What do snowflakes, clouds, and this picture have in common? They’re all “fractals:” cool patterns where each shape is made of smaller versions of itself. In a cloud, the big fluffs and tiny fluffs look the same. One famous fractal is the Sierpinski triangle pattern we see here. 3 triangles make a bigger triangle with a hole in the middle. Then that whole thing becomes one of 3 triangles in an even bigger one…and it just keeps growing!
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.”