# What are the most beautiful fields of geometry?

• Samardar
Broken]In summary, fractals are beautiful and complex geometrical images that can be imagined by even a layman. They are also a field of mathematics that is fascinating and can be associated with intrinsic mathematical beauty.f

#### Samardar

Hello , I have something called Asperger's Syndrome and I would like to find a narrow topic or highly specialized field to study in the future as an Aspiring pure mathematician. But I have little or no experience in Mathematics , that's why I'm asking this question. I obsess about geometrical pictures that could be imagined by even a layman.

It would also be nice if you provided an example simply of what the field does in simple terms and how you experience it's beauty when encountering it. Or you could even provide a favorite picture of the mathematics involved and how it actually looks like!

http://en.wikipedia.org/wiki/Fractal" [Broken] maybe

(This particular fractal is http://en.wikipedia.org/wiki/Mandelbrot_set" [Broken])

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Fractals

I agree, I don't know much math but fractal geometry is intriguing. http://topdocumentaryfilms.com/fractals-hunting-the-hidden-dimension/

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As obscure as they are , they are beautiful.
Any thing else?

I think Riemann surfaces and Complex manifolds in general are fairly beautiful. I have not studied it at all but I imagine complex algebraic geometry could be something you'd like. Here is an often recommended book (which I have not read):

https://www.amazon.com/dp/0821802682/?tag=pfamazon01-20

edit: Of course I don't think you'll be able to read this given you say you have no experience with mathematics. But I guess its food for thought. I think to really see the beauty of mathematics you need to have studied a fair amount of it.

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If you want tangible/visible beauty then Fractal geometry is probably best. If you want intrinsic mathematical beauty then what deluks917 suggests is probably appropriate. Perhaps the more intense mathematical regions of theoretical physics (i.e string theory) have some beautiful hidden geometry (also check Garrett Lisi's E_8 Lie Group theory of everything), there are some very profound - though abstract - symmetries within the geometry of theoretical physics; however it is hardly as tangible and visually beautiful as Fractals are.

I know I'm going against the norm of this thread so far but I think Fractals are disgusting looking, they literally make me a bit nauseous. I think it has to do with the rigidness of them, although the images done in Apophysis are very beautiful.

Beauty, to me, exists in curved objects and surfaces. It amazes me how math can compute lengths, areas, and volumes of curvy things, actually it is pretty much the reason why I'm drawn to math. I still remember being a kid and throwing a hose on the ground then trying to measure it's length when curvy with a ruler. When I found out that math can compute that length I'm pretty sure I made this face -->

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Thanks! I'll definitely look up images of them :D

Here's what I found:

These are tangible or visual aspects of
Curved objects and surfaces:

Spirograph
Spirograph (Denys Fisher) produced mathematical curves using disks with holes strategically placed in the plastic circle.

In other words the CURVATURE at a point tells you how fast is the curve turning at that point.

Intrinsic mathematical beauty is what mathematicians look for , it's probably what derives pleasure from their work.
Here's one image but the formulas are probably more interesting , it's of Riemannian Surfaces:

Heres' one of Complex Manifolds:

And lastly one of Complex Algebraic Geometry:

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They are profound but more in formulas and equations :
String theory:

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Wow look at that massive thing:

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Here's a Riemann surface of an inverse polynomial of degree six. Take a vertical line down it, that line hits the surface six times. Keep in mind that's a plot of a function like y=2x+1 but just more complicated. These are called "multi-valued functions" in Complex Analysis. They are as diverse in form as life on earth. In order to work with them, you would need to first study Calculus, Differential Equations, and then Complex Analysis.

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I've studied calculus , someone recommended me a text in elementary complex analysis , I have little knowledge of advanced topics and applications of differential equations.

But I'm still very basic in knowledge , thanks for posting!

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