PiDay 2020: A catalogue of formulas involving pi, with analysis

I have prepared a new paper containing a catalogue of 72 summation formulas, integral formulas and iterative algorithms for Pi. The catalogue contains both classical and modern formulas, ranging from Archimedes’ 2200-year-old algorithm to intriguing formulas found by Ramanujan and the quadratic, cubic, quartic and nonic algorithms of Jonathan Borwein and Peter Borwein, the latter of which double, triple, quadruple and nine-times, respectively, the number of correct digits with each iteration.

The catalogue of formulas and iterative algorithm is followed by results of carefully designed computer implementations, which enable one to compare the relative speed of these formulas.

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Pi Day 2020: A new crossword puzzle

Yes, it is that time of year — Pi Day (March 14, or 3/14 in North American month/day date notation) is here.

So in honor of the occasion, I have constructed a new crossword puzzle — see below. This puzzle honors several of the key persons through history who have made significant contributions to the theory and computation of Pi.

This puzzle conforms to the New York Times crossword conventions. As far as difficulty level, it would be comparable to the NYT Tuesday or Wednesday puzzles (the NYT puzzles are graded each week from Monday [easiest] to Saturday [most difficult]).

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Universe or multiverse? The war rages on

Credit: Quanta Magazine


A growing controversy over the multiverse and the anthropic principle has exposed a major fault line in modern physics and cosmology. Some researchers see the multiverse and the anthropic principle as inevitable, others see them as an abdication of empirical science. The controversy spans quantum mechanics, inflationary Big Bang cosmology, string theory, supersymmetry and, more generally, the proper roles of experimentation and mathematical theory in modern science.

The “many worlds interpretation” of quantum mechanics

Since the 1930s, when physicists first developed the mathematics behind quantum mechanics, researchers have found that this theory appears to

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Do probability arguments refute evolution?


Both traditional creationists and intelligent design writers have invoked probability arguments in criticisms of biological evolution. They argue that certain features of biology are so fantastically improbable that they could never have been produced by a purely natural, “random” process, even assuming the billions of years of history asserted by geologists and astronomers. They often equate the hypothesis of evolution to the absurd suggestion that monkeys randomly typing at a typewriter could compose a selection from the works of Shakepeare, or that an explosion in an aerospace equipment yard could produce a working 747 airliner [Dembski1998; Foster1991; Hoyle1981;

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New factorization advances: Is your bank account safe?

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Factorization and cryptography

Until a few decades ago, number theory, namely the study of prime numbers, factorization and other features of the integers, was widely regarded as the epitome of pure mathematics, completely divorced from considerations of practical utility. This sentiment was expressed most memorably by British mathematician G.H. Hardy (best known for mentoring Ramanujan and results on the Riemann Zeta function), who wrote in his book A Mathematician’s Apology (1941),

I have never done anything “useful”. No discovery of mine has made, or is likely to make, directly or indirectly, for good

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Jim Simons: The man who solved the market

Gregory Zuckerman, author of The Greatest Trade Ever, has published a new book highlighting the life and work of Jim Simons, who, at the age of 40, walked away from a very successful career as a research mathematician and cryptologist to try his hand at the financial markets, and ultimately revolutionized the field. Zuckerman’s new book is titled The Man Who Solved the Market: How Jim Simons Launched the Quant Revolution.

Upbringing and academic career

Simons’ background hardly suggested that he would one day lead one of the most successful, if not the most successful, quantitative hedge fund operation in

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The scientific debate is over: it is time to act on climate change

Credit: IPCC

The facts of climate change

At this point in time, the basic facts of climate change are not disputable in the least. Careful planet-wide observations by NASA and others have confirmed that 2018 was the fourth-warmest year in recorded history. The only warmer years were 2016, 2017 and 2015, respectively, and 18 of the 19 warmest years in history have occurred since 2001. Countless observational studies and supercomputer simulations have confirmed both the fact of warming and the conclusion that this warming is principally due to human activity. These studies and computations have been scrutinized in great

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Quantum supremacy has been achieved; or has it?

IBM’s “Q” quantum computer; courtesy IBM

Google’s quantum computing achievement

For at least three decades, teams of researchers have been exploring quantum computers for real-world applications in scientific research, engineering and finance. Researchers have dreamed of the day when quantum computers would first achieve “supremacy” over classical computers, in the sense that a quantum computer solving a particular problem faster than any present-day or soon-to-be-produced classical computer system.

In a Nature article dated 23 October 2019, researchers at Google announced that they have achieved exactly this.

Google researchers employed a custom-designed quantum processor, named “Sycamore,” consisting of programmable quantum

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Pi as the limit of n-sided circumscribed and inscribed polygons

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Credit: Ancient Origins

Introduction In a previous Math Scholar blog, we presented Archimedes’ ingenious scheme for approximating $\pi$, based on an analysis of regular circumscribed and inscribed polygons with $3 \cdot 2^k$ sides, using modern mathematical notation and techniques.

One motivation for both the previous blog and this blog is to respond to some recent writers who reject basic mathematical theory and the accepted value of $\pi$, claiming instead that they have found $\pi$ to be a different value. For example, one author asserts that $\pi = 17 – 8 \sqrt{3} =

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New paper proves 80-year-old approximation conjecture

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Log_10 of the error of a continued fraction approximation of Pi to k terms

Approximation of real numbers by rationals

The question of finding rational approximations to real numbers was first explored by the Greek scholar Diophantus of Alexandra (c. 201-285 BCE), and continues to fascinate mathematicians today, in a field known as Diophantine approximations.

It is easy to see that any real number can be approximated to any desired accuracy by simply taking the sequence of approximations given by the decimal digits out to some point, divided by the appropriate power

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