Dimension

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Dimension

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A line segment has one dimension, namely length.

Euclidean Dimension 1
length 1 unit length 2 units
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A square has 2 dimensions, length width.
Euclidean Dimension 2

• length 1
  length 2
• width 1 width 2
• Area 1 12 Area 4 22

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A cube has 3 dimensions. What are
they?
Volume 13 Volume 23 What is E, the
Euclidean dimension of a cube?
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A line

• A line has 1 dimension, length.
• It is infinitely long.
• It is also infinitely thin, but we give its
  drawing thickness to make it visible

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A plane

• A plane is a flat surface that is infinitely long
  and infinitely wide. It has 2 dimensions.

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Space

• Space has 3 dimensions

Infinite length
Infinite height (or depth)
Infinite width (or breadth)
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Euclidean Dimension E
Solid space
Plane Line Point
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There Are Other Types of Dimensions
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Fractal Dimension

• What does it look like?

• It is shown as an exponent

• That exponent is a generally a fraction

It is a fractional dimension
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D Fractal Dimension

• In 1977 Mandelbroit called fractional dimension
  (Hausdorff Besicovitch Dimension) a fractal
  dimension
• The Fractal Geometry of Nature (1977, 1983), p 15
  B,

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How do you find the fractal dimension?

• Because fractals are generally self-similar,
  we can use the
  self-similarity
  dimension. P. 37, The Fractal Geometry of Nature,
  1977,1983

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What does self-similar mean?

• Instead of comparing
• two separate shapes,

we compare a part of a shape to the whole.
Self-similar The part is the same shape as the
whole thing.
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• Let N the number of rescaled objects in the
  generator that replace the initiator.
• N

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• Let N the number of rescaled objects in the
  generator that replace the initiator.
• N 2

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• Let m how many times larger the figure in the
  initiator is than the the same figure in the
  generator.
• (Think m magnification)

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Find the fractal dimension D

• N mD
• N 2
• M 3
• 2 3D so 3D 2

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Find the fractal dimension D3D 2

• We know 30 1
• We know 31 3
• D must be between 0 and 1

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Using logs to find D

• Often our m is written as 1/r
• m 1/r
• N mD
• N (1/r)D
• D log N/log(1/r)

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Mandelbrots Definition of a Fractal
A fractal is by definition a set for which the
Hausdorff Besicovitch dimension strictly exceeds
the topological dimension. Mandelbrot, 1977,1983,
p 15