Bijective Mapping of an Interval to a Square

This Demonstration shows a discretized version of a bijective map between points in the unit interval and points in the unit square, with and . Each point on the square corresponds to a unique point on the interval and vice versa. This map shows that these one- and two-dimensional sets of points have the same cardinality.


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Bijective mappings from the interval to the square are necessarily discontinuous. For example, Snapshots 1 and 2 show the large change in the location of on the line corresponding to a small change in location near the center of the square. Similarly, Snapshots 2 and 3 show a small change on the line that corresponds to a large change on the square. The dark blue region of the square corresponds to points mapped to the dark blue region of the interval, that is, points on the interval that are less than or equal to .
This map uses the binary representation of each number. The map consists of splitting the binary digits of into two groups: the digits in odd locations form the binary digits of and the even locations form the binary digits of . The discrete version of the mapping uses 10 binary digits of , and hence five binary digits for each of and .
For example, the point maps to , and vice versa.
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