# Visual research problems in geometry

I am considering doing research in mathematics to be my career (and my life) someday.

I'm a visually oriented person in general; for example, I prefer chess over cards because when I play chess, I do all my thinking by looking at the board and analyzing it, but when I play cards, I have to remember things and calculate things because the details are not visible or visual. That's why I did very well with traditional plane geometry problems at school.

I was good at problems that can be visually explained or visually modeled, like proving the equality of two line segments or two angles just by looking at the figure. It has always been more interesting for me than Algebra where I had to write down terms and rearrange them to reach the solution.

Now I am wondering if there is a branch of modern advanced mathematics that works the same way to be my research interest.

I am looking for the kind of problems that I can call "visual puzzles": problems that can be solved by looking at them.

Is there such a field in modern mathematics that I can do research in?

I realize the importance of algebra and mathematical logic, and I know that I must use them, and I like to use them.

I am considering discrete geometry, but I am not sure if its problems are really visual.

I have been looking for the advanced branches of geometry in the universities research pages and I downloaded many research papers and books just to look at the advanced fields of geometry from inside and see how it "looks" like. I didn't understand anything for sure. :-) I found topics like non-euclidean geometry, differential geometry, topology and Riemann geometry, among others.

What really disappointed me is that I couldn't find a lot of figures!

I need your help to find the most interesting field for me.

Thank you.

• it's an interesting question. would you say you're confortable "visualizing" more abstract geometrical objects ? like the klein bottle, or a fibration of the sphere with circles (try to picture attaching a circle at each point of a sphere, but in a larger space than R^3 so that they don't intersect...), or the fact than "glueing" a point at infinity on a plane gives a sphere... if so, go for geometry ! (don't worry if you don"t though that comes with practice) – Glougloubarbaki Nov 24 '11 at 0:38
• thank you for the fast response , the examples you have mentioned are far from my interests . i am more interested in less sophisticated objects like circles , polygons , triangles , convex hulls ... etc . thanks again . – Akram Hassan Nov 24 '11 at 0:43
• Mathematics at the research level is about sophisticated objects. – lhf Nov 24 '11 at 1:06
• Try and have a look at a copy of Indra's Pearls: The Vision of Felix Klein. Not easy or elementary, but it has wonderful pictures. – lhf Nov 24 '11 at 2:47
• @JoelReyesNoche: The link to Steve Sigur's webpage is now broken, but has been updated to sites.paideiaschool.org/steve_sigur/interesting2.htm. – J W Oct 11 '14 at 12:36

Discrete geometry (tilings, tessellations, packings, etc.) seems to be for you. If you like looking at nice pictures, see, for example, some of the questions of Joseph O'Rourke (at MathOverflow and at Mathematics StackExchange). (I first learned about him from his book Art Gallery Theorems and Algorithms.)

Tristan Needham, Visual Complex Analysis, Oxford Univ. Press.

"One of the saddest developments in school mathematics has been the downgrading of the visual for the formal. ... to replace our rich visual intuition by silly games with 2 x 2 matrices has always seemed to me to be the height of folly. It is therefore a special pleasure to see Tristan Needham's Visual Complex Analysis with its elegantly illustrated visual approach. Yes, he has 2 x 2 matrices—but his are interesting." —Ian Stewart

Here's one figure from the book, p.135:

You can almost guess the theorem from the figure: The two spheres $$S_1$$ and $$S_2$$ are orthogonal iff the two circles $$C_1$$ and $$C_2$$ are orthogonal.

• This looks like a wonderful book. I ordered a copy yesterday. Thanks for the reference. – bubba Feb 28 '13 at 2:42
• @bubba: You will not be disappointed! – Joseph O'Rourke Mar 1 '13 at 1:11

If you are really visually oriented person as you said, I would advise you using it and develop it. Take a look at Multidimensional Geometry - Parallel Coordinates: Visual Multidimensional Geometry and Its Applications by Alfred Inselberg

• Thanks for the reference to my book above. I'll be happy to respond to questions/remarks Alfred Inselberg aiisreal@math.tau.ac.il www.cs.tau.ac.il/~aiisreal/ – user23826 Jan 27 '12 at 14:19

A lot of modern geometry has become very abstract, and far removed from the familiar 2D and 3D objects in our everyday lives (or, at least, the connections are no longer obvious). Since the objects of study are so abstract, it's pretty difficult to draw pictures of them.

One field that is still firmly connected to plain old 3D space is called Computer-Aided Geometric Design (CAGD). It's the study of the curves and surfaces that are used in engineering, manufacturing, animation, and games. It's a mixture of mathematics and computer science. The mathematics involved is differential geometry, algebraic geometry, approximation theory.

Take a look at the journal Computer-Aided Geometric Design (CAGD), or at any of the books by Gerald Farin.