Robinson
>which is I said you need to learn more and have clearly concepts.
I think I had a small stroke while writing that
Luis James
Define dimension.
For affine n-space over an arbitrary ring R, we have [math]\dim \mathbb{A}_R^n = n + {\dim _k}R[/math].
When R is a field this gives dimension n, but for instance if [math]R = \mathbb{Z}[/math] then it gives n+1. Yet [math]R = \mathbb{Z}[/math] is an infinite amount of zero dimensional points, just like [math]\mathbb{Q},\mathbb{R},\mathbb{C}[/math] etc.
Cooper Cox
OP you are stupid. Imagine a plane: it's an infinite slate within two dimensions. Now take a 1 ft x 1 ft square, and infinitely fill up the plane. You are still in 2 dimensions, but you're using infinite number of squares.
Take a point, and place it at (0,0). Then take another point and place it at (1,1). Then take anotehr point and place it at (2,2). Continue this for infinitely many points, placing them at (n, n) for n -> infinity. You are using an infinite amount of points, yet there are gaps between every point so it's not a line.
Nolan Cook
>other forum >MS Paint Found your problem. >question would be how exactly a n dimensional thing can be made of infinite n-1 things It can't. Other forums are probably thinking of calculus where what you're saying is still technically wrong but is roughly accurate. eg. in integration you get an nD object by adding together infinitely many infinitesimally thin slices. The slices are nD but they're so thin they're approximately n-1D. Like in Paint your 'line' is actually 2D already, whatever length it has it's at least 1 pixel thick. If it was a real 1D line it'd have 0 thickness and even an infinite number of 0 thickness things together still have 0 thickness.
Carson Nelson
This poster right here, just explained what you are thinking of OP. If you have a cube and take a cross section of it (i.e. you slice it), you will find that you can make a set of infinitely many 2D objects. This holds in general for any dimension.
Jason Ward
But that's exactly my fucking point. My question is, how? How can an infinite 2D squares make a 3D cube?
Jordan Miller
There are different sizes of infinity. (If you marked a point at every integer would that not be an infinite number of points? But definitely not a line.)
Measure theory (sets of measure zero) might help.
Cantor's diagonal argument also.
Then for a real mind-fuck, the Heine–Borel theorem seems relevant.
Oliver Foster
Cantor dust: not a line, infinite number of points. Not just infinite, but cardinality of the continuum infinite. There are just as many points in Cantor dust as there are on the real number line, and yet it's not a line.
Gabriel Carter
I guess, think about systems of linear equations. If you have n unknowns, you need n independent equations to solve for the unknowns. If you take one of the equations f(x1,x2,...,xn)=0 and solve for say, x1, in terms of the other variables to get x1=g(x2,x3,...,xn) then substitute this for all occurrences of x1 in your other equations, you end up with n-1 equations in n-1 unknowns. (dim goes down by 1)
Keywords: Vector Spaces, Linearly Independent Basis, Degrees of Freedom
Christian Phillips
>My question is, how? How can an infinite 2D squares make a 3D cube? because that's exactly how 3D cube is defined (a cartesian product of a 2D square with an interval).
Josiah Garcia
Why not just use the irrationals. ;^)
Rationals are pretty cool since they have measure 0 but are still dense.
Because Cantor dust has measure 0, while the set of irrationals between 0 and 1 has measure 1.
Evan Hughes
don't talk shit about michael
Chase Cruz
Damn, my bad.
I guess I'm not of his intended layman audience. I didn't say he is annoying. (his characteristic) I said he annoys me. (my characteristic pertaining to him)
Ethan Diaz
And if you have an infinite amount of dicks in your mouth, you're OP.
Caleb Moore
>If you have an infinite amount of one dimensional lines, you have a two dimensional square no you don't, you have a circle
Oliver Garcia
>friendly user extends olive branch to misguided anime/philosophy poster >get called troll, rude, and arrogant
a glorious example of why i hate animeposters
Asher Myers
>If you have an infinite amount of zero dimensional points, you have a one dimensional line. Fuck off, weeb.
