Whidbey Island And Bagel Mathematics

re: Whidbey Island And Bagel Mathematics

Jens — Wed, 15 Dec 2004 13:12:00 GMT

Google is your best friend.

1. [http://www.kayakplace.com/places/plnchnl.htm] Manitoulin is the largest island in the Great Lakes, and, for that matter, the largest island in a freshwater lake anywhere.

2. [http://www.geocities.com/statesaz2/Michigan.html] Ryan Island in Siskiwit Lake on Isle Royale in Lake Superior is the largest island in a lake on an island in a lake (bet you have to reread that!).

re: Whidbey Island And Bagel Mathematics

Bruce Johnson — Wed, 15 Dec 2004 15:08:00 GMT

Being from Ontario, this is an easy one.

The largest freshwater island is Manitoulin Island in Lake Huron.

The largest island in a lake on an island in a lake is Treasure Island in Mindemoya Lake, also on Manitoulin Island.

re: Whidbey Island And Bagel Mathematics

BradC — Wed, 15 Dec 2004 15:36:00 GMT

Actually, according to http://users.erols.com/jcalder/SUPERLATIVESV2.html the largest island on an island is in Indonesia:

"Island on an island

The largest island on another island is Pulau Samosir, in Lake Toba on Sumatra, Indonesia. Samosir is 245 sq mi / 630 sq km, and is inhabited.

There are several places where islands on islands in turn have lakes with islands - triple islands."

re: Whidbey Island And Bagel Mathematics

Eric Lippert — Wed, 15 Dec 2004 17:21:00 GMT

Indonesia isn't on a lake.

re: Whidbey Island And Bagel Mathematics

Dave Anderson — Wed, 15 Dec 2004 17:45:00 GMT

"Ryan Island in Siskiwit Lake on Isle Royale in Lake Superior is the largest island in a lake on an island in a lake..."

But is it the longest island in a lake on an island in a lake, or even the largest island in a lake on the longest island in a lake?

Could it be the largest island in a lake on an island in the longest lake?

re: Whidbey Island And Bagel Mathematics

Eric Lippert — Wed, 15 Dec 2004 21:31:00 GMT

Now my brain hurts.

I had always heard that Treasure Island was the largest island-in-a-lake-in-an-island-in-a-lake. How big is it compared to Ryan Island, anyone know?

re: Whidbey Island And Bagel Mathematics

Brent Vukmer — Thu, 16 Dec 2004 03:10:00 GMT

Thanks for a great article, Eric.

Ometepe (in Lake Nicaragua) is supposed to be the world's largest island in a freshwater lake. It gets additional cool points for having two volcanos; one is still active .. the other has a lagoon inside it :)

re: Whidbey Island And Bagel Mathematics

Deadprogrammer — Thu, 16 Dec 2004 05:10:00 GMT

This reminded me of a Soviet era joke seemingly local to the city where I grew up, Odessa, Ukraine. It goes like this : what's the longest street in Odessa? Bebel Street. Why? (it's a smallish street) Because it begins in Odessa and ends in Siberia (due to #12 Bebel street aka local KGB headquarters).

re: Whidbey Island And Bagel Mathematics

Dr. Matt — Thu, 16 Dec 2004 17:47:00 GMT

Just as a terminology note, when you talk about the longest shortest path you're talking about what graph theorists call <i>diameter</i>. The usual construction goes something like:

* Define the distance between any two points (which you're doing using lengths of paths)
* Define the <i>eccentricity</i> of a point as the maximum of all distances between that point and other points in the graph.
* The diameter is then the maximum eccentricity. There's a related concept -- the radius -- which is the minimum eccentricity: the shortest longest shortest path, in a sense.

re: Whidbey Island And Bagel Mathematics

Phil Jollans — Thu, 16 Dec 2004 20:00:00 GMT

Two things occur to me about the length of an island.

The definition is clever, but seems to fail on the example you give of a ring shaped island. Your algorithm would result in only half of the 'true' length. For example, if the circumference is 100 miles, then the shortest longest distance between points would be 50 miles. Making a small break in the ring would double the length of the island.

The second point is that, if I am not mistaken, this is basically the mathematical concept of Limit Superior (lowest highest number) or Limit Inferior (highest lowest number) (http://en.wikipedia.org/wiki/Limit_superior_and_limit_inferior). Maybe you could submit your example to brighten up the rather dry matehmatical description. On the other hand, I could be wrong about this.

