Some musings on the meanings of information content

Frank Oppenheimer was a physicist who made pioneering studies of cosmic rays via high-altitude balloons in the late 1940s. He was later the founding directorof the San Francicso Exploratorium. There he hired the physicist Larry Shaw, who became the creator of Pi Day.

Today we discuss what information—if any—is conveyed by the digits of pi.

Oppenheimer’s older brother Julius is better known as J. Robert Oppenheimer. He is the title subject of a major Hollywood biopic directed by Christopher Nolan, to be released in July. The film will cover not only his work on the atomic bomb but also the revocation of his security clearance over ties to the Communist Party, of which Frank had been a card-carrying member. Frank lost his faculty position at the University of Minnesota and was blackballed from teaching physics anywhere until gaining a high school position in 1957 and joining the University of Colorado in 1959.

We are thinking of another movie, Everything, Everywhere, All at Once, which just won the Best Picture Oscar. It involves the multiverse as a macguffin. Could pi have a different value in another universe? As card-carrying Platonists, our instinct is to just say no. But a related question lends some subtlety.

Does Pi Equal 0/(2i)?

I considered titling this post, “Does Pi Equal Zero?” I realized that, if anything, it is that equals zero—if you are raising to that number. Then .

What I really mean is to ask:

Does have zero information content?

Here is a sense in which the answer is basically yes: Pi comes from a short program. Perhaps the simplest one is

via the Madhava series. There are less-simple programs that converge more quickly—plus we long ago covered this and this—but speed of approximation does not matter for the information content of the specification.

We can also say that the specification is just . Or more clearly, a circle with a diameter: . That is literally minuscule. By Claude Shannon’s information theory, this little speck of spec is the information content of pi.

Making Contact With Our Question

Our question is joined by Carl Sagan’s novel Contact, especially the last two pages. Having been tipped by the aliens to look for evidence of super-intelligence in fundamental constants, Ellie Arroway sets a super-computer onto and receives a statistically significant pattern in its expansion. It gives a raster image using only the digits and of a nearly perfect circle. Arroway regards this also as a personal message symbolizing her full-circle journey.

To quote the last page:

In whatever galaxy you happen to find yourself, you take the circumference of a circle, divide it by a diameter, measure closely enough, and uncover a miracle—another circle, drawn kilometers downstream of the decimal point. There would be richer messages further in.

This causes a “wait-a-sec…” reaction. If has minuscule information content, then how can any meaningful block of information be contained in it? By meaningful we mean that the block has more information than the effort needed to specify the block.

Hereby hangs a tale. It is almost universally believed that every finite sequence of digits appears somewhere in the base-10 expansion of . It is believed further that is normal, meaning that for every and , every word in appears with frequency converging to in the base- expansion of . Thus, almost certainly contains Waiting for Godot in the original French in base 16, taking every two hex digits as an ASCII character.

But you have to wait a long time to find it—and specifying an offset where it begins expends as much information as is gained by reading it. There is no net gain of information from the tumbling hexadecimals. This reality is consistent with the zero-information picture, in the same sense that the vacuum of space teems with dancing particles and antiparticles that ultimately cancel.

Baking a Base Into Pi

Contact, however, includes a twist different from having strings in base 10 or base 16. The raster message appears when is written in base eleven. We could quibble that Sagan should have written, “…kilometers downstream of the undecimal point.”

Base eleven must count as an “unusual” base. Maybe a choice like base 57—the first Grothendieck prime—would be more unusual. Choosing a base is a sneaky way to inflate the resulting expansion of with more information—the information from specifying the base.

Taking this idea further, we could consider bases that are fractional, or algebraic irrational, or even transcendental. We can do expansions in base itself. In that base, has a simple representation: .

We may, however, have to regard cherry-picking the base as a similar statistical fools’ errand to picking the offset (in whatever base). Base eleven is simple enough to allow that the elective bias of choosing it is minimal, so that the raster-circle message would retain its significance. But that is in a novel, anway. In reality, the ability to select the base raises the bar of how much information is needed to glean in order to infer structure rather than chance.

E Replies

The transcendental partner of in the unifying equation replies to this statistical nullification with thunder:

e = 2.718281828…, you fools.

This is in our native base-ten representation. Perhaps this is the true message from the cosmos, ordaining that we have ten fingers and ten toes, so that no artifice of choosing the base is needed to reveal it.

There is a real-life claim of a significant find in the base-ten expansion of pi. It is by Arne Bergstrom in a 2014 paper titled, “Carl Sagan’s Conjecture of a Message in .” It is in the number , whose primacy over we argued two years ago today. From its abstract:

The present article looks for markers that might possibly support such a hypothesis, and surprisingly finds a sequence of seven successive zeros (actually seven successive nines rounded off) at a depth of 3,256 digits into the representation of in the special case of base ten. Finding such a sequence of zeros within the first 1,000 digits has a probability of 1 in 10,000. No such occurrences happen even remotely for at any base other than ten, nor even remotely in corresponding representations of other common transcendental numbers, such as , which appear in physical applications.

The abstract goes on to concede, however, that “these effects are most probably just numerical coincidences without physical relevance.”

The reference to physics brings us full-circle. Non-Euclidean spaces can have notions of “circle” defined by equidistance and of “diameter” whose ratio is not . If we ever find that we live in a space that in large scales is curved, however slightly, might this dislodge our conviction that Euclidean space is salient? If so, then we might regard the value as having elective bias after all. This change in attitude would be accompanied, so I conjecture, by the pre-actuated infusion of a huge amount of genuine information in the digits of—the Euclidean value of—.

Open Problems

Where does this leave us in regard to the information content of ? How does it differ in this regard from the measure-one set of real numbers in the unit interval that are both algorithmically and statistically random?

[fixed pi in base pi = 10, not 1]