Take a picture to a photocopier and set it to scale by a factor of three. Then all lengths in the picture triple in size (factor of three to the one-th power) and all areas increase by a factor of nine (three to the two-th power). But there are objects that scale by fractional powers!

## 9 thoughts on “What is Fractional Dimension and What is a Fractal? (TANTON Mathematics)”

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If you only do the erase step in the koch curve (you know, each time you only erace the middle third), you get this:

size = 2*size*(1/3)^d

And therefore d = 0.631

(cool)

Thanks, really cool! Im 14 and i love this

2:33 thank you i would not have been able to concentrate if you didnt delete that little bit of pink left god damn my anxiety attacks

Interesting that the Sierpinski triangle in 3D (using tetrahedrons instead of triangles) is 2 dimensional.

“to the one-th power” just means to the power of one. So “factor of three to the one-th power” means “factor of 3^1″ which is a factor of 3 (any number to the power of 1 is itself).

What does “factor of three to the one-th power” mean?

Actually, a factor of a half won’t change matters. Area scales – no matter what specific formula it follows for a particular shape – by k^2 and that is all that is being used.

E.g. If the sides of a square are doubled, area changes by factor 4.

If the sides of a triangle are doubled, the area changes by a factor of 4.

If the radius of a circle is doubled, the area chanegs by a factor of 4.

If the sides of an irregular dodecagon are doubled, area changes by a factor of 4.

with the triangles, wouldnt it affect the area because the area of a triangle if half x b x h so wouldnt the half change anything??

We have 3 dimensional photocopiers here…