Tombstone (typography)

In mathematics, the tombstone, halmos, end-of-proof, or Q.E.D. symbol "∎" (or "□") is a symbol used to denote the end of a proof, in place of the traditional abbreviation "Q.E.D." for the Latin phrase "quod erat demonstrandum", meaning "which was to be demonstrated".[1] In magazines, it is one of the various symbols used to indicate the end of an article.[2]

Various forms of the end-of-proof symbol

In Unicode, it is represented as character U+220E ∎ END OF PROOF (HTML ∎). Its graphic form varies, as it may be a hollow or filled rectangle or square.

In AMS-LaTeX, the symbol is automatically appended at the end of a proof environment \begin{proof} ... \end{proof}. It can also be obtained from the commands \qedsymbol, \qedhere or \qed (the latter causes the symbol to be right aligned).[3]

It is sometimes called a "Halmos finality symbol" or "halmos" after the mathematician Paul Halmos, who first used it in a mathematical context in 1950.[4] He got the idea of using it from seeing it was being used to indicate the end of articles in magazines. In his memoir I Want to Be a Mathematician, he wrote the following:[5]

The symbol is definitely not my invention — it appeared in popular magazines (not mathematical ones) before I adopted it, but, once again, I seem to have introduced it into mathematics. It is the symbol that sometimes looks like ▯, and is used to indicate an end, usually the end of a proof. It is most frequently called the 'tombstone', but at least one generous author referred to it as the 'halmos'.

See also

Notes
  1. "The Definitive Glossary of Higher Mathematical Jargon — Q.E.D." Math Vault. 2019-08-01. Retrieved 2019-11-04.
  2. Foster, A. J. "Tombstones in Typography | AJ Foster". aj-foster.com. Retrieved 2019-11-05.
  3. "LaTeX/Theorems - Wikibooks, open books for an open world". en.wikibooks.org. Retrieved 2019-11-05.
  4. Halmos, Paul R. (Paul Richard), 1916-2006. (1950). Measure theory. New York: Van Nostrand. pp. 6. ISBN 0387900888. OCLC 529634.CS1 maint: multiple names: authors list (link)
  5. Paul R. Halmos, I Want to Be a Mathematician: An Automathography, 1985, p. 403.

References


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