The Cantor set has many interesting and initially unintuitive properties: it is a fractal, perfect, nowhere-dense, totally disconnected, closed set of measure zero. Yet two such sets can be combined to give a simple interval.
The (standard) Cantor set
is the limit of the following iteration. Starting with an interval, take out its middle third, leaving two closed intervals at each end. Repeat on each subinterval; then continue to any depth, doubling the number of intervals each time.
This can be generalized to
by taking out the fraction
at each stage; this leaves the intervals
and
at the first stage. (Other generalizations are to take out the second and fourth fifths at each stage, etc., or to use a sequence of fractions, but not here.)
Two Cantor sets
and
using the fractions
and
are constructed one unit apart. All of the points of
are joined to all of the points of
by lines; these sets of lines are approximated by overlapping bands (parallelograms) that get thinner and more numerous as the depth increases.
The cross-section of the bands by the horizontal line
give approximations to the set
, which is
for
and
for
. So this set
is a kind of blend of
and
or a convex interpolation between
and
. For
,
is
, the average of the two sets, or its scaled sum.
What is the nature of
? As you can see here (or in the Demonstration
The Sum of Two Cantor Sets),
is the whole unit interval. If
or
, then
appears to be fractal.
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