sextuples

Sextuples are six-tuples, a concept in mathematics describing an ordered collection of six elements. They are a specific instance of an n-tuple and are used to specify a sequence with exactly six coordinates. When the entries come from a set A, the collection of all possible sextuples from A is denoted A^6, the 6-fold Cartesian product of A with itself. The name follows the common convention of pairing the Latin root for six with the suffix -tuple.

A sextuple is written as (a1, a2, a3, a4, a5, a6). The order of the components is essential, and each component may belong to different sets or be drawn from the same set. The entries need not be distinct unless the context imposes a constraint, so (1, 1, 1, 1, 1, 1) is valid if the elements allow it.

Two sextuples are equal precisely when all corresponding components match: (a1, a2, a3, a4, a5, a6) = (b1, b2, b3, b4, b5, b6) if and only if ai = bi for all i from 1 to 6. This distinguishes sextuples from unordered six-element sets, where the arrangement of elements is not recorded.

In applications, sextuples describe coordinates in six-dimensional space, specify inputs to multi-argument functions f: X1 × X2 × X3 × X4 × X5 × X6 → Y, and appear in algebraic structures that involve six-tuple data. The term is often used interchangeably with six-tuple or 6-tuple in mathematical literature, with “sextuple” emphasizing the six-element length.

See also: tuple, six-tuple, quintuple, septuple. Etymology: from Latin sex (six) plus the suffix -tuple.