Question

Dear Mitch,

My husband, who is standing right behind me, is doing his best to keep from laughing because I'm writing to you with a question that I __think__ is a math question (or at least one of those "problem-solving" questions that they do now in math. My husband, though, he thinks the whole issue comes down to personal taste, and NOT math, and so I want to see who's right. And if* I'm *right then I would definitely appreciate it if you would explain the math to me, so I could understand it. Here goes:

We're having Thanksgiving dinner in our home, and we'll probably have a big group (we have 4 grown children, and two of those children have children of their own. Then there are my in-laws, my Dad, his wife, and maybe my brother and his fiancée. Maybe, but we'll have to see about those two. Anyway, so, last week I had a birthday and my husband knew I'd been dreaming about getting a new table for our dining area (the one we've been using came with the house when we moved in and it's too small and scratched up and was always supposed to be "temporary". Well, that 'temporary' period has lasted twenty years, and so now when my husband told me to pick one out for my birthday, I figured I could hold off one more week to decide exactly what I want and still have it in time for the Thanksgiving.

(Now my husband's* really* laughing because he thinks I'm rambling and won't be able to get my question out before the holiday.) So here it is:

I've always liked round tables more than square or rectangular ones. Most people I know prefer the rectangular ones. To me, round tables have a nice feel to them because there's not just one 'head' of the table and everyone faces each other more but I keep hearing from people that they think round tables are fine for everyday use, but not quite as formal as square or rectangular dining tables for holidays. (Oh, and now my husband's laughing even louder, but I can ignore that for now because I'm pretty sure I'm right on this one.)...Wait, hold on, he's asking me something. Oh, he said, "Think about it, have you ever attended a big formal dinner at a round table? Even on television or in the movies... can you think of a scene where that occurs?"

And I had to admit that none came to mind.

*Still*, my heart is set on a round table. So can you tell me if there is something in the mathematics of shapes that I am missing, anything that could make a difference once the thing were to arrive, because I'd much rather hear about it now than after we spend the money and bring it home...

I realize it's holiday time and you must be as busy as anyone, but if you would find a moment to give even a brief reply to this long question by Tuesday so one of the local movers will be able to deliver it before the holiday, it would be GREAT! Because I really do want to have Thanksgiving dinner on a new table...

So now that I've blabbered away, I guess the question is this: I say that to make an intelligent, well-informed choice, there is some form of math involved as well as personal taste, but my husband says that may be true for small round tables but for large round tables it's a different story and comes down to nothing but personal taste.

And so I've become curious and have been reading up on the pros and cons of one shape over another.

By the way, we really look forward to reading your answers, even my son who's in 5^{th} grade. Please help. Am I missing something, and if so, is it so important that I reconsider and find a way to be satisfied with a basic rectangle? And is this a dumb question?

Yours truly,

Sherri W.

Brooklyn Heights, NY

Answer

Dear Sherri,

Well! Let me first say this: Your question is the opposite of 'dumb'. The OPPOSITE. In fact, you hit upon a mathematical concept whose implications I know first-hand; I once spent a couple of weeks building a round table to replace the rectangular one in the dining area of a home I inhabited, and it was well worth it! Had it not been for an unfortunate robbery, I would still be enjoying my smoothly polished pine circle! Now, I am going to try to answer your sprawling question as concisely as I can, so that readers don't lose patience with our combined efforts. So here goes:

First, you are absolutely correct to sense that there are mathematical concepts involved. There is personal taste too, of course, but when it comes to furniture that you rely upon to function the hidden math becomes increasingly important.

You clearly are not someone who would be satisfied with a breathtakingly beautiful table that you couldn't easily use.

In as small a nutshell as I can manage: The main issue with tables is the relationship between its perimeters (how much of it is edge) and area (how much surface it has. Why? Because you can only seat people around a table's edge, and of all shapes circles have the most surface to edge ratio. Therefore, with a round table you are unlikely that you will ever end up with such a crowded surface that it becomes a challenge where to place serving dishes, etc. For small to medium-sized tables, this can be a huge advantage. The other advantage of circles over rectangles is, as you pointed out, each person can naturally face more of the other diners and, therefore, include a higher portion of the group in interactions.

HOWEVER, the reason that large formal events are rarely to be seen at large circular tables is this: Since the area is large compared to the perimeter, and does so more and more as circles growl, a large circle can quickly become deeper than anyone's arm-length. So you are left with lots of surface area, but it is in the center and no one can reach any of the food or dishes there!

One solution, for those who must have a round table and must have a large one is this: They are sometimes sold with large "lazy-Susan's" built in to the centers so that the food, etc., can be rotated around the table without anyone having to stand on anyone else's head. I have seen such a table myself at the late Jackson Pollack's house, and I recall one that long ago was at a Rockefeller vacation home in upstate New York. But some people would rather not have soups, etc., spun toward them while they are eating.

And that is that!

Round or Square, Happy Thanksgiving!

Hope this helps,

Mitch

P.S. There is an excellent children's book on this very subject. It is used in classrooms of students of ALL levels to explore the interesting relationship between area and perimeter. It is called SPAGHETTI AND MEATBALLS FOR ALL!, A MATHEMATICAL STORY, by Marilyn Burns, illustrated by Debbie Tilley, published by Scholastic Press. It tells the tale of how one family deals with increasing perimeter of their seating arrangement as more and more people show up for dinner. Check it out!