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.