Whidbey Island And Bagel Mathematics

Whidbey Island And Bagel Mathematics

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I was highly amused to read on Raymond Chen's blog the other day that mathematicians are hard at work solving the problem of how to most evenly distribute poppyseeds over a bagel. The reason I was highly amused was not just the whimsical description of what is actually a quite practical and difficult problem.

And yes, believe it or not, it is a practical problem; if you can figure out how to distribute points evenly around an arbitrary shape then you can use that information to develop more efficient computer algorithms to solve complex calculus problems that come up in physics all the time. There are also applications in computer graphics, I'd imagine; 3-D modeling frequently requires generating well-behaved finite approximations of a surface.

But I digress. The other reason I was highly amused is that Whidbey Island is the longest island in the United States.

OK, obviously I should explain that one, as the connection between mathematicians, bagels and Whidbey Island may not be immediately apparent.  But first, yes, I know that Whidbey Island is NOT the longest island in the United States -- not that anyone here in Washington State will admit to it.  As this page points out, by any reasonable measurement Whidbey Island is shorter than Long Island, Hawaii and almost two dozen islands off the coast of Alaska.  Whidbey Island's real claim to fame is that it is the longest island in the United States that is also in the Pacific Ocean but not part of Alaska or Hawaii -- big whoop.

Anyway, when I first moved here I did a bicycle trip around the southern lobe of Whidbey Island, and plenty of people told me lies about Whidbey Island being the longest island in the United States.  Which got me wondering how exactly that would be measured.

This led to a spirited email debate amongst my various math geek friends. The page I referenced above gives three definitions for island length, all vague and imprecise:

  • if the island is "narrow and straight" then "it's easy" to determine what the length is.
  • if the island is "rounded" then the longest distance to "opposite shores" is the length.
  • if the island is "bent", like Whidbey, then draw a bent line "down the middle" and measure that.

I didn't like any of these then, and I don't like them now.  There should be one unambiguous standard for length that doesn't rely upon vague and imprecise notions like "opposite shores", or "down the middle".

The standard should also work for really weird-shaped islands.  Consider for example an island which is a narrow ring.  Say, a circle with a circular lake in the middle.  None of the algorithms above are satisfactory. That island should be longer than an island with the same size and shape but no lake in the middle. In the island with the lake in the middle you can't walk in a straight line from one side to the other, so the island should be longer. Similarly, islands like Whidbey that are bent should be longer than straight islands.  Or, put another way, taking a long, straight island and bending it into a spiral should not change its length drastically.

Here's how I define the length of an island:  consider every pair of points on the perimeter of the island.  Clearly there are an infinite number of pairs of points, but we won't let that deter us.  Pick a pair.  Now consider every possible path, no matter how twisty-turny, that connects those two points, with the restriction that the path must be entirely on the island at all times.  Clearly, there's an infinite number of those too.

But of that infinite number of paths between the two points, at least one of them has to be the shortest.  (There could be two or more equal shortest paths, if, say, you could go both ways around an internal lake to get from point A to B, but whatever, pick one of them.)  Of course, for most of them it will be a straight line, but on weird shaped islands, it might have to bend here and there.

Now, find the pair of points where the shortest distance between them is as long as possible.  That is, find the two points on the shore where, even if you took the shortest path between those points, you'd still have to walk farther than if you took the shortest path between any other pair of points.

That distance, the length of the longest shortest path, is the length of the island. 

Of course, finding those points and the shortest path between them might be a difficult problem if the island is a really strange shape.  But that's just a technical detail that we can wave away; we have an impractical but formal definition, and that's what matters to me when I'm wearing my mathematician hat.

What does this have to do with the bagel algorithm?

Imagine all the possible distributions of, say, a thousand points around a "manifold".  (Which is just a fancy way of saying "shape" for our purposes; a more technical definition would take too long.)  Suppose the manifold is the surface of the earth.  Obviously there are an infinite number of such distributions, and an infinite number of them are going to be really, really sucky from the perspective of spreading them around evenly. 

Like, any configurations in which all of the points are on Whidbey Island, for example. 

For each of those distributions of a thousand points, pick every pair, and again, consider every path between each pair, where the path stays on the manifold. For every pair, find the shortest path, and then find the pair that has the shortest shortest path. 

If all the points in a given configuration are in Whidbey Island, maybe the shortest shortest path is one centimeter, maybe it's one meter, but you are guaranteed that the shortest shortest path is quite a bit smaller than the length of the island!

Now work out the shortest shortest path for every possible configuration of points.  At least one of those configurations has got to have the longest shortest shortest path, and that configuration is the desired configuration.  It's the configuration of points where the smallest possible distance between the two points that are closest together cannot be made larger without making some other pair of points become even closer together than the original pair.  You simply can't get any better than the configuration with the longest shortest shortest path. 

Of course, finding that configuration out of the infinite number of possibilities is a non-trivial problem. But clearly the "find the length of the island" problem is just a very simple version of the bagel problem.  It's got only two points and a Whidbey-Island shaped manifold, rather than a thousand points and a bagel-shaped manifold. If you can solve one, you can solve the other!

And while we're on the subject of island superlatives and ridiculous phrases like "the longest shortest shortest path", I should point out that Canada is home to not only the largest island in a freshwater lake in the world, but also home to the largest island in a lake that is on an island in a lake! Anyone care to hazard a guess at what they are?

  • Sure, it would be madness IF I WERE A CROW. Crows can travel through the three-dimensional manifold that is the Earth's lower atmosphere.

