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 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=116S=60.8

The Creek is Dimensional

N=1 S=1 N=31 S=19.32 N=17 S=11.33

N=74S=32.85 N=121S=54.75

The Cloud is Dimensional

N=16 S=12.24 N=1 S=1 N=35 S=25.34

N=75S=52.75

The Crack is Dimensional

N=1 S=1 N=26 S=13.08

N=68S=30.41

N=129S=54.73

The Leaf is Dimensional

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