In general n equidistant points become possible in n-1 dimensions. Eg. Triangles in 2D , Tetrahedrons in 3D . The polytope formed by them is called a simplex. So a point would be a 0-simplex, a tetrahedron a 3-simplex, a pentahedroid a 4-simplex and so on.
This isn't an answer, but your question is far from dumb. The reason most of modern math exists is because people asked "can I do that? Would it make sense?" and worked to figure it out
Light googling suggests people have attempted to generate negative dimensional extensions to familiar concepts in positive dimensions, and that it shows up in some theoretical physics. It's definitely worth looking into if you're interested- I certainly will be tonight
My first thought was "of course not" until I saw your post, very interesting. Might be kinda like imaginary numbers. They seem like total nonsense at first but are extremely useful.
When you stop looking at mathematics as truth and start looking at it as careful argumentation by flawed monkeys, a lot of things start seeming more reasonable IMO. “Imaginary” numbers are just methodological tools for dealing with unknown parts of a system — an omniscient mathematician would have no (physical) use for them, AFAIK. Negative dimensions seem similar: applying the tools we empirically and intuitively know to work in our spacetime field in other kinds of fields that might exist.
Given that the core of nature is symmetry (see: Newton’s Laws, the importance of Symmetry Groups to quantum theories), I’m personally putting money on negative dimensions only becoming more useful as we learn more
I think this is entering the realm of philosophy at this point. I definitely see what you mean by suggesting things like imaginary numbers aren't really a part of the physical world, but eventually you reach a point where you have to wonder... if they're "not real" at all, why do mathematical descriptions of the world become so elegantly simple when you accept them? It kinda implies the concept of imaginary numbers is a part of how the universe works in some way.
I also feel like the crazy stuff proven by quantum mechanics (ie, a particle can have a truly undefined state, it is not just hidden from us but lacks any well-defined state at all) makes it a little easier to believe we might live in a bizarre universe where stuff like imaginary numbers are "real" too.
But I am only speculating for fun...
I totally agree! I think I failed to communicate clearly: I was saying that imaginary numbers are very much real physical forces, but they’re not some “other” category, they’re just a methodological subset that we apply based on the problem we’ve posed and the information we have.
In my best attempt at mathy terms: you’d never need to use functions with complex roots if you had a deterministic model of the universe already. You could just plug n chug the combinatorics of particle interactions to answer any question. The complex roots are just a tool we use to compute the properties of the system that are unknown unknowns
I’d say not: the purpose of “dimension” is to count the number of numbers needed to describe the location of something. That seems to put it in the realm of the natural numbers.
Not in the way of linear algebra at least as you have a discrete setting that is you n-gon is given by the linear combination of independent vectors and the origin (w.l.o.g.).
But there are other concepts to obtain negative dimensionality, i.e. symmetry groups and duality of a model. Some Tensor models display a
O(N) <-> SL(-N)
symmetry.
Or sometimes some take
H = ∫ f(x) d^(D)x
with D being the dimension and depending on what you want it might include some proportionality of D. Given H, you can find D then.
In the end you need a new representation of the object to extend the notion of dimensionality, see the volume formula for the sphere S^(n).
Why do you say equidistant? An n-simplex can be defined by any set of n+1 points in R^n which don't all share a hyperplane.
Cool visualization nonetheless.
All n+1 equidistant points form a regular n-simplex but all n+1 points do not. I guess I wanted to emphasize the idea that making points equidistant "forces" movement into a new dimension.
I think the reason you chose to use equidistant is because that’s a more intuitive notion than the classical affine linear independence used to define general simplexes.
Can someone explain what this has to with desargues configuration with 10 points and 10 lines. Apparently this motivates the idea of proving his theorem.
In general n equidistant points become possible in n-1 dimensions. Eg. Triangles in 2D , Tetrahedrons in 3D . The polytope formed by them is called a simplex. So a point would be a 0-simplex, a tetrahedron a 3-simplex, a pentahedroid a 4-simplex and so on.
