More Experimental Topology and Experiments in Topology

I visited my friends Peter and Liz (I stayed a few nights) and came back with a laundry list of things to post about:

1) They have some really cool polyhedra and mathematical quilts, so sometime I am going to have to go over there with a camera and click some photos. Peter’s latest quilt project is one tiled with spidrons, which is in the design phase now.

2) Peter was a little upset that I failed to credit him for pointing out to me originally that and expanded base is (see synchronicity) and asked if there were any other examples of this. Anyway, thanks, Peter, for pointing this out and inspiring so much recreational math.

3) I finally carried out that idea I had that I mentioned back in the post more experimental topology about making a five pointed star. It took some trial and error (hey! that’s what experimental topology is all about. Right?) but here is what I came up with:

Start with ten strips of paper, with one end cut into points. The points should come to an angle of approximately .

Draw a line down the center of each and tape them together so that the tips are all touching:

Now start taping opposite ends together, like so:

When you have four of the five opposite pairs taped together, the last one needs one full twist. If you’ve been proceeding in a clockwise direction when taping opposite pairs together, you should make one full twist in the last pair by turning the strip closest to you in a counter-clockwise direction. Anyway, in the end you should get something like this:

Now, cut along all of those center lines, and what do you get?

A mess, but if you untangle the mess, you should get:

A pair of stars that are linked!

Topologically this is the Hopf link.

4) Peter also loaned me a couple of books:

Experiments in Topology

I haven’t read much yet, but it has some cool stuff, like if you have a strip of paper, one inch thick, what is the shortest Moebius strip that you could make?

Also he lent me

Goedel, Escher, Bach: an Eternal Golden Braid

(gosh, that’s a really small image of the book, but, oh well). I’m about a fifth of the way into this book. Excellent so far. Won the Pulitzer Prize! I’m learning lots about the works of Goedel, Escher and (believe it or not) Bach.

That’s all for now, but soon I will have to blog about my visit to my kids’ classes for career day, which was last week….Stay tuned.

Thinking about infinity

As promised, a post about how natural it is for humans to think of infinity (see also My Grandma).

Let me say first that there are some very big and important questions about infinity that mathematicians and philosophers are still trying to answer (and I’m just talking about mathematical infinity here!). Unfortunately I have nothing to say here about those questions. This post is very naive.

I hadn’t planned on doing it so soon but John Lienhard did a piece on infinity on the Engines of Our Ingenuity. So it seemed like an appropriate time.

As I said, the question that I’m interested in here is: How natural is it for people to think about infinity? I want to consider three cases: my son, myself and my grandmother.

My Son

One thing that has me thinking about thinking of infinity is that my son has been talking about infinity lately. He is 6 years old and in kindergarten. He has asked me several times “Dad, is infinity a real number.” This catches me off guard and I usually mumble “umm…no. It’s an extended real.” Of course this is not at all helpful and usually prompts a “Huh?” from him.

I just asked my 4-year-old daughter what infinity is. She said that her teacher hasn’t told her yet.

What I wonder is, where did he even learn the word infinity? Where and when did he learn what it means? I don’t remember ever specifically teaching him what infinity is. Then again, I do remember him asking me questions about what the biggest number is. I told him that if you had a number that you thought was biggest you could always add one to it. . . which leads me to. . .

Myself

I don’t remember where or when I learned the words infinity or infinite. On the other hand I remember exactly where and when I learned the word finite. It was in the fifth grade. I immediately understood what it meant: not infinite. The word finite itself seemed very funny to me.

I know that I had started thinking about infinity much sooner. When I was, oh probably 6 or 7, I guess. I remember trying to come to grips with the following paradox: it was absolutely impossible for me to comprehend space going on forever and it was absolutely impossible for me to comprehend space “stopping” at some point (what would be just beyond that point?). It seemed that one of these alternatives had to be true (I hadn’t yet learned that there are compact, boundaryless 3-manifolds).

So how natural is it for us to think about infinity? First of all, from studies that psychologists have done with newborns it appears that we are born thinking about numbers (at least the numbers “one.” “two,” and “three or more.” I don’t think that thinking about infinity comes that early.

So here is my hypothesis (prompted by the above two anecdotes): at some point around 6 years of age it is natural for people to think “do numbers (or space) ever end? If they do, why can’t you just go one step further? But it seems impossible that they could go on forever. The resolution of this paradox, for most of us at least, is to construct for ourselves and accept the concept of infinity.

I could be totally wrong on this. I am no psychologist or philosopher or anything or the sort.

My Grandma

Of course the one monkey-wrench in the works of my hypothesis is my grandma. I can’t be sure whether she understood the concept of infinity or not (see My Grandmother).

So this post is more of a question and request: What are your fist memories of thinking about infinity? How natural do you think it is for us to think about infinity? Please share your ideas.

An Artist-Mathematician

I recently learned that Eric, a friend and colleague of mine, was an artist for 7 years before getting a Ph.D. in math. He was a sculptor and printmaker. Here is some of his work:

Since learning that he was an artist I’ve only got a brief chance to pick Eric’s brain at a barbecue. If I’m remembering correctly, he said that his undergraduate degree was in math. Then he worked as an artist for 7 years, which lead him back to math and graduate school.

In Eric’s own words “Most of my artwork was designed using random numbers and elementary mathematics.”

Maybe we can get Eric to tell us more sometime. . .