N-tuples in the context of Four-dimensional space

⭐ Core Definition: N-tuples

In mathematics, a tuple is a finite sequence or ordered list of numbers or, more generally, mathematical objects, which are called the elements of the tuple. An $n$ -tuple is a tuple of $n$ elements, where $n$ is a non-negative integer. There is only one 0-tuple, called the empty tuple. A 1-tuple and a 2-tuple are commonly called a singleton and an ordered pair, respectively. The term "infinite tuple" is occasionally used for "infinite sequences".

Tuples are usually written by listing the elements within parentheses " $( )$ " and separated by commas; for example, $(2, 7, 4, 1, 7)$ denotes a 5-tuple. Other types of brackets are sometimes used, although they may have a different meaning.

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N-tuples in the context of Four-dimensional

Four-dimensional space (4D) is the mathematical extension of the concept of three-dimensional space (3D). Three-dimensional space is the simplest possible abstraction of the observation that one needs only three numbers, called dimensions, to describe the sizes or locations of objects in the everyday world. This concept of ordinary space is called Euclidean space because it corresponds to Euclid 's geometry, which was originally abstracted from the spatial experiences of everyday life.

Single locations in Euclidean 4D space can be given as vectors or 4-tuples, i.e., as ordered lists of numbers such as $(x, y, z, w)$ . For example, the volume of a rectangular box is found by measuring and multiplying its length, width, and height (often labeled $x$ , $y$ , and $z$ ). It is only when such locations are linked together into more complicated shapes that the full richness and geometric complexity of 4D spaces emerge. A hint of that complexity can be seen in the accompanying 2D animation of one of the simplest possible regular 4D objects, the tesseract, which is analogous to the 3D cube.

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N-tuples in the context of Four-dimensional space

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⭐ Core Definition: N-tuples

N-tuples in the context of Four-dimensional