Phil

re: Whidbey Island And Bagel Mathematics

Eric Lippert — Thu, 16 Dec 2004 20:29:00 GMT

> Making a small break in the ring would double the length of the island.

Yes, it would. It would double the longest shortest distance between two points. A long, skinny island bent into a near-circle is LONGER than a long, skinny island squashed into a circle with a lake stuck in the middle.

You're describing a desirable property of the length metric, not an undesirable property.

> this is basically the mathematical concept of Limit Superior

Yes, you are absolutely right. The existence of a shortest path depends on the Axiom of Continuity, which states that bounded sets of numbers have superior and inferior limits, and thereby defines the real numbers.

re: Whidbey Island And Bagel Mathematics

Norman Diamond — Fri, 17 Dec 2004 01:08:00 GMT

> 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.

Well if it's longer than Guam then it's a moderately big whoop. (I think the "if" in that statement is longer though.)

re: Whidbey Island And Bagel Mathematics

Marcus Tucker — Fri, 17 Dec 2004 15:25:00 GMT

I would have thought that the length of an island is the length of the longest side of the smallest rectangle which bounds the island. But perhaps not.

re: Whidbey Island And Bagel Mathematics

Eric Lippert — Fri, 17 Dec 2004 15:34:00 GMT

Go to www.SpiralJetty.org.

Suppose that we detached Spiral Jetty from the mainland, making it an artificial island in the Great Salt Lake.

If we say that the length of Spiral Jetty is the length of the longest side of the smallest rectangle, we'd get a value FAR shorter than the 1500 feet it takes to walk from one end of the spiral to the other.

"Longest side of smallest rectangle" -- where by "smallest" you mean "the rectangle with the smallest longest side", not "the rectangle with the least area" -- works well for islands that are everywhere convex. It underestimates concave islands.

re: Whidbey Island And Bagel Mathematics

Marcus Tucker — Fri, 17 Dec 2004 17:00:00 GMT

Personally I don't think of the length of an island as being affected to its convexity/concavity. The shortest distance between the two furthest points by land is neither here nor there. Surely it is the distance between the two points on the coastline which are furthest apart as the crow flies, which is in fact identical to the rectangular bounding box's longest side?

Otherwise, following your definition you could theoretically have a spiral island (like Spiral Jetty) which is 500Km "long", while an island much bigger in size (such as Manhattan) would be "shorter". Surely that's madness?!

re: Whidbey Island And Bagel Mathematics

Eric Lippert — Fri, 17 Dec 2004 18:00:00 GMT

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.

re: Whidbey Island And Bagel Mathematics

Zac — Fri, 17 Dec 2004 22:50:00 GMT

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.

complex boundaries

Anton Sherwood — Sat, 18 Dec 2004 21:00:00 GMT

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.

re: Whidbey Island And Bagel Mathematics

Al Dunbar — Tue, 21 Dec 2004 04:19:00 GMT

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.

re: Whidbey Island And Bagel Mathematics

david dowd — Mon, 27 Dec 2004 10:47:00 GMT

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.

re: Whidbey Island And Bagel Mathematics

Eric Lippert — Mon, 27 Dec 2004 12:22:00 GMT

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?

re: Whidbey Island And Bagel Mathematics

david dowd — Sat, 01 Jan 2005 08:36:00 GMT

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).

re: Whidbey Island And Bagel Mathematics

Eric Lippert — Wed, 05 Jan 2005 19:47:00 GMT

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!

re: Whidbey Island And Bagel Mathematics

Eric Lippert — Wed, 05 Jan 2005 21:00:00 GMT

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.

re: Whidbey Island And Bagel Mathematics

Chris Moorhouse — Tue, 20 Sep 2005 08:48:34 GMT

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.

re: Whidbey Island And Bagel Mathematics

Scott Moonball — Sat, 29 Apr 2006 02:21:36 GMT

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.....

re: Whidbey Island And Bagel Mathematics

Reloysmiff@hotmail.com — Mon, 05 Mar 2007 00:10:51 GMT

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?

re: Whidbey Island And Bagel Mathematics

luckylooke — Sun, 25 Oct 2009 15:50:13 GMT

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