/sci/ - Science & Math » Thread #11332207

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Anonymous Sun 26 Jan 03:50:44 2020 No.11332207 View Reply Original Report

Quoted By: >>11332212 >>11332249 >>11332613 >>11332657 >>11332679 >>11334503 >>11334538

Does higher dimensional (4 and above) geometry have any uses?

Please, actual uses, like being able to solve simple to explain problems in a simple to explain way.

Or having a predictive value when looking at physical or social phenomena.

Anonymous

Anonymous Sun 26 Jan 2020 03:53:17 No.11332212 Report

Quoted By:

>>11332207
one way of looking at the (noneuclidean) 4D spacetime manifolds in GR is to embed them in a 5D euclidean space

Anonymous

Anonymous Sun 26 Jan 2020 04:19:45 No.11332249 Report

Quoted By:

>>11332207
visually, maybe not.
mathematically yes.
machine learning algorithms and other data manipulation techniques use higher dimension matrices to do magic

Anonymous

Anonymous Sun 26 Jan 2020 08:15:14 No.11332613 Report

Quoted By: >>11332677

>>11332207
It's useful for data analysis where there are more than 3 variables. For example, you could measure height, weight, age, and finger length, then plot each individual on a 4-dimensional graph.

Anonymous

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Anonymous Sun 26 Jan 2020 08:47:15 No.11332657 Report

Quoted By: >>11332677

>>11332207
Maybe not strictly geometry, but it can certainly be interpreted geometrically,
principal component analysis and basis change are incredibly useful for data analysis.
I could not find a very impressive image, but pic related shows two most significant features in a probably >> 3 dimensional data set. These two components can then be used to generate new shapes that are likely to be/realistic human scapulas.
t. undergrad that has taken one modelling course

Anonymous

Anonymous Sun 26 Jan 2020 08:56:30 No.11332677 Report

Quoted By: >>11334526

>>11332613
What's the point of having a graph you can't read?
Can you apply geometric principles to find things out that you wouldn't be able to otherwise, or find them out more efficiently?

>>11332657
I'm a mathlet. Can you elaborate a bit on what principal component analysis and basis change are and how 4 dimensional geometry relate to this, plus how they can be used to do things?

Anonymous

Anonymous Sun 26 Jan 2020 08:56:38 No.11332679 Report

Quoted By: >>11332730

>>11332207
literally all modeling done on complex systems

Anonymous

Anonymous Sun 26 Jan 2020 09:01:32 No.11332688 Report

Quoted By: >>11332730

Simulation in general. Dimension refers to number of degrees of freedom. Hilbert spaces are often infinite-dimensional. You're using it for Fourier analysis or similar techniques.

Anonymous

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Anonymous Sun 26 Jan 2020 09:32:43 No.11332730 Report

Quoted By:

>>11332679
Can you elaborate?

>>11332688
I'm seeing a lot of words here, but I don't know what they mean.

Anonymous

Anonymous Sun 26 Jan 2020 23:14:14 No.11334503 Report

Quoted By:

>>11332207
"dimensions" don't have to just be spacial dimensions. They can be the Tomato, Potato, Lettuce, and Fig "dimensions" when trying to figure out how to optimal ways to pack trucks.

Anonymous

Anonymous Sun 26 Jan 2020 23:21:51 No.11334526 Report

Quoted By:

>>11332677
As far as principal component analysis goes imagine you have an n dimensional dataset. (The orchid one in R uses something like petal length, petal width, sepal length, sepal width,...).
You model your data set as a series of n vectors of n dimensions each with a weight(we'll call it a_n). This is does automatically with an sklearn function usually. You can then sort these vectors in order of a_n with the biggest a_n first. These a_n are analagous to what % of the variance of the data is in that vector. So if I wanted to look at the core of my data I could take the first 3 of these vectors (our principal components). Model data points as a linear combination of them and plot them on a 3d graph. Reducing my data from n dimensions to 3 while maintaining as much as possible of the variance in qualities of the dataset.

Anonymous

Anonymous Sun 26 Jan 2020 23:23:52 No.11334538 Report

Quoted By:

>>11332207
Think of a point in n dimensions as uniquely determined by some set of n sliders which can describe particular values of a set of parameters. Point would be a particular state described by such parameters. Depending on the propeties of the space, states can change in some ways often well described by a particular geometry rules of the space. Sort of.

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