For Solid ("Euclidean Geometry") Spheres

Mass ---> Perimeter^3 ...(Dimension D=3)

Double the size--->Octuple Mass


For crumbled paper balls (non-Euclidean objects)

 

M-->P^DM

SO:

LOG (M) = DM LOG(P) + C

 

DM: Fractal Dimension, 2<DM<3

M: MASS(Scale), P: PERIMETER(string)


NEWSPAPER

Perimeter (cm) Mass (g) Log(Perimeter) Log(Mass)
30.6 22,7 1.486 1.356
22.3 11.1 1.348 1.045 Fractal Dimension D= 2.4525
16.7 5.4 1.223 0.732
12.1 2.7 1.083 0.431
10.5 1.4 1.021 0.146


Printer Paper

Perimeter (cm) Mass (g) Log(Perimeter) Log(Mass)
13.4 4.95 1.127 0.694
10 2.45 1.000 0.389 Fractal Dimension D= 2.1241
6.5 1.2 0.813 0.079
5.3 0.61 0.724 -0.214
3.7 0.32 0.568 -0.495


Play-Doh Balls are expected to be "almost Euclidean" : D = 3

Play-Doh

Perimeter (cm) Mass (g) Log(Perimeter) Log(Mass)
28.7 473.5 1.458 2.675
26.5 385.8 1.423 2.586 Fractal Dimension D= 2.9047
22.9 265.7 1.360 2.424
17.6 115.4 1.245 2.062
13.3 52 1.124 1.716