Does analysis form a bridge to geometry?

[FallenApple]

So I always thought that geometry is somewhat different from the rest of math. I mean, most of math is about numbers and relations. While geometry is about space.

Does analysis connect the two? For example, the hypotenuse of a triangle is just a truncated portion of the number line that has been completely filled with the real numbers. And shapes can be plotted in a cartesian coordinate system where the functions are shown to be continuous, basically a mapping of the number line. So basically shapes under that logic would be made of dense numbers.

 

[fresh_42]

Your description of geometry is already a reduction of the field to the classical Euclidean geometry. But even this has a lot of boundaries to other fields: e.g. calculus (trigonometry), topology (simplices), algebra (linear transformations and symmetries, Galois theory), algebraic geometry (the nature of solutions of polynomial equations), differential geometry (the nature of solutions of differential equations). Euclidean geometry can be taught (and often is) as purely self-contained. However, as with all scientific fields, there is no strict boundary between them. It only depends on your interest or taste which way you look at it. Another important way to classify geometry is the historical development: As soon as geometry has been taken into real world (Gauss has worked as cartographer!), curvature is no longer zero.

So you might as well consider geometry under an analytic aspect, but I'd start with trigonometric functions instead of set theoretic or topological terms like density.

 

[dkotschessaa]

If I had to illustrate it, I think topology is the 3 way link (at least) or middle of a venn diagram with analysis, algebra, and geometry on the outside.

Of course they all have their links. Subject overlap is always a bit messy.

-Dave K

 

[cpsinkule]

I think all of the branches of math can be somewhat separated from each other (although they all use set theory and logic at their core), but that is a very dry way of thinking about it. The most beautiful parts of mathematics, I think, is when you can apply all of those areas together. Certainly you can't get very far in geometry without analysis.

 

[micromass]

Certainly you can't get very far in geometry without analysis.

Sure you can. I know many many geometry books that don't use any analysis.

 

[cpsinkule]

Sure you can. I know many many geometry books that don't use any analysis.

Do they use algebra?

 

[micromass]

Do they use algebra?

There are also many that don't. But most modern books use algebra.

 

[cpsinkule]

There are also many that don't. But most modern books use algebra.

How far can you go with pure axiomatic geometry?

 

[micromass]

How far can you go with pure axiomatic geometry?

It's been a major field of mathematics for thousands of years... So pretty far I guess.

 

[cpsinkule]

It's been a major field of mathematics for thousands of years... So pretty far I guess.

Do you have a book recommendation for that? I've never really considered pure geometry other than what's needed for other fields.

 

[micromass]

Do you have a book recommendation for that? I've never really considered pure geometry other than what's needed for other fields.

It's really interesting. But most modern books use the language of abstract algebra at the very least. Even for things that were done centuries ago, and where things like abstract algebra or calculus were not available. So if you really want a book that uses the axiomatic approach strictly, you'll have to go to quite old books. Nevertheless, here is a selection of books I physically have in my bookcase and which you might find interesting:

- Coxeter, regular polytopes. Coxeter is a big mathematician and has written many gometry boks. This is one I personally like a lot. It's not really very "axiomatic" though, but it falls under classical geometry

- Euclid Elements together with Hartshorne's Euclid and Beyond: Euclid is of course very classical. It's a must read for any mathematician in my opinion. Hartshorne offers a perfect companion to Euclid. His book is quite algebraically inclined though. You might say it offers an abstract algebra interpretation of the Elements. I wish there was a similar book that did Appolonius' conics.

- Moise, elementary geometry from an advanced viewpoint: also is a modern interpretation of Euclid in a similar but different spirit than Hartshorne. More easy to digest than Hartshorne though.

- Richter-Gebert Perspectives on Projective geometry: very cool book with many neat insights. Not reaaally a textbook in the usual sense though. Uses the language of linear algebra freely.

- Hartshorne, Foundations of Projective geometry: fully axiomatic treatment of projective geometry. Very cool book.

 

[dkotschessaa]

Do you have a book recommendation for that? I've never really considered pure geometry other than what's needed for other fields.

Euclid. :)

 

[lavinia]

I think of axiomatic plane geometry as a way to minimally describe the relations of lines in planes, the way lines separate the plane, and how lines intersect. These do not deal directly with the structure of space but undoubtedly are based on a intuition of what it is like.

When analytical ideas of space started to be defined more formally, mathematicians realized that axioms of geometry can be made to correspond to manifolds endowed with metric relations and such manifolds that satisfy an axiom system are possible models of the axioms. These models of geometry could be considered to be a link between axiomatic geometry and analysis.

The ancients understood that geometry can be discovered through measurement. They understood that notions of length and angle link the axioms to actual space. But they never understood what space itself was.

 

[cpsinkule]

Euclid. :)

i did that one a long time ago ;P

 

[ShayanJ]

@micromass, I remember a very long time ago, I saw a book on geometry that started with assuming a set of points which could be anything(e.g. matrices) and then defining lines and then going on to more complicated concepts and theorems about them. But I can't remember what book it was. Do you know any book like that?