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Self-Recursion


Self-recursion is a recursion that is defined in terms of itself, resulting in an ill-defined infinite regress.

The formula for the volume of a cylinder leads to the mathematical joke/self-description: "What is the volume of a pizza of thickness a and radius z?" Answer: pi z z a. This result is sometimes known as the second pizza theorem.

A self-recursive statement due to J. vos Post is, "This sentence contains ten words, eighteen syllables, and sixty-four letters" (Hofstadter 1985, pp. 26-27).

Math Horizons (2006) gives a number of "self-answering problems" that can be regarded as a form of self recursion. For example:

1. In 1978, Raymond Smullyan wrote a book about logical puzzles. What is the name of this book?

2. I am the square root of -1. Who am i?

3. What would the value of 190 in hexadecimal be?

4. Twenty-nine is a prime example of what kind of number?

5. The reciprocal of sqrt(2) is half of what number?

6. How many consonants are in "one"? How many in "two"? And how many in "three"?

7. What do you do to the length of an edge of a cube to find its volume?


See also

Kolakoski Sequence, Look and Say Sequence, Recursion, Regression, Self-Recursion, Tupper's Self-Referential Formula

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References

Carroll, L. "What the Tortoise Said to Achilles." Mind 4, 278-280, 1895.Gardner, M. "Infinite Regress." Ch. 22 in The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 220-229, 1984.Gleick, J. Chaos: Making a New Science. New York: Penguin Books, p. 179, 1988.Hofstadter, D. R. "On Self-Referential Sentences," "On Self-Referential Sentences: A Follow-Up," and "On Viral Sentences and Self-Replicating Structures." Chs. 1-3 in Metamagical Themas: Questing of Mind and Pattern. New York: BasicBooks, pp. 1-69, 1985."Self-Answering Problems." Math. Horizons 13, No. 4, 19, Apr. 2006.

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Self-Recursion

Cite this as:

Weisstein, Eric W. "Self-Recursion." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Self-Recursion.html

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