Hammersley Point Set
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18-3-2021
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Hammersley Point Set
The two-dimensional Hammersley point set of order 
is defined by taking all numbers in the range from 0 to 
and interpreting them as binary fractions. Calling these numbers 
, then the corresponding 
are obtained by reversing the binary digits of 
. For example, the 
for the Hammersley point set of order 2 are given by 
, 
, 
, and 
, or (0, 1/2, 1/4, 3/4). Reversing the bits then gives the second component, leading to the set of points (0, 0), (1/2, 1/4), (1/4, 1/2), and (3/4, 3/4).
The point set can be generalized by truncating 
bits from each coordinate. The result is known as a binary 
-net, with 
representing the dimension (in this case, 
). Examples of these sets are illustrated above for 
, 
, and various degrees of truncation.
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