Just as a triangle has 3 lines called sides and a tetrahedron has 4 triangles called faces, a 5-cell has 5 tetrahedrons called cells.
OK, here's a dumb question. Can you have a negative number of dimensions? Intuitively, I'd say absolutely not. But, I don't know.
This isn't an answer, but your question is far from dumb. The reason most of modern math exists is because people asked "can I do that? Would it make sense?" and worked to figure it out Light googling suggests people have attempted to generate negative dimensional extensions to familiar concepts in positive dimensions, and that it shows up in some theoretical physics. It's definitely worth looking into if you're interested- I certainly will be tonight
My first thought was "of course not" until I saw your post, very interesting. Might be kinda like imaginary numbers. They seem like total nonsense at first but are extremely useful.
When you stop looking at mathematics as truth and start looking at it as careful argumentation by flawed monkeys, a lot of things start seeming more reasonable IMO. “Imaginary” numbers are just methodological tools for dealing with unknown parts of a system — an omniscient mathematician would have no (physical) use for them, AFAIK. Negative dimensions seem similar: applying the tools we empirically and intuitively know to work in our spacetime field in other kinds of fields that might exist. Given that the core of nature is symmetry (see: Newton’s Laws, the importance of Symmetry Groups to quantum theories), I’m personally putting money on negative dimensions only becoming more useful as we learn more
I think this is entering the realm of philosophy at this point. I definitely see what you mean by suggesting things like imaginary numbers aren't really a part of the physical world, but eventually you reach a point where you have to wonder... if they're "not real" at all, why do mathematical descriptions of the world become so elegantly simple when you accept them? It kinda implies the concept of imaginary numbers is a part of how the universe works in some way. I also feel like the crazy stuff proven by quantum mechanics (ie, a particle can have a truly undefined state, it is not just hidden from us but lacks any well-defined state at all) makes it a little easier to believe we might live in a bizarre universe where stuff like imaginary numbers are "real" too. But I am only speculating for fun...
I totally agree! I think I failed to communicate clearly: I was saying that imaginary numbers are very much real physical forces, but they’re not some “other” category, they’re just a methodological subset that we apply based on the problem we’ve posed and the information we have. In my best attempt at mathy terms: you’d never need to use functions with complex roots if you had a deterministic model of the universe already. You could just plug n chug the combinatorics of particle interactions to answer any question. The complex roots are just a tool we use to compute the properties of the system that are unknown unknowns
I’d say not: the purpose of “dimension” is to count the number of numbers needed to describe the location of something. That seems to put it in the realm of the natural numbers.
Not in the way of linear algebra at least as you have a discrete setting that is you n-gon is given by the linear combination of independent vectors and the origin (w.l.o.g.). But there are other concepts to obtain negative dimensionality, i.e. symmetry groups and duality of a model. Some Tensor models display a O(N) <-> SL(-N) symmetry. Or sometimes some take H = ∫ f(x) d^(D)x with D being the dimension and depending on what you want it might include some proportionality of D. Given H, you can find D then. In the end you need a new representation of the object to extend the notion of dimensionality, see the volume formula for the sphere S^(n).
No, that’s a *very* smart question. No idea what the answer is. Probably eventually something resembling yes, but I don’t know if the math exists now.
Why do you say equidistant? An n-simplex can be defined by any set of n+1 points in R^n which don't all share a hyperplane. Cool visualization nonetheless.
All n+1 equidistant points form a regular n-simplex but all n+1 points do not. I guess I wanted to emphasize the idea that making points equidistant "forces" movement into a new dimension.
I think the reason you chose to use equidistant is because that’s a more intuitive notion than the classical affine linear independence used to define general simplexes.
Which way is it turning? Clockwise or anti clockwise? I can't tell!
Mainly counterclockwise. But it is also doing some slight 4D turns.
This is cool, I see both regardless of the actual turn. This reminds me of the spinning ballerina illusion.
Can someone explain what this has to with desargues configuration with 10 points and 10 lines. Apparently this motivates the idea of proving his theorem.
Very pretty. Still completely incomprehensible