Infinity symbol | |
|---|---|
| In Unicode | U+221E ∞INFINITY (∞) |
| Different from | |
| Different from | U+267E ♾PERMANENT PAPER SIGN U+26AD ⚭MARRIAGE SYMBOL U+221D ∝PROPORTIONAL TO |
Theinfinity symbol (∞) is amathematical symbol representing the concept ofinfinity. This symbol is also called alemniscate,[1] after thelemniscate curves of a similar shape studied inalgebraic geometry,[2] or "lazy eight", in the terminology oflivestock branding.[3]
This symbol was first used mathematically byJohn Wallis in the 17th century, although it has a longer history of other uses. In mathematics, it often refers to infinite processes (potential infinity) but may also refer to infinite values (actual infinity). It has other related technical meanings, such as the use of long-lasting paper inbookbinding, and has been used for its symbolic value of the infinite in modern mysticism and literature. It is a common element ofgraphic design, for instance in corporate logos as well as in earlier designs such as theMétis flag.
The infinity symbol and several variations of the symbol are available in variouscharacter encodings.
The lemniscate has been a common decorative motif since ancient times; for instance, it is commonly seen onViking Age combs.[4]
The English mathematicianJohn Wallis is credited with introducing the infinity symbol with its mathematical meaning in 1655, in hisDe sectionibus conicis.[5][6][7] Wallis did not explain his choice of this symbol. It has been conjectured to be a variant form of aRoman 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 variantofω, the lower-case form ofomega, the last letter in theGreek 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 byLeonhard Euler was typeset with an open letterform more closely resembling a reflected and sideways S than a lemniscate (somethinglikeS),[10] and even"O–O" has been used as a stand-in for the infinity symbol itself.[7]
In mathematics, the infinity symbol is often used to represent apotential infinity.[11] For instance, in mathematical expressions withsummations andlimits such as
the infinity sign is conventionally interpreted as meaning that the variable growsarbitrarily large towards infinity, rather than actually taking an infinite value, although other interpretations are possible.[12]
The infinity symbol may also be used in some contexts for anactual infinity, infinite entities taken asmathematical objects per se, especially when only a single infinite value is considered. Ingeometry andtopology, the infinity symbol may be used to represent apoint at infinity. This usage includes, in particular, the infinite point of aprojective line,[13] and the point added to atopological space to form itsone-point compactification.[14] Inmeasure theory, the value of a measure is often taken as anextended real number, allowing as a value; this usage allows sets of infinite measure to be distinguished from nonmeasurable sets.[15] However, other notations are also used for some other kinds of infinite values. For example, (aleph-nought) denotes the smallest infinitecardinal number (representing the size of the set ofnatural numbers), and (omega) denotes the smallest infiniteordinal number (representing the ordering of the natural numbers).[16]
In areas other than mathematics, the infinity symbol may take on other related meanings. For instance, it has been used inbookbinding to indicate that a book is printed onacid-free paper and will therefore be long-lasting.[17] Oncameras and theirlenses, the infinity symbol indicates that the lens'sfocal length isset to an infinite distance, and is "probably one of the oldest symbols to be used on cameras".[18]
In modern mysticism, the infinity symbol has become identified with a variation of theouroboros, 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.[20]
In the works ofVladimir Nabokov, includingThe Gift andPale Fire, the figure-eight shape is used symbolically to refer to theMöbius 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 entitledPale Fire explicitly refers to "the miracle of the lemniscate".[21] Other authors whose works use this shape with its symbolic meaning of the infinite includeJames Joyce, inUlysses,[22] andDavid Foster Wallace, inInfinite Jest.[23]
The well-known shape and meaning of the infinity symbol have made it a commontypographic element ofgraphic design. For instance, theMétis flag, used by the CanadianMétis people since the early 19th century, is based around this symbol.[24] 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 andFirst Nations traditions,[25][26] but also evoking the geometric shapes of Métic dances,[27],Celtic knots,[28] orPlains First Nations Sign Language.[29]
Arainbow-coloured infinity symbol is also used by theautism rights movement, as a way to symbolize the infinite variation of the people in the movement and of human cognition.[30] TheBakelite company took up this symbol in its corporate logo to refer to the wide range of varied applications of the synthetic material they produced.[31] Versions of this symbol have been used in other trademarks, corporate logos, and emblems including those ofFujitsu,[32]Cell Press,[33] and the2022 FIFA World Cup.[34]
The symbol is encoded inUnicode atU+221E ∞INFINITY[35] and inLaTeX as\infty:.[36] An encircled version is encoded for use as a symbol foracid-free paper.
| Preview | ∞ | ♾ | ||
|---|---|---|---|---|
| Unicode name | INFINITY | PERMANENT PAPER SIGN | ||
| Encodings | decimal | hex | dec | hex |
| Unicode | 8734 | U+221E | 9854 | U+267E |
| UTF-8 | 226 136 158 | E2 88 9E | 226 153 190 | E2 99 BE |
| GB 18030 | 161 222 | A1 DE | 129 55 174 56 | 81 37 AE 38 |
| Numeric character reference | ∞ | ∞ | ♾ | ♾ |
| Named character reference | ∞ | |||
| OEM-437 (Alt Code)[37] | 236 | EC | ||
| Mac OS Roman[38] | 176 | B0 | ||
| Symbol Font encoding[39] | 165 | A5 | ||
| Shift JIS[40] | 129 135 | 81 87 | ||
| EUC-JP[41] | 161 231 | A1 E7 | ||
| EUC-KR[42] /UHC[43] | 161 196 | A1 C4 | ||
| EUC-KPS-9566[44] | 162 172 | A2 AC | ||
| Big5[45] | 161 219 | A1 DB | ||
| LaTeX[36] | \infty | \acidfree | ||
| CLDR text-to-speech name[46] | infinity sign | infinity | ||
The Unicode set of symbols also includes several variant forms of the infinity symbol that are less frequently available in fonts in the blockMiscellaneous Mathematical Symbols-B.[47]
| Preview | ⧜ | ⧝ | ⧞ | |||
|---|---|---|---|---|---|---|
| Unicode name | INCOMPLETE INFINITY | TIE OVER INFINITY | INFINITY NEGATED WITH VERTICAL BAR | |||
| Encodings | decimal | hex | dec | hex | dec | hex |
| Unicode | 10716 | U+29DC | 10717 | U+29DD | 10718 | U+29DE |
| UTF-8 | 226 167 156 | E2 A7 9C | 226 167 157 | E2 A7 9D | 226 167 158 | E2 A7 9E |
| Numeric character reference | ⧜ | ⧜ | ⧝ | ⧝ | ⧞ | ⧞ |
| Named character reference | ⧜ | ⧝ | ⧞ | |||
| LaTeX[36] | \iinfin | \tieinfty | \nvinfty | |||
Media related toInfinity symbols at Wikimedia Commons