Ad infinitum is a Latin phrase meaning "to infinity" or "forevermore".
In context, it usually means "continue forever, without limit" and thus can be used to describe a non-terminating process, a non-terminating repeating process, or a set of instructions to be repeated "forever," among other uses. It may also be used in a manner similar to the Latin phrase "et cetera" to denote written words or a concept that continues for a lengthy period beyond what is shown. Examples include:
- "The sequence 1, 2, 3, ... continues ad infinitum."
- "The perimeter of a fractal may be iteratively drawn ad infinitum."
- The 17th century writer Jonathan Swift mocked the idea of self-similarity in natural philosophy with the following lines in his poem "On Poetry: A Rhapsody":
- ...
- "So nat'ralists observe, a flea
- Hath smaller fleas that on him prey;
- And these have smaller fleas to bite 'em.
- And so proceeds Ad infinitum."
- ...
- The Victorian era mathematician Augustus De Morgan expanded on this with a similar verse:
- "Great fleas have little fleas upon their backs to bite 'em,
- And little fleas have lesser fleas, and so ad infinitum.
- And the great fleas themselves, in turn, have greater fleas to go on,
- While these again have greater still, and greater still, and so on."
This text appears in the nursery rhyme The Siphonaptera almost unchanged.