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.
Both the infinity symbol itself and several variations of the symbol are available in various
character encodings.
History
John Wallis introduced the infinity symbol to mathematical literature.
The lemniscate has been a common decorative motif since ancient times; for instance it is commonly seen on
Viking Age combs.[4]
The English mathematician
John Wallis is credited with introducing the infinity symbol with its mathematical meaning in 1655, in his De sectionibus conicis.[5][6][7] 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.[8] Another proposes instead that it was based on the notation CIↃ used to represent 1,000.[9] Instead of a Roman numeral, it may alternatively be derived from a variant of ω, the lower-case form of
omega, the last letter in the
Greek alphabet.[9]
Perhaps in some cases because of typographic limitations, other symbols resembling the infinity sign have been used for the same meaning.[7] 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),[10] and even "O–O" has been used as a stand-in for the infinity symbol itself.[7]
Usage
Mathematics
In mathematics, the infinity symbol is typically used to represent a
potential infinity.[11] For instance, in mathematical expressions with
summations and
limits such as
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.[12]
When quantifying
actual infinity, infinite entities taken as objects per se, other notations are typically used. For example, (
aleph-nought) denotes the
cardinal number representing the size of the set of
natural numbers, and (
omega) denotes the smallest
ordinal number which is larger than all natural numbers.
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.[15] 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".[16]
Symbolism and literary uses
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.[18]
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".[19] Other authors whose works use this shape with its symbolic meaning of the infinite include
James Joyce, in Ulysses,[20] and
David Foster Wallace, in Infinite Jest.[21]
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.[22] 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,[23][24] but also evoking the geometric shapes of Métic dances,[25],
Celtic knots,[26] or
Plains First Nations Sign Language.[27]
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.[28] 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.[29] Versions of this symbol have been used in other trademarks, corporate logos, and emblems including those of
Fujitsu,[30]Cell Press,[31] and the
2022 FIFA World Cup.[32]
Encoding
The symbol is encoded in
Unicode at U+221E∞INFINITY[33] and in
LaTeX as \infty: .[34] An encircled version is encoded for use as a symbol for
acid-free paper.
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.[45]
^Rucker, Rudy (1982). Infinity and the Mind: The science and philosophy of the infinite. Boston, Massachusetts: Birkhäuser. p. 1.
ISBN3-7643-3034-1.
MR0658492.
^
abClegg, Brian (2003). "Chapter 6: Labelling the infinite". A Brief History of Infinity: The Quest to Think the Unthinkable. Constable & Robinson Ltd.
ISBN978-1-84119-650-3.
^Shipman, Barbara A. (April 2013). "Convergence and the Cauchy property of sequences in the setting of actual infinity". PRIMUS. 23 (5): 441–458.
doi:
10.1080/10511970.2012.753963.
S2CID120023303.
^Crist, Brian; Aurello, David N. (October 1990). "Development of camera symbols for consumers". Proceedings of the Human Factors Society Annual Meeting. 34 (5): 489–493.
doi:
10.1177/154193129003400512.
^Bahun, Sanja (2012). "'These heavy sands are language tide and wind have silted here': Tidal voicing and the poetics of home in James Joyce's Ulysses". In Kim, Rina; Westall, Claire (eds.). Cross-Gendered Literary Voices: Appropriating, Resisting, Embracing. Palgrave Macmillan. pp. 57–73.
doi:
10.1057/9781137020758_4.
^Natalini, Roberto (2013). "David Foster Wallace and the mathematics of infinity". In Boswell, Marshall; Burn, Stephen J. (eds.). A Companion to David Foster Wallace Studies. American Literature Readings in the 21st Century. Palgrave Macmillan. pp. 43–57.
doi:
10.1057/9781137078346_3.