It was a night like any other night, except here I was climbing the platonic peaks like Romeo on a second date. (ugh) I was there for the dame. She had eyes like imaginary numbers and curves that went on forever. Said she wanted to go home. Said I could help. Said the pay was good. Didn't say anything about climbing a... Voice: "Who's there?" Manny Brot: "Manny Brot, private eye." Voice: "What are you doing here?" "A pretty number sent me to find a stolen dingus." Voice: "Well, to enter the cave, you must answer my riddles three." What was it with riddles, and why do they always come in threes? "Is it an egg?" "No. Why would it be an egg?" "It's usually an egg." "What can I hold in my hand, but has zero area?" "Is it a dodo egg?" "It's not an egg!" I took out the rock that had nearly brained me before and gave it a hard ponder. The size of the rising bump on my conk said to me that this thing had area, and a lot of it. 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 zero area. A bounded shape with zero area. Now, it's not often that I surprise myself, but my own two mitts had created something crazy, and new. "Very good. (ahem) Now, show me a shape with finite area, but an infinitely long perimeter." "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 ... " "It would go on for ... " "Wait 'til I'm through, and then you can talk. It would go on forever." "Are you through?" "Yeah." "So show me that shape then." Mmm ... I hadn't been this stuck since the Rubik's Cube fiasco of '58. All the shapes I knew had perimeters. Circles: 2πr. Triangles: sum of their sides. What's this? An angle. An angle from heaven. 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 could 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. Now that is a piece of work. "Oh, you're good. (ahem) Riddle three: Show me a picture that if I magnify it under my microscope, I'll keep seeing the original picture, no matter how much I zoom in." "You're a strange little man." "Thank you." I was out of ideas, so I looked at my muse, my complex Dora. Voice: "Who's the dame?" And then it hit me. "She's a heart breaker, my fractal femme fatale. Will she do?" "Yes, she'll do just fine." (lightning) It was dark, and at first I thought the cave was empty, but then I noticed: the box. The dame had played me like a triangle. She had told me she wanted to go home. (Lightning) What she really wanted was to bring her home here. The fractals spread everywhere. Most of them the same no matter how deep you looked at them, like Dora's mugshot. Some had infinitely long perimeters, others were objects with no area or volume, all of them created through infinite repetition. So, you wanted to know what fractals are? Well, kid, they're the stuff that dreams are made of. (Music)
