Infinity symbol

The infinity symbol (∞) is a mathematical symbol representing the concept of infinity. This symbol is also called a lemniscate,Template:R/superscript after the lemniscate curves of a similar shape studied in algebraic geometry,Template:R/superscript or "lazy eight", in the terminology of livestock branding.Template:R/superscript

This symbol was first used mathematically by John Wallis in the 17th century, although it has a longer history of other uses. In mathematics, it often refers to infinite processes (potential infinity) rather than infinite values (actual infinity). It has other related technical meanings, such as the use of long-lasting paper in bookbinding, and has been used for its symbolic value of the infinite in modern mysticism and literature. It is a common element of graphic design, for instance in corporate logos as well as in older designs such as the Métis flag.

The infinity symbol and several variations of the symbol are available in various character encodings.

History

The lemniscate has been a common decorative motif since ancient times; for instance, it is commonly seen on Viking Age combs.Template:R/superscript

The English mathematician John Wallis is credited with introducing the infinity symbol with its mathematical meaning in 1655, in his De sectionibus conicis.Template:R/superscriptTemplate:R/superscriptTemplate:R/superscript Wallis did not explain his choice of this symbol. It has been conjectured to be a variant form of a Roman numeral, but which Roman numeral is unclear. One theory proposes that the infinity symbol was based on the numeral for 100 million, which resembled the same symbol enclosed within a rectangular frame.Template:R/superscript Another proposes instead that it was based on the notation CIↃ used to represent 1,000.Template:R/superscript Instead of a Roman numeral, it may alternatively be derived from a variant of Template:Char, the lower-case form of omega, the last letter in the Greek alphabet.Template:R/superscript

Perhaps in some cases because of typographic limitations, other symbols resembling the infinity sign have been used for the same meaning.Template:R/superscript One paper by Leonhard Euler was typeset with an open letterform more closely resembling a reflected and sideways S than a lemniscate (something like S),Template:R/superscript and even "O–O" has been used as a stand-in for the infinity symbol itself.Template:R/superscript

Usage

Mathematics

In mathematics, the infinity symbol is typically used to represent a potential infinity.Template:R/superscript For instance, in mathematical expressions with summations and limits such as

$$\sum _{n=0}^{\mathrm{\infty }}\frac{1}{2^n}=\underset{x\to \mathrm{\infty }}{\lim }\frac{2^x-1}{2^{x-1}}=2,$$

the infinity sign is conventionally interpreted as meaning that the variable grows arbitrarily large towards infinity, rather than actually taking an infinite value, although other interpretations are possible.Template:R/superscript

When quantifying actual infinity, infinite entities taken as objects per se, other notations are typically used. For example, ${\mathrm{\aleph }}_0$ (aleph-nought) denotes the smallest infinite cardinal number (representing the size of the set of natural numbers), and $\omega$ (omega) denotes the smallest infinite ordinal number (representing the ordering of the natural numbers). However, in geometry and topology, the infinity symbol may be used to represent a point at infinity, especially when there is only one such point under consideration. This usage includes, in particular, the infinite point of a projective line,Template:R/superscript and the point added to a topological space to form its one-point compactification.Template:R/superscript

Other technical uses

Minolta MD 4.5/300mm IF lens

Side view of a camera lens, showing infinity symbol on the focal length indicator

In areas other than mathematics, the infinity symbol may take on other related meanings. For instance, it has been used in bookbinding to indicate that a book is printed on acid-free paper and will therefore be long-lasting.Template:R/superscript On cameras and their lenses, the infinity symbol indicates that the lens's focal length is set to an infinite distance, and is "probably one of the oldest symbols to be used on cameras".Template:R/superscript

Symbolism and literary uses

Strength tarot card, depicting a woman crowned by an infinity symbol, holding shut a lion's mouth

The infinity symbol appears on several cards of the Rider–Waite tarot deck.Template:R/superscript

In modern mysticism, the infinity symbol has become identified with a variation of the ouroboros, an ancient image of a snake eating its own tail that has also come to symbolize the infinite, and the ouroboros is sometimes drawn in figure-eight form to reflect this identification—rather than in its more traditional circular form.Template:R/superscript

In the works of Vladimir Nabokov, including The Gift and Pale Fire, the figure-eight shape is used symbolically to refer to the Möbius strip and the infinite, as is the case in these books' descriptions of the shapes of bicycle tire tracks and of the outlines of half-remembered people. Nabokov's poem after which he entitled Pale Fire explicitly refers to "the miracle of the lemniscate".Template:R/superscript Other authors whose works use this shape with its symbolic meaning of the infinite include James Joyce, in Ulysses,Template:R/superscript and David Foster Wallace, in Infinite Jest.Template:R/superscript

Graphic design

The well-known shape and meaning of the infinity symbol have made it a common typographic element of graphic design. For instance, the Métis flag, used by the Canadian Métis people since the early 19th century, is based around this symbol.Template:R/superscript Different theories have been put forward for the meaning of the symbol on this flag, including the hope for an infinite future for Métis culture and its mix of European and First Nations traditions,Template:R/superscriptTemplate:R/superscript but also evoking the geometric shapes of Métic dances,Template:R/superscript, Celtic knots,Template:R/superscript or Plains First Nations Sign Language.Template:R/superscript

A rainbow-coloured infinity symbol is also used by the autism rights movement, as a way to symbolize the infinite variation of the people in the movement and of human cognition.Template:R/superscript The Bakelite company took up this symbol in its corporate logo to refer to the wide range of varied applications of the synthetic material they produced.Template:R/superscript Versions of this symbol have been used in other trademarks, corporate logos, and emblems including those of Fujitsu,Template:R/superscript Cell Press,Template:R/superscript and the 2022 FIFA World Cup.Template:R/superscript

Encoding

Eight variations of the infinity symbol

The $\mathrm{\infty }$ symbol in several typefaces

The symbol is encoded in Unicode at Template:UnicharTemplate:R/superscript and in LaTeX as \infty: $\mathrm{\infty }$.Template:R/superscript An encircled version is encoded for use as a symbol for acid-free paper. Template:Charmap

The Unicode set of symbols also includes several variant forms of the infinity symbol that are less frequently available in fonts in the block Miscellaneous Mathematical Symbols-B.Template:R/superscript Template:Charmap

See also

• Aleph number
• History of mathematical notation
• Lazy Eight (disambiguation)