Much of math can be made to be as challenging as the student wants.
The leech tree is one example. The student can simply be asked to assign the correct weights to the appropriate edges. Or one could be asked to write and prove a theorem about leech trees.

The same is true of another famous math problem.

It is so with the Cantor Set. How is the Cantor Set formed ? Well, one begins with all of the numbers from 0 to 1 (inclusive). Then one throws out the middle third. Then one continues this same process of throwing out the middle third out of what remains. And one does this forever.

So, one would begin with the numbers from 0 to 1. We divide them into 3 sections. Thus, we have 0 to 1/3 , 1/3 to 2/3, and 2/3 to 1. Then, one throws out the middle third: 1/3 to 2/3. So, those numbers are gone. Now, we throw out the middle third of the sections that remain: 0 to 1/3 and 2/3 to 1. So, we divide them into thirds and throw out the middle third. For example, 0 to 1/3 becomes 0 to 3/9 and we would divide it in the following way: 0 to 1/9 and 1/9 to 2/9 and 2/9 to 3/9. Then, we throw out 1/9 to 2/9. And we do the same thing with the numbers from 2/3 to 1. So, that becomes 6/9 to 9/9. And those three sections are 6/9 to 7/9 and 7/9 to 8/9 and 8/9 to 9/9. We throw out the middle third, all the numbers from 7/9 to 8/9.

And we keep following this pattern.

So, what are some fractions that will never be thrown away no matter how many times we throw away the middle third ?

How many of these numbers can you name ?

How many numbers are left in this set after you keep throwing away the middle third infinitely many times ?

How would you prove that ?

Some great comments. Yes, we do not throw out the endpoints. The end points are all that remain. Now, we can create a mapping from the natural numbers (an uncountable set) to our endpoints or vice versa. After we show this, then we may conclude that the Cantor Set is uncountable . It does have measure zero.
I love the leech tree by the way.
Examples of fractions that remain are one third, 2/3, 1/9 ....

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#### Comment Preferences

• ##### I <3 Math diaries!!!(4+ / 0-)

The fact that there is no end to the number of numbers is what makes this set so cool...

Our country can survive war, disease, and poverty... what it cannot do without is justice.

• ##### Yay ! Uncountable Set ! Yay Cantor ! (2+ / 0-)
Recommended by:

I think this is accessible (naming members) but can be more challenging (proving theorems about the Cantor Set - although proving that it is an uncountable set is not very challenging).

I have always liked the Cantor Set.
:-)
Thanks !

• ##### You may find the leech tree of interest. (2+ / 0-)
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Subset of Graph Theory Subset of Discrete Math.

• ##### How about...(3+ / 0-)
Recommended by:
Vote4Obamain2012, rb608, mommyof3

If I plotted each step on an ordinary X-Y graph.
The X axis would be the number of calculations (steps) and the Y axis would represent the total remaining numbers.
At step one, X would be zero and Y would be 1.
At step two, X would be 1 and Y would be 2/3 of 1, or 2/3.
At step three, X would be 2 and Y would be 2/3 of 2/3, or 4/9.
At step four, X would be 3 and Y would be 2/3 of 4/9 or 8/27.
And so on and so on.
The resulting curve would approach zero on the Y axis but never actually reach zero because there would always be a value for Y no matter what value of X.

Big ol' fleas have little fleas
That cling to their backs
And bite 'em.
And little fleas have tiny fleas
And so on

The whole world of loneliness, poverty, and pain make a mockery of what human life should be. I long to alleviate this evil, but I cannot, and I too suffer. Bertrand Russell

• ##### Uh-oh...(3+ / 0-)

My 'proof' only means there would be an infinite number of numbers left after infinite calculations but not their names (values).
It's been a long time since I tried to do this kind of thing.
Good stuff to keep an old man's mind active though.
Thanks for a pleasant distraction from all the other things going on around here.

The whole world of loneliness, poverty, and pain make a mockery of what human life should be. I long to alleviate this evil, but I cannot, and I too suffer. Bertrand Russell

[ Parent ]

• ##### Well, there are an infinite number of numbers(2+ / 0-)
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So the limit of the Cantor Set still has an infinite number of numbers.

I recall reading (in James Gliek's "Chaos"?) that the Cantor Set resembles the pattern of data errors in telephone transmissions.

That is an amazing result, since at the time Cantor proposed this set of operations many mainstream mathematicians considered it a useless monstrosity.

This concept can be extended into 2 dimensions -the Serepinski Carpet is an example, athough the process can be applied to other regular polygons - and 3 dimensions (the Menger Sponge is an example)...

Derided as a "monstrosity", the Menger sponge in it's limit has zero volume, but an infinite surface area.  A real world example of this (obviously not extended to it's infinite limit) are the lungs that all of use have and (presumably) use...

Non-linear math is actually the math of nature.  The linear crap we get taught though grad school is really the "monstrosity".

• ##### For completeness, . . . (3+ / 0-)

. . . it's important to make clear that when you throw out a middle third, you do not throw out the endpoints of that middle third. That is, in the first step you do not throw out the points 1/3 or 2/3, even though you throw out everything between them.

And for purposes of praeteritio, I'm not going to point out what a terrific double entendre the subject of this comment is.

• ##### i love this(1+ / 0-)
Recommended by:
Vote4Obamain2012

alas, i'm enough years removed from my most advanced math classes to be wanting for even a hint of a clue for how to approach your problem. (the leech tree, also, is entirely new to me. apart from a semester of calculus, ditto stats, and some more stats in grad school, my post secondary studies were largely math-free)

but tangentially / apropos of nearly nothing, out of boredom earlier tonight, pre band rehearsal, waiting for my late bandmates ( ... musicians. sigh ...), i took out a pad, began division of fibonacci numbers by the one that preceded it, and drew a graph of the sine-like curve as it dampens and converges around phi.

"everybody's got something to hide except for me and my monkey." -john lennon

• ##### aha, it's coming back to me(1+ / 0-)
Recommended by:
Vote4Obamain2012

So any number that can be written decimal-style (but base 3) with only 0's and 2's. survives every cut. Since there's a one-to-one correspondence of that batch with all the sequences of 0's and 1's, which represent all the numbers in the interval [0,1] base 2, you have an uncountable infinity. Yet the measure is trivially zero.

Michael Weissman UID 197542