The similarity dimension of the snowflake curve is finite. This is arrived at in the following manner..... by the factor of 4 which is (1/2)^2. For a cube, acting similarly, decreasing its size by a factor of 2(HERE IT MUST BE 8 INSTEAD OF 2, RIGHT), results in smaller cubes each with the volume 1/8=(1/2)^3 of the "parent" cube. We can detect a commonality in these three examples......

If I am wrong, please tell me why?, thanks a lot

I think that the ambiguity can be resolved by replacing the word "size" with the word "side". For it's the side whose soze changes by a factor of 2.

Many thanks for pointing out the typo.