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Sums of Generalized Cantor Sets
Sums of Generalized Cantor Sets
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.
C
1/3
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.)
C
a
1-2a
[0,a]
[1-a,1]
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.
C
a
C
b
a
b
C
a
C
b
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.
y=c
f(a,b,c)=(1-c)+c={(1-c)α+cβ:α∈,β∈}
C
a
C
b
C
a
C
b
C
a
c=0
C
b
c=1
f(a,b,c)
C
a
C
b
C
a
C
b
c=1/2
f(a,b,c)
1/2(+)
C
a
C
b
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.
f(a,b,c)
f(1/3,1/3,1/2)
a<1/3
b<1/3
f(a,b,c)