This is something that confuses me a lot. So… Dimensions greater than 3. How does that work? The way my partner explained it to me, if you can define 5 parameters about an object, it exists in 5 dimensions. So… I can imagine a (somewhat confusing) graph along 5 axes, if they’re just random stuff… But is that a object?
Once, I played 5D tic-tac-toe my friend programmed. Before getting excited, it was just rows of playing fields, like sets of cross-sections of that hypothetical object. But… What kind of object is that?
It seems like this is maybe a different concept to what I think as, say, three dimensional objects. I get what happens when you make a 4D object, even if I can imagine it only in “cross sections”. But what’s above that? It would sort of follow that 5D would be some kind of spectrum of alternative times, is that a thing in theory? And above that? Or does it stop with 4? Why?
Someone probably hurled a ton of concepts my way that I didn’t really have the framework to understand properly and my brain is just making a mess of it. Ummm… Help?
In: 10
You may as well stop trying to visualize it. You literally can’t. Everything you visualize is going to be from the frame of reference of your existence in 3-D space with 1-D time.
There are various ways to present the concept of extra-spacial dimensions, but you can’t accurately visualize them because you have no frame of observation from which to do so. Similar to visualizing an all new, never before seen color. You usually just think of a rare color or rare-combination that you don’t ever remember seeing. But its always in the context of colors that you know.
Edit to add a bit more based on your overall question. It also depends on what kind of extra dimension you are adding. A dimension of time is different than a dimension of space (Assuming they are actually different things and what we think of as time isn’t spacial movement through a 4th spacial dimension in one direction). A second dimension of time would allow you to move “laterally” to our dimension of time (again, no idea what that would “look” like). Space would give you a different direction to move in (no idea what it would actually look like).
A hypercube or tesseract is, to my mind, a first step towards getting one’s head around a potential extra spacial dimension, https://en.m.wikipedia.org/wiki/Tesseract
I believe string theory alludes to over 10 dimensions and membranes where entire universes could exist in a sense right next to each other but completely unable to ever interact
Mathematically you can have as many spatial dimensions as you like. For instance a [five cube](https://en.m.wikipedia.org/wiki/5-cube) has five dimensions. [Superstring theory](https://en.m.wikipedia.org/wiki/String_theory#Extra_dimensions ) requires 9 spatial dimensions to work.
While you can claim as many dimensions as you like, as far as we know our universe only has three spatial dimensions. Anything with more dimensions is a mathematical description of what a higher dimensional object would look like *if* such higher dimensions existed.
Visualise a line. That’s a 1 dimensional object. Along the line you can only move say left and right.
Now, take that line and at each end extend it out at 90 degrees. You now have a square. That’s a 2 dimensional object. In the square you can move left and right, and forwards and backwards.
Now, take that square and at each corner extend it at 90 degrees from each side. You now have a cube. That’s a 3 dimensional object. In the cube you can move left and right, forwards and backwards, and up and down.
Now, take that cube and at each corner extend it at 90 degrees from each side. Oh, you can’t. We live in 3 dimensional space. You can’t even visualise it because your brain only evolved to visualise 3 dimensional space. But that’s what a 4 dimensional object would look like. At each step you would extend it out at 90 degrees from each side.
A dimension is basically something you can measure and track about a system. They don’t have to be locations is space. They don’t stop. In school I had to work with n (arbitrary) as well as infinite dimensions. Now, I routinely work with 6 and 4 dimensional objects.
Don’t try to visualize them, they don’t map to 3d space. They don’t represent something “real” like 3d space does, they’re just mathematical constructs. Any attempt to do so is limiting and often involves sacrificing dimensions. It’s common to drop dimensions and explain it in fewer where it makes sense(ie explaining the 4d as a 3d). Then you just say the same thing happens but in higher dimensions (because the math extends to work in higher dimensions).
A very basic (if oversimplified) way to think about the dimensionality of a space is simply how many numbers you need to specify a point on it. The dimensions may not be easy to visualize as physical spaces, but they nonetheless allow us to apply ideas from geometry to more complicated/abstract things.
The common “extra dimensions” people talk about are spacetime (4 dimensions) and the compactified dimensions of string theory, but a more abstract use of lots of dimensions is common in statistics and machine learning. If you think about a data set representing a bunch of people, with different variables–say for example, for each person you have age, height, weight, income,…–each of those n variables represents a dimension, and in this n-dimensional space, we can use concepts (like distance, colinearity, etc.) that we’ve abstracted from our common experience with 2d and 3d geometry.
(You could try to imagine n orthogonal axes and plotting each of these data points (people) in that space. but it’s pretty hard.)
By the way, have you ever played that card game Set? It’s essentially tic-tac-toe in 4 dimensions.
Here’s a practical example.
Suppose you’re trying to predict housing prices in a particular city. Every house has a sales price, and a number of relevant features (living area, number of bedrooms, number of bathrooms, year built, distance to nearest school, etc.). Each of these *k* features can be thought of as an independent dimension, so that every house can be represented as a point in *k*-dimensional space.
This is a useful way to think about things because clusters of points that are close together in this space will represent houses that are similar, and that would be expected to have similar sales prices.
This formulation leads to some straightforward mathematical machinery that allows solving fairly complex problems.
As your partner said, a dimension is like a parameter or measurement.
Dimensions are really just parameters/ranges of numbers used to express something.
For example, we see in 3D – three dimensions. That’s because every shape we can see can be expressed in a combination of 3 spatial directions (left/right, forward/back, and up/down).
An example of a problem that is three-dimensional would be, say, trying to find the fastest and most fuel efficient route to drive to work.
One dimension would be the measure of how fuel efficient your car is at any speed (10 mpg @ 5 mph and 30 mpg @ 60 mph).
Another dimension would be the average speed of a route (one route may average 30 mph while another may average 35).
The last dimension would be a measure of long the route is.
What you’ve done is created a 3D shape that takes into account all of those factors. This shape would probably look like a twisted and mangled sheet. By looking at that shape, you can see where your shortest and fastest route is, or the route that has the highest average speed and lowest gas usage.
However, finding that alone probably isn’t what’ll get you the fastest and shortest route that uses the least amount of fuel. For that, you’d need to extend up a dimension by finding a way to account for all of the parameters.
For example, you could score each combination of factors based on your desired outcome (shorter routes are scored higher than longer ones, faster score higher than slower, less gas scores higher than more gas).
By doing this, you’ve invented a new dimension (one that ranks how good a route is based on all three factors). If you were to try to graph the shape, you would find it would be literally impossible. We only see 3 dimensions, so there’s no way to express a fourth.
But you can do some clever math to find where the ranking of a route is the highest. This math might tell you that a route that’s 10 miles long, averages 50 mph, and uses an average of 27 mpg is the best that you can take.
So, while extremely difficult to imagine, dimensions are their math are really useful.
Einstein theorized that our universe is a 4D combination of space and time called spacetime.
String Theory can explain basically the entire universe, but it needs 9 dimensions to do so.
Machine learning is literally just doing what we did to find the best results from a combination of different factors.
Simply put it never stops but your brain can’t imagine more than 3 spatial dimensions
We are pretty sure out universe has 3 though
One explanation I heard on YouTube says that you can create/visualize dimensions by making a perpendicular line. For example a straight line is one dimensional because it’s a line and can only go forward or backward. A rectangle has a 90 degree angle and an object on the rectangles area can go up down forward and backward. A cube’s edges for 3 90 degrees(x axis, y axis, z axis) a point on a cube can go forward backward up down left and right. But when it comes to 4D it’s literally impossible, well at least seems impossible to add another 90° on the cubes edge.
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