When Triangles Make You Dizzy

by Laura Overdeck

What do snowflakes, clouds, and this picture have in common? They’re all “fractals.” Fractals are cool patterns where each shape is made of tiny versions of itself. Then those new big shapes become the tiny shapes in an even bigger pattern, and so on. That’s why it’s hard to tell how close or far away the clouds are: big fluffs and tiny fluffs look the same. One of the most famous fractals is the Sierpinski triangle pattern we see here. 3 triangles are lined up to make a big triangle with a hole in the middle. Then that whole thing becomes one of 3 triangles in an even bigger one…if you have enough paper and crayons, you could cover your whole town!

Wee ones: How many sides does a triangle have?

Little kids: If you grab 3 little triangles, how many sides do they have in total?  Bonus: How many of those sides face outward to make the bigger triangle sides?

Big kids: If you make a big triangle out of 3 little triangles, each of which contains 3 smaller triangles, each of which contains 3 teeny triangles, how many teeny triangles do you have?  Bonus: How many triangles are there in total, including the big one?

The sky’s the limit: 6,561 teeny triangles work out perfectly to make one really giant triangle. How many rounds of triangles will it have? You can count the first “layer” as the 3 biggest triangles making the whole outline of the pattern.

Wee ones: 3 sides.

Little kids: 9 sides.  Bonus: 6 sides, because 3 face inward.

Big kids: 27 teeny triangles.  Bonus: 40 triangles in total, since you add on the 9 smaller ones, the 3 bigger than those, and then the 1 biggest.

The sky’s the limit: 8 layers, since 6561 = 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3. That’s also called 3 to the 8th “power.”

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