    I, being a human, tend to WALK from place to place on the two-dimensional manifold that is the solid surface of the earth. The question of how long something is therefore has to do with the farthest I could possibly have to walk.
  • Ryan Island is the largest island in the largest lake on the largest island in the largest (freshwater?) lake (Superior).

    Treasure Island is the largest island in a lake on an island in a lake (Huron).

    Very important distinctions, especially for people talking about longest shortest shortest paths.
  • I'm reminded of the alleged existence of a third-order enclave in Cooch Behar: 0.7 hectare of India within a piece of Bangladesh within a piece of India surrounded by Bangladesh.
  • Nice arguments. Reminds me, though, of other arguments that spring basically from a simple concept (i.e. "longest") being applied where it is not applicable (i.e. "island"),i.e.:

    - The greenest pasture. Think about it, anyone owning a pasture will want to adopt a definition that makes his (hers?) the greenest. But, in reality, the claim of a pasture being the greenest, is meaningless. But no more so than the longest island.

    Although most people can understand what is probably meant by these things, that is more an indication of our abilities with "fuzzy logic", than anything else. Humans have an innate ability to understand much that we are unable to define with exactitude. At the same time, we seem to have an affinity for trying to come up with an adequate definition for things that, owing to our aforesaid abilities, need no such thing.
  • How to measure an island of any shape. I'm NOT a mathematician, but surely the easiest way would be to imagine a circle which just touches two points on a real island's perimeter (i.e., not a circular one) and does not include any part of it. Is not the circle's diameter the maximum length of the island?
    Path lengths should have no bearing on the concept of overall length, even for 'bent' islands or islands with lakes in them.
  • Sure, that's an EASY way, it just doesn't measure anything I'm interested in.

    You say that path length "has no bearing". I say that it's the only important thing. Clearly you mean something different by "length" than I do. By "length" I mean "if I were put on this island at point A, how long would I have to walk to point B if I walked the most efficient route?"

    On more practical grounds, I assume you mean the SMALLEST circle that contains the entire island. OK, then consider an island in the shape of an equilateral triangle. By your measurement of length, the length of the island is LONGER than the longest straight-line path you can walk on the island!

    That certainly seems wrong, doesn't it?

  • Yes, you're quite right about the equilateral triangle (and I'll bet there are real islands approximating this), so the rectangular box is a better all-inclusive method (though you'll not agree with that, of course). My point NOW is that the argument began about the length ranking of a real geographic island (Whidbey) not a theoretical topological one - so the common man's definition of length (whatever that may be)should be used, not PATH LENGTHS which are your bag and might include infinitely long spiral ones. Even if you would limit path lengths to the real world, you should certainly include the third dimension (climbing mountains versus traversing flatlands).
  • I don't see how my definition includes any infinitely long paths. I don't think there are any islands of finite area which have infinitely long paths, unless of course the islands in question are allowed to have infinitely thin fjords or peninsulas.

    The question of a third dimension is a good one, which is covered by my definition but NOT covered by most definitions. Most definitions consider only the outline of the island when determining length. You'll note that I was careful to call out that the path must be on the surface of the island.

    The rectangular box method -- the longest side of the smallest rectangle which fits the island -- has a number of failings, not just for concave islands.

    Consider for example, a long, narrow strip of an island, say, 14.1 kilometres long. Clearly the longest side of the smallest rectangular box that contains that island will be 14.1 km.

    Now consider another island in the shape of a square, 10 km on a side. By the rectangle metric, the length of that island is 10 km. But our original island would surely fit entirely into the square island, as the diagonal! How is it that you can make an island, everywhere convex, BIGGER, and it gets SHORTER? That doesn't make any sense!

  • Now that I think about it a bit more, one could come up with a spiral manifold that was everywhere of finite width and yet of infinite length.

    I have no problem with that. Such an impossible island is infinitely long. Just like an infinitely long thin strip of an island is infinitely long.
  • Taking the third dimension into account introduces some issues concerning those mountains, though. When given two islands of identical outline, does it make sense that one would be longer than the other merely by reason of it having a taller mountain in the middle? I think not. It also occurs to me that, by your as-the-donkey-trundles path definition, an island can have it's length altered by means of digging a tunnel. We'd have to consider then if a record-holding island should loose it's status by dint of having a convenient highway tunnel, or even a bridge connecting two points. And what if the bridge is of the folding-up variety?

    The circular island with the lake in it's middle has similar problems. Given two islands of identical diameter, why should one be considered longer because it has a larger lake in the middle? If we were to install a hydroelectric project on an island, why should we consider it's length to have changed?

    To get around these issues, and other impediments like variations in the curvature of the earth (which unfairly impacts some big Canadian islands, I might add), we need some way of having that surface path ignore the intervening topography.
  • Long Island is longer no matter how you slice it.  However, the Supreme Court by a count of 9-0 declared Long Island to be a peninsula and not in fact a "true" island.  It looks like an island to me, but.....
  • What if the an almost complete ring shaped island is connected at one end by cave or tunnel, to an island that is a complete ring?

  • Eric did not know that You are interested in differential manifolds, metrics and so on:)

    so what is the diameter (which You renamed as the length of an island...) of moebius band?:)

    a pleasant read for my mind on a weekend. as allways

    You should write some book really

    luke

    Thanks for the kind words. I've written several books and have another one on the back burner that I'm considering doing. Maybe this winter. -- Eric

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