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Sunday, May 3, 2009

Twitter and the Holographic Principle

{link » Limit on information density}
Entropy, if considered as information, is measured in bits. The total quantity of bits is related to the total degrees of freedom of matter/energy. In a given volume, there is an upper limit to the density of information about the whereabouts of all the particles, which compose matter in that volume, suggesting that matter itself cannot be subdivided infinitely many times, and that there must be an ultimate level of fundamental particles. As the degrees of freedom of a particle are the product of all the degrees of freedom of its sub-particles, were a particle to have infinite subdivisions into lower-level particles, then the degrees of freedom of the original particle must be infinite, violating the maximal limit of entropy density. The holographic principle thus implies that the subdivisions must stop at some level, and that the fundamental particle is a bit (1 or 0) of information.
{link » Entropy (information theory)}
The entropy rate of English text is between 1.0 and 1.5 bits per letter, or as low as 0.6 to 1.3 bits per letter, according to estimates by Shannon based on human experiments.
Armed with the holographic principle, as briefly outlined in the above excerpts, we are in a position to estimate the value-added contribution of an individual tweet to the repository of human knowledge. This is what I aim to do in this post.

As everyone knows, if an infinite number of monkeys were typing on an infinite number of keyboards, sooner or later one of them would eventually replicate Shakespeare's Hamlet. It follows that millions of Twitter users can do better. The question is, "How much better"?

Let's make the very reasonable assumption that the entropy-rate of English text is 1 bit per letter (or, more generally, any character on a standard keyboard, including the "space"). Let us stipulate, further, that 140 arbitrary characters, namely a tweet, constitutes a value-added unit of human knowledge. Furthermore, we make the simplifying assumption that the Earth is an effective black hole (a technical term) of information storage, whose event horizon would have a radius equivalent to that of a cranberry.

Now, according to Leonard Susskind:
Adding one bit of information will increase the area of the horizon of any black hole by one square Planck unit.
A Planck unit of area is the incredibly small number of 10-70 square meters! The scientific notation means that this size area is calculated as the reciprocal of the number "1 followed by 70 zeros", in units of square meters!

Moving right along, we proceed to guesstimate the surface area of a cranberry. If we take the nominal radius of a cranberry to be 10 millimeters, its area is 1257 square millimeters, which for convenience we round up to 1400 square millimeters (for obvious reasons). Dividing by 140 bits per unit of value-added human knowledge, we get 10 square millimeters of effective storage area available for Planck units of human knowledge on Earth.

Now we compute the ratio of 10 square millimeters to one square Planck unit of area, viz.
(10 square millimeters) / (1 million square millimeters per 1 square meter) = 10-5 square meters;

10-5 / 10-70 = 1065 units (maximum) of human knowledge that can be stored on Earth
Finally, the reciprocal of the above computed ratio allows us to conclude that the value-added unit of human knowledge contributed by a typical tweet is the remarkably small number, namely: 10-65.

Seems hardly worth the effort, though admittedly, your mileage may vary.

The Holographic Principle

Credit & © E. Winfree, et al. (Caltech)

Post #739 Twitter and the Holographic Principle

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