A measure of how completely a pattern fills a space
A calculation of an object’s fractional dimension
Coastline paradox
Lewis Fry Richardson observed
that the measured length
of a coastline or other natural border
depends on the length of the measuring stick.
Covering and packing
Construct a figure’s Hausdorff dimension
by counting open balls needed to cover a figure.
Construct a figure’s packing dimension
by counting open balls needed to fit inside a figure.
For each construction, use smaller and smaller balls.
Minkowski–Bouligand dimension
Count how many boxes of a grid
are needed to cover a figure
and calculate the limit
as the size of the box
approaches zero.
A.K.A. the box-counting dimension.
More than one, less than two
How do you measure
the dimension of straight line?
It has a length but no width.
But draw a bent line in a plane,
and define the path
between any two points on the line
to have an infinite length.
How can one say
it has no width?
Self-similarity
If you have
self-similarity,
when we zoom in
the same patterns appear.
This might not be
a good thing.
When the length of our measuring stick approaches zero,
the length of the coast of Great Britain approaches infinity,
which makes the length of a coastline seem indeterminable;
however, the Hausdorff dimension of the coastline of Great Britain
is 1.25, quite a bit more determinable.
When the length of our measuring stick approaches zero, the length of the coast of Great Britain approaches infinity, which makes the length of a coastline seem indeterminable; however, the Hausdorff dimension of the coastline of Great Britain is 1.25, quite a bit more determinable.
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