Measuring the Dimensions of Natural Objects

James Cahill

Background

• This project is aimed at measuring the dimensions of seemingly one-dimensional objects like cracks, rivers and coastlines. For objects such as rivers, we will ignore the area of the river and consider it to be a single line, as found on a map or as seen from really, really far away.

Process

• Box Counting Method– Place object in a single large box that leaves little

to no extra space on the ends.– Apply a grid with smaller boxes over object, and

count the number of boxes the object is in.– Repeat the second step with increasingly

increasingly smaller boxes, and least twice more.

The Math

• The number of boxes = N• The scale factor (s) = Big box / Little box• Plot graph with log(N) on the y-axis and log(s)

on the x-axis.• Create a best-fit line of the points. The slope

of that line is the dimension of the object.

But Does It Really Work?

N=1

S=1

N=5

S=5

N=10

S=10

N=20

S=20

The Line is 1.000 Dimensional!

Canacadea Creek, Alfred

Cloud in Sunny Alfred Sky

Crack in an Alfred Sidewalk

A Leaf from October Tree

SO WHAT WILL THEY BE?

N=1

S=1

N=17

S=11.02

N=38

S=20.84

N=116 S=60.8

The Creek is 1.166 Dimensional

N=1

S=1

N=31

S=19.32

N=17

S=11.33

N=74 S=32.85

N=121 S=54.75

The Cloud is 1.206 Dimensional

N=16

S=12.24

N=1

S=1

N=35

S=25.34

N=75 S=52.75

The Crack is 1.092 Dimensional

N=1

S=1

N=26

S=13.08

N=68 S=30.41

N=129 S=54.73

The Leaf is 1.222 Dimensional

1.166 1.2061.222 1.092

• The box counting method is a slow, painstaking, but all together fairly accurate way to find the dimensions of natural objects. The idea behind it is that we take and average of the smaller and larger N values, and hope that it smoothes out any wrinkles in the results. Unlike mathematical fractals like the Koch Curve, our rivers and cracks lose definition at high magnification, so there comes a point when smaller S values are completely pointless, as the boxes are smaller than the thickness of the line we are examining.

Concludatory