Infinity symbol (∞)
∞ is the mathematical sign for infinity, a figure eight lying on its side. The English mathematician John Wallis introduced it in 1655, and why he chose the shape is disputed. It is also called the lemniscate, and outside mathematics it is carried on the Métis flag and by autistic and neurodiversity advocates.
Meaning
Wallis first used ∞ in De sectionibus conicis (On Conic Sections), published in 1655, where he wrote “esto enim ∞ nota numeri infiniti”, “let ∞ be the sign of an infinite number”; he used it again in Arithmetica infinitorum. Its other name, lemniscate, is also the name of a figure-eight curve Jacob Bernoulli described in 1694 and called by the Latin lemniscus, “a pendant ribbon”. Unicode encodes the sign as U+221E INFINITY, among its mathematical operators. The Métis Nation carries it on its flag, and the autistic and neurodiversity movements have made it their emblem.
Origin and history
Wallis held the Savilian chair of geometry at Oxford from 1649 until his death (Edward Worth Library). In De sectionibus conicis he imagined a flat figure made of infinitely many thin parallelograms, each one 1/∞ of the whole height, “an infinitely small part”, and added in brackets that ∞ was to be the sign of an infinite number.
Why he chose the shape is a matter of conjecture. MacTutor, the University of St Andrews history of mathematics, says he chose it to stand for a curve that can be traced infinitely many times. Florian Cajori’s History of Mathematical Notations records a different conjecture, made by the palaeographer Wilhelm Wattenbach: that Wallis, a classical scholar, took over a late Roman sign for 1,000. A Roman hand abacus in the Bibliothèque nationale in Paris supports it, with a sign close to ∞ over its thousands column. Alexander Humez, Nicholas Humez and Joseph Maguire, in Zero to Lazy Eight, allow another source: a lower-case omega, the last letter of the Greek alphabet and a long-standing figure for the end.
Bernoulli’s curve came thirty-nine years after Wallis’s sign. In 1694 he described in Acta Eruditorum a curve “shaped like a figure 8, or a knot, or the bow of a ribbon”, and named it lemniscus.
The Métis infinity flag
The Métis flag is a white ∞ on a blue or a red field. The Métis Nation of Ontario says the symbol stands for the immortality of the nation and the joining of two cultures, that its historical origins are still debated, and that the flag is carried today as a symbol of continuity and pride. The Gabriel Dumont Institute’s Virtual Museum of Métis History and Culture gives two accounts of the design: some Métis see in it the Plains sign language for “Métis”, made with the sign for a cart; the more common reading is that it adapts the saltire of Scotland. The Institute records North West Company partners giving the Métis a red infinity flag in spring 1815, and Peter Fidler describing on 31 May 1816 a blue flag “about 4 feet square and a figure of 8 horizontally in the middle”, a century and a half before Canada’s maple leaf flag.
Over the Magician’s head
In A. E. Waite’s tarot of 1910, the sign floats above the Magician. Waite describes it as “the mysterious sign of the Holy Spirit, the sign of life, like an endless cord, forming the figure 8 in a horizontal position”; in the Two of Pentacles the same endless cord joins the dancing young man’s two pentacles.
Autism and neurodiversity
Since Autistic Pride Day began in 2005, the infinity sign has been the emblem autistic and neurodivergent people choose for themselves: a rainbow ∞ for the neurodiversity movement and a gold one for autism.
Where you’ll see it today
whatdoesthissymbolmean.comOmegaOne conjecture derives Wallis's ∞ from a lower-case omegaSee also
888 angel number, Autism infinity symbol, Neurodiversity symbol, The Magician, Sigils, Schrödinger’s cat.
Sources
- Earliest Uses of Symbols of Calculus, Jeff Miller, Earliest Uses of Various Mathematical Symbols, MacTutor History of Mathematics, University of St Andrews, “Infinity”
- Infinity, J. J. O’Connor and E. F. Robertson, MacTutor History of Mathematics, University of St Andrews
- John Wallis, J. J. O’Connor and E. F. Robertson, MacTutor History of Mathematics, University of St Andrews
- Wallis, Mathematics at the Edward Worth Library, Edward Worth Library, Dublin
- Lemniscate of Bernoulli, MacTutor History of Mathematics, University of St Andrews
- Mathematical Operators: Range 2200–22FF (code chart), The Unicode Standard, Version 18.0, Unicode Consortium
- Métis Symbols: History, Meaning and Significance, Métis Nation of Ontario, Métis Flag
- The Metis Infinity Flag, Gabriel Dumont Institute, Virtual Museum of Métis History and Culture
- The Pictorial Key to the Tarot, by A. E. Waite, William Rider & Son, London, 1910 (the sacred-texts.com transcription, on the Internet Archive), The Magician; Two of Pentacles
- Why the autism jigsaw puzzle piece is such a problematic symbol, Aimee Grant, Swansea University, 10 April 2025, The Conversation
- Autistic Pride, Autistic Empire
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Last reviewed September 30, 2026 by Rory Hansen. How we research