Length, scale and dimension

Learning Goals

• Understand some of the issues underlying the definitions of arclength, area etc.

• See how the scaling properties of measure lead to the definition of fractal dimension.

Learning Objectives:

• Apply the covering definition of arclength using a physical model• Analyze the behavior of arclength at different length scales• Connect the power laws of scaling to the idea of fractal dimension

Properties of arclength

Why is this true, and how general is it?

1. Usual formula for straight lines2. Makes sense for smooth curves, polygonal curves,

and more complicated shapes.3. Preserved by rigid motions of the plane4. Expanded linearly by dilations

5. Area expands quadratically, volume cubically.

What definitions of arclength can apply to a wide variety of shapes?

Arclength of smooth curves

We learned this formula in calculus:

But it only applies to differentiable curves!

Arclength by disk-coverings1. Choose a length scale

2. Cover your arc by disks of diameter , where is as small as possible.

3. Let be your estimate for arclength.

4. Take a limit as 5. Note, this applies to any set, not just arcs! (but does the limit exist?)

• Our shapes: – a circle, – the Koch Snowflake, – the Coastline of Norway.

• Apply the circle covering definition using– Pennies (19mm)– Jujubes (10 mm)– Split-peas (7 mm)– Beads (5 mm)

• Are the arclength estimates stabilizing? • How does the covering number grow as element size

decreases?

Estimation activity:

Arclength estimate versus length scale:

4 6 8 10 12 14 16 18 20400

500

600

700

800

900

1000

1100

1200

1300

1400

CircleSnowflakeNorway

(mm

)

(mm)

Observations about the data

• Except for the circle, grows as decreases. • This means that can grow faster than linearly

in .

• How fast does it grow? • How do we measure that?

Interlude…

• The students go home with this charge: Collect more data on the Koch Snowflake or the coast line of Norway, and try to determine which functional relationship holds between N and epsilon.

Power law?

If there were a power law of the form

We could write

And taking logs,

Hence a slope on a log-log plot will help us estimate d. (slope 0 corresponds to d=1.)

Slope estimates d-1. Circle: d ~ 1. Snowflake: d ~ 1.27. Norway: d ~ 1.35

-1.4 -1.3 -1.2 -1.1 -1 -0.9 -0.8 -0.7 -0.62.5

2.6

2.7

2.8

2.9

3

3.1

3.2

CircleSnowflakeNorway

Summary• If the covering number grows like

we say the figure has fractal dimension d.

• The limit of

is arclength when d=1 , area when d=2, and “Hausdorff measure of dimension d” in general.

• The Hausdorff measure of an object of dimension d scales like when its diameter is magnified by

Challenges/Further thoughts• The Koch curve has dimension exactly log 4/log 3. Explain this to your

own satisfaction!

• A coastline need not have a well-defined dimension, but over a specific range of lengths this might be a good approximation.

• All our definitions were rough approximations. There are a number of non-equivalent variations.

– Look up: Hausdorff dimension, box dimension, Lebesgue measure.

• What properties of physical processes might lead to the formation of fractals?