Charles James
dumb animeposter
Jose White
is intuitive, but it isn't a rigorous (acceptable explanation in modern math) > In math, dimensions is the minimum required coordinates for specify a point. A point P in a space with n dimensions can be represented by a vector [math]\vec{p}[/math] (list, array, ordered list) with n entries. [math]\vec{p}=(p_1, \ldots, p_n)[/math] where [math]n \in \mathbb{N}[/math] > So a point in 3D Space can be represented by a vector [math]\vec{p}=(p_1, p_2, p_3) = (x, y, z) [/math] 4D Relativity Space-Time add a extra entry for time [math]\vec{p}=(p_1, p_2, p_3, p_4) = (t, x, y, z)[/math] > In Edward Witten version of String Theory work in 11 dimensions instead of 4. > Economics & Statistics can work with several more dimensions >Example 1 imagine a vector [math]\vec{p}=(p_1, p_2, p_3, p_4, p_5, p_6)[/math] which each coordinate represents a data form acountry. [math]p_1[/math] is the Net exports [math]p_2[/math] is the Military budget [math]p_3[/math] is the National Investment in Education [math]p_4[/math] is the Taxes [math]p_5[/math] is the Government Expense in Social security & Health >Example 2 Imagine a vector [math]\vec{p}=(p_1, p_2, p_3, p_4, p_5, p_6, p_7)[/math] where [math]p_1[/math] is the amount of Apples, [math]p_2[/math] of Oranges, [math]p_3[/math] of Bananas, [math]p_4[/math] of Pineapple, [math]p_5[/math] of Grapes, [math]p_6[/math] of Avocado, [math]p_7[/math] of watermelon. > How we can visualize dimensions higher than 3? There are many Techniques for it: 1) Projection en.wikipedia.org/wiki/Projection 2)Contour Lines en.wikipedia.org/wiki/Contour_lin 3)Statistical Charts en.wikipedia.org/wiki/Chart 4)Tesseract en.wikipedia.org/wiki/Tesseract 5)Color Gradient en.wikipedia.org/wiki/Color_gradient > Actually I saw also the point->line->plane->cube explanation before in books about design, psychology & architecture. It's based on Ancient Greek Math (Euclid, Pythagoras). But this is archaic & obsolete.
Jayden Davis
You need to understand the construction used to create these situations. What you do is take the cube, and divide it along an axis n time. This results in several prisms with the cross section of a square, and a height proportional to the side length divided by n. When n approaches infinity, the thickness approaches 0, and so you can consider this set of infinite squares to form a cube. This does not, however, apply the other way round. For instance, at what point do the infinite squares that you stack exceed the height of the cube, and become a rectangular prism again? The construction method holds for any height, including a height of 0. You need to be very careful when dealing with infinity.
Carson Gonzalez
...
Xavier Torres
>In math, dimensions is the minimum required coordinates for specify a point.
Only that doesn't make sense unless you space is k^n for some field k.
Mason Myers
And even then dimension is relative to the base field. C^n has dim=n over C but dim=2n over R.
John Lopez
>Only that doesn't make sense unless you space is k^n for some field k. what is a manifold
Daniel Phillips
because you can have an infinite amount of solutions to a homogeneous system. that is, any number R can solve it. how many elements in R exist? infinite amount even within a given bound. that's why. this is how smoothness is formed.
i think you need to learn some Analysis. this is a big issue for most people due to lack of understanding how the reals are formed.
Adam Powell
A paracompact hausdorff space locally isomorphic to R^n or C^n.
The point definition doesn't work. A "point" on S^2 would be (x,y,z) viewing S^2 embedded in R^3, but S^2 is 2-dimensional.
Not to mention all other types of spaces which are not manifolds.
William Ortiz
Nanyako is made for Yukari.
Carter Wright
>If you have an infinite amount of zero dimensional points, you have a one dimensional line. Er.. not true.
Jason Martin
If you follow your logic, then an infinite amount of zero dimensional points can represent infinite dimensions
