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Patterns in pi in "Contact"

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Patterns in pi in “Contact”

Submission Notes

Patterns in pi in “Contact”

In Carl Sagan’s novel Contact, the main character (Ellie Arroway) is told by an alien that certain megastructures in the universe were created by an unknown advanced intelligence that left messages embedded inside transcendental numbers. To check this, Arroway writes a program that computes the digits of π in several bases, and eventually finds that the base 11 representation of π contains a sequence of ones and zeros that, when properly aligned on a page, produce a circular pattern. She takes this as an indication that there is a higher intelligence that imbues meaning in the universe.

I always thought that Sagan was pulling a fast one on us. Given that transcendental numbers (or, for that matter, irrational numbers) are infinite, non-repeating sequences of digits, they contain any possible sequence of numbers, including the one Arroway found. It’s hard for me to infer anything philosophical/spiritual from the fact that you can find this sequence if you look hard enough (to make a somewhat facile comparison, if I look hard enough in my sock drawer, I will find both socks of any given pair, but you can’t take this as evidence for a higher intelligence in the universe). But then, Sagan did know one or two things about math, so maybe I’m missing something here. Are there any circumstances in which finding a particular sequence in a certain position of π would make mathematicians go “wow!“?

Key Responses

Normal Numbers and Probability

The assumption that all transcendental numbers contain every possible sequence of digits is not necessarily true. A number whose digits contain, with equal frequency, all possible sequences of the same length, is a normal number. There are transcendental numbers that are known to be normal, and others that are known to be not normal, generally because they have been specifically constructed as such. However, whether pi is normal has been long suspected, and appears to be the case when you look at the digits already computed, but it has never been proven one way or the other.

Pi is (believed to be) a universal constant that does not depend on local physics or units of measurement, making it one of the best places to put such a message. However, there is a risk that other sentient races would use a different circle constant (such as tau, τ=2π) and any race that goes down that route would never see such a message.

Early Occurrence is Key

The real point of Sagan’s idea is not that such a pattern exists somewhere in pi, but that it was found much earlier than statistically expected. If a 121-digit pattern (11×11) consisting mostly of zeros and ones appears unusually early in the digits of pi—particularly if it forms a meaningful pattern like a circle—that would be statistically improbable enough to suggest intelligent design rather than mere chance. The key insight is that while any pattern will eventually appear in an infinite sequence, finding it sooner than probabilistically expected would be remarkable.

The Measure Zero Argument

Most real numbers are normal numbers (the set of non-normal numbers has measure zero), suggesting that pi likely contains all possible finite digit sequences. However, this property is not proven for pi, leaving open the mathematical possibility that Sagan’s scenario could hold special significance.

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