(insert Layton's Mystery Journey-like case intro here)
Sector Intersection II

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Zerro, do we REALLY have to do this? And what's with this bag of-

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(stops Three from touching the content inside the bag) DON'T. TOUCH. THAT.

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Why?

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They're FRACTALS.

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What's a fractal?

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A fractal is a curve or geometrical figure, each part of which has the same statistical character as the whole. They are useful in modelling structures (such as snowflakes) in which similar patterns recur at progressively smaller scales, and in describing partly random or chaotic phenomena such as crystal growth and galaxy formation.

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:O

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:O

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...I have no idea what you said.

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Okay. Let's not talk about that and start putting these things back where they belong.

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And where is that?
Sector III

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HERE.
(camera pans out to show the entire Sector III)

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:O

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:O

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Stop making that face!

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There are three stands, so that means three fractals are required!

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How'd you know we need fractals?

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There's this video tape here that says "ƨlɒɈɔɒɿʇ ϱniƨƨim ǝʜɈ ʇo ǝƨɒɔ ǝʜɈ". I took a while to decode it and I got "the case of the missing fractals". See? (holds up Three's mirror next to the video tape to reveal "the case of the missing fractals")

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:O

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:O

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:O

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Great, now I'M making that face.

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I guess you can say...

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Anyway, let's just watch the video.
(Zerro inserts tape into VCR, which is connected to a TV in the sector)

Click here if you want to see the video yourself!

(extremely loud intro starts)

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:O

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:O

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:O

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:|
MANNY BROT: PRIVATE EYE
~in~
THE CASE OF THE MISSING FRACTALS

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Told ya.

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It was a night like any other night, except here I was climbing the platonic peaks like Romeo on a second date.

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(gets hit by a tetrahedronal rock) Ugh.

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I was there for the dame. She had eyes like imaginary numbers and curves that went on forever.

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Said she wanted to go home. Said I could help. Said the pay was good.

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Didn't say anything about climbing a...

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Who's there?

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Manny Brot, private eye.

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What are you doing here?

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A pretty number sent me to find a stolen dingus.

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Well, to enter the cave, you must answer my riddles three.

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THAT'S MY NUMBER! MAYBE I AM THE-

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(slaps Three)

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What was it with riddles, and why do they always come in threes?

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Is it an egg?

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No. Why would it be an egg?

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It's usually an egg.

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(shakes her head) What can hold in my hand, but has nearly zero area?

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...

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Is it a dodo egg?

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IT'S NOT AN EGG!

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I took out the rock that had nearly brained me before and gave it a hard ponder.

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The size of the rising bump on my conk said to me that this thing had area, and a lot of it.

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But what if I carved out a triangle from this side here? As any mook could see, this triangle had a quarter of the area of the full triangle. I did the same thing again with each of the smaller triangles. Again, a quarter of the remaining area -- gone. And I just kept going. After an infinite number of cuts, I was satisfied that my triangle had nearly zero area. A bounded shape with almost zero area.

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Now, it's not often that I surprise myself, but I own two mitts had created something crazy, and new.
[PAUSE]

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I think I have that fractal...

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What?

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HERE IT IS! (holding the fractal) SIERPINSKI'S TRIANGLE!
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(goes back to the three stands) Each stand has a number, labelled "!", "@", and "#".

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I assume that you put this into the stand with the exclamation point, because the "!" is what you get when you hold the Shift key, and then press "1", on a standard keyboard.
(the "!" lights up)

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I think that indicates that we got the right fractal!

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Now, where was I?
[PLAY]

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Very good. (ahem)

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Now, show me a shape with finite area, but an infinitely long perimeter.

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Let me get this straight. If I want to make a snip in the border of this shape, smooth it out, and lay it on the ground...

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...it would go on for-

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Wait 'til I'm through, and then you can talk.

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It would go on forever.

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Are you through?

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Yeah.

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So show me that shape then.

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Hmm... I hadn't been this stuck since the Rubik's Cube fiasco of '58.
[PAUSE]

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Wait, hold on. The Rubik's Cube hasn't been invented until 1974, so there COULDN'T have been a Rubik's Cube fiasco in 1958! Weird. 'Kay, let's continue...
[PLAY]

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All the shapes I knew had perimeters. Circles: 2πr. Triangles: sum of their sides.
(it starts snowing)

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What's this?

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An angle.

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An angle from heaven.

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What if I were to pinch each side, like so. A third of the way through, just so. And do it again, and again, and again. After each pinch, the perimeter got a third longer because where there had been three line segments, now there were four. As for the area, every pinch made more triangles, that's true. But those triangles were getting smaller and smaller. You can say that the area was converging, approaching a fixed number, while the perimeter was just getting bigger and bigger, uncontrollably ballooning like an overindulgent birthday clown. After infinity pinches, flimflam, there it was: finite area, but infinite perimeter.

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Now that is a piece of work.
[PAUSE]

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That fractal, I have... THE KOCH SNOWFLAKE!


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(places the Koch snowflake into the stand labelled "@", and the symbol lights up)

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Now we need the third and final fractal. But which one?

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You'll have to find out... IN THE NEXT EPISODE.
TO BE CONTINUED

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I hate cliffhangers.

Click here if you want to see part two.

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