mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, an interleave sequence is obtained by merging two

sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...

s via an

in shuffle The faro shuffle (American), weave shuffle (British), or dovetail shuffle is a method of shuffling playing cards, in which half of the deck is held in each hand with the thumbs inward, then cards are released by the thumbs so that they fall to th ...

. Let

S

be a

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

, and let

(x_i)

and

(y_i)

i=0,1,2,\ldots,

be two

s in

S.

The interleave sequence is defined to be the sequence

x_0, y_0, x_1, y_1, \dots.

Formally, it is the sequence

(z_i), i=0,1,2,\ldots

given by :

z_i := \begin x_ & \text i \text \\
 y_ & \text i \text  \end

Properties

* The interleave sequence

(z_i)

is convergent

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...

the sequences

(x_i)

and

(y_i)

are convergent and have the same limit. * Consider two

real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...

s ''a'' and ''b'' greater than zero and smaller than 1. One can interleave the sequences of digits of ''a'' and ''b'', which will determine a third number ''c'', also greater than zero and smaller than 1. In this way one obtains an

injection Injection or injected may refer to: Science and technology * Injective function, a mathematical function mapping distinct arguments to distinct values * Injection (medicine), insertion of liquid into the body with a syringe * Injection, in broadca ...

from the square (0, 1)×(0, 1) to the interval (0, 1). Different radixes give rise to different injections; the one for the binary numbers is called the

Z-order curve In mathematical analysis and computer science, functions which are Z-order, Lebesgue curve, Morton space-filling curve, Morton order or Morton code map multidimensional data to one dimension while preserving locality of the data points. It i ...

or Morton code..

References

{{PlanetMath attribution, id=449, title=Interleave sequence Real analysis Sequences and series