Distinction 1: Surfaces

This post is a part of an experiment for me to improve the distinctions I use in my world. I have daily conversations with interesting and well-educated people who tell me how a plane flies or how digital cookies are used in a login process. If I then try to share what I've learned with others, I'm unable to reproduce what I've learned in an eloquent and clear way. And so I decided to see if writing things down will help me refine my own stories.

Disclaimer: Don't read these posts as if I know what I'm talking about. I'm a lay person in the fields I'm writing about and you should approach these posts as such. In fact, if you know more on what I'm talking about and have something to add or correct, please do so.

So, what is a surface?

(from the perspective of math and, in particular, Algebraic Geometry)

Attributed to mint imperial of Flickr / CC BY-NC-SA 2.0

You might be wondering why I'm writing a post on what a surface is. Well, my fiancé is a mathematician who studies surfaces, and even though I have spoken to him many times about his work, I realized recently that I do not have a good lay-person distinction for the objects which he studies.

A couple weeks ago, I was speaking with a non-mathematical friend, Guillermo Wechsler, about what a surface was. He claimed that a surface was two-dimensional. I, having heard my fiancé, a mathematician in Algebraic Geometry, talk about the hundreds of different variables he was working with to describe the surface he was studying, had a different idea about what a surface was. I said...”A surface can be multi-dimensional – imagine a sphere on which you have a point with three coordinates. As there are three variables involved in describing that point, it is a three dimensional surface.”  Right?

Wrong.

A week or so ago, I recapped this conversation to my fiancé. And...I learned quickly that I didn't know what I was talking about. What I had been calling a surface was based on a common sense, non-rigorous definition. What I learned from the conversation is the following:

1. A surface is a two-dimensional object. Plain and simple. If something is three-dimensional, it is called a three-fold. If something is four-dimensional, it is called a four-fold. My first mistake was a tautological one.
2. A sphere is a surface. It is not a solid object but instead a hollow casing of a ball. But even with this made explicit to me, I had a hard time seeing a sphere as a surface. I imagined a sphere with radius 1 centered around the origin (0,0,0) of the x, y, and z-axes. In order to describe the point that intersects the y-axis, one would write (0,1,0)...three variables and so three dimensions. Tony said, “Nope. You can describe that same point on a sphere simply with longitude and latitude." The dimensions of any particular object are those that the object can be reduced to. And so a sphere is two-dimensional.
3. A surface can be produced at the intersection of multiple multi-dimensional objects. So what about the hundreds of variables Tony has to work with in order to describe his two-dimensional Del Pezzo surfaces of Degree 1? Well, a surface can live in a three-dimensional, or four-dimensional, or n-dimensional space. Imagine a one-dimensional line. It can live on a plane or in a three-dimensional world.

Now, you can also take higher dimensional objects and have them intersect to produce a lower-dimensional object. Take two planes in a three-dimensional space, with equations in the form of ax+by+cz+d=0. If these planes intersect, then you produce a line. So while a line is one-dimensional, you can produce it by equating functions of higher dimensions.

Tony, who works with functions that contain hundreds of variables, is doing something in the same vein – equating these functions in order to produce a two-dimensional object, a surface of a certain type. And this is what a lot of algebraic geometry is.  It is the process of studying geometric objects that are defined by polynomial equations. And at its most basic, it is the equating of multi-dimensional polynomials to describe a polynomial.

Comments

[Tony]

I like the post a lot! I thought that a surface was intuitively 2-d. Did you think I was simply abusing language?

Also, planes in 3-d have equations of the form:

ax + by + cz +d = 0 (no squares).

Love,
Tony

[Sarah Cove]

Tony,

Thanks for the equation fix. I've made the changes in the post.

What do you mean by abusing language? Is it a question relating to what caused my confusion?

I think that part of the confusion came from confusing a common understanding of a surface with the dimension of a non-flat surface.

When I thought of the surface of a table - the very top layer of it - I imagined a flat, two-dimensional surface. This is where the common understanding and your mathematical distinction were relatively interchangeable for a lay person.

Now let's take the surface of the ocean. There are many peaks and valleys produced by the waves and I thought of this as a three-dimensional surface. If I was asked to take a section of that ocean and map a point on that section, that point for me would consist of three variables: width, length, and height. And, combining my common understanding with this mathematical story, it seemed to be a three-dimensional surface to me.

Now, while higher dimensions were hard to visualize, I thought that one could abstract this to four or more dimensions. With the wave example, I thought that all one would have to do to make a surface four-dimensional would be to add a variable such as speed to the equation. Your point would then have three location variables and one speed variable to have it exist on a four-dimensional "surface" - i.e. the top layer of the ocean.

Does that make sense?

[J.Venkataramana Raju]

This is a comment on that last question-'Does that make sense?
Well the thing you are talking about-Waves explained with additional information, no longer does it remain as a surface then, it becomes an abstraction leading to the concept of a manifold...Geometry would then be only an intuition. For instance in Economics one would look at at a 10-dimensional object(say) to understand how your consumption (or expenditure rather) is affected by ten different variables in a given economy. [See the works of Donald Saari, Steve Smale etc in this direction]
As another different example the Space-time concept of Einstein is yet another manifold wherein the three location variables you are talking about are clubbed with the time variable to understand the dynamics of the Universe in a relativistic sense.