The 528th digit of Pi

Gotfried Wilhelm von Leibniz

Two posts ago I mentioned that the best simple fractional approximation of the value of p is 355/113 = 3.14159292..., which was discovered in the West in the 16th century. Later, better approximations were obtained, but no longer in the form of a fraction, rather as the sum of a series. Several infinite series of terms are known whose sum is p. So it is enough to add a sufficiently large number of terms to obtain as many digits of p as we want, as long as we have time to do the sums. The first to propose one of these series was the French mathematician François Vieta. As his series was quite complicated, we give here the much better known series proposed in 1673 by the German mathematician and philosopher Gotfried Wilhelm von Leibniz:

The more terms we add of this series, the closer we will come to the value of p. The following table shows the advances made over time in the calculation of the successive approximations of this number, using different series, formulas or procedures.

The mystery of the Great Pyramid

The Great Pyramid of Giza, also called Pyramid of Cheops or Pyramid of Jufu, was built to be the tomb of the pharaoh Jufu (called by the Greeks Cheops), of the fourth dynasty, the high point of the Ancient Egyptian Empire. The reign of Jufu is usually dated in the 26th century before Christ, over 4500 years ago.

The current height of the Great Pyramid is 138.8 meters, but the pyramid is truncated, having lost its top. It is easy to calculate that its original height was about 8 meters higher: 146.7 meters, or 280 Egyptian cubits. The base of the pyramid is a square with a side of 230.34 meters, or 440 Egyptian cubits.

Observe a curious point: the semi-perimeter of the pyramid (twice the side of the base) is equal to 880 cubits. If we divide it by the height of the pyramid, we get the following:

(880/220) = (22/7) = 3.142857...

Are the digits of Pi real?

Martin Gardner

In an article published in Discover magazine in 1985, Martin Gardner wrote this:

As it happens, the thousandth decimal of pi is 9... The question: Was [this assertion] true before the 1949 calculation? To those of the realist school, the sentence expresses a timeless truth whether anyone knows it or not... [Others] prefer to think of mathematical objects as having no reality independent of the human mind.

This problem is quite old, as we have been discussing it for over two thousand years. The question about whether mathematical objects really exist or are a pure creation of our mind is a particular case of another problem, much more general, that debates whether ideas and concepts (like the dog species) really exist, or just this dog and that dog exist. This is the problem of universals, famous in the Middle Ages, which has not yet been solved to everyone’s satisfaction. In fact, at present, this debate is more virulent than ever.

What’s a scientific theory

Karl Popper

Although it is fashionable to assert that Karl Popper’s theories about the evolution of science are outdated, his definition of what is a scientific theory is unassailable:

A theory is scientific if and only if it is possible to design an experiment that proves that this theory is false.

A paradigmatic case is the Copenhagen Interpretation of Quantum Mechanics. In 1935, Einstein, Podolsky and Rosen designed an experiment that could prove this theory false. A few months later, Niels Bohr published another article in the same magazine, in answer to the previous article. Almost 30 years later, as I explained in another post in this blog, the EPR experiment, which up to that point had been mental, could be carried out and confirmed Bohr’s predictions, rather than Einstein’s. As this theory was able to resist an attempt to prove it false, it must be considered a scientific theory.

Of course, this success of the theory does not imply that it should automatically be considered correct or true. Scientific theories (always according to Popper) never become so. This theory has successfully withstood an attempt to prove it false, but the next attempt could do it.

The standard cosmological model

Map of the Cosmic Background Radiation

In 1927, the Belgian priest and astronomer Georges Lemaître discovered Hubble’s law.

Yeah that’s right. Hubble did not discover the law until 1929. What happened was that Lemaître published it in French in a low-impact journal (Annales de la Société Scientifique de Bruxelles), while Hubble published it two years later in English in the Proceedings of The National Academy of Sciences, received much more publicity and his name got associated with the discovery.

Combined with Einstein’s cosmological equation, Lemaître-Hubble’s law implies that the universe is expanding. In an article published in 1931 in Nature, Lemaître drew the consequence by proposing the Big Bang theory, so called in derision by its opponent Fred Hoyle in 1950. The name caught on.

In 1948, Ralph Alpher, George Gamow and Robert Herman made two surprising predictions, based on the Big Bang theory: the average composition of the mass of the cosmos (three quarters hydrogen and one quarter helium), and the existence of the cosmic background radiation. Both were confirmed during the sixties. From that point, the Big Bang theory became the standard cosmological theory.