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 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 $\pi = 17 – 8 \sqrt{3} = 3.1435935394\ldots$.

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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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How many habitable exoplanets are there, really?

The TRAPPIST-1 system of exoplanets, approximately 40 light-years away

Exoplanets galore

As of the present date (August 2019), more than 4000 exoplanets have been discovered orbiting other stars, and by the time you read this even more will have been logged. Several hundred exoplanets were announced in a July 2019 paper (although these await independent confirmation). All of this is a remarkable advance, given that the first confirmed exoplanet discovery did not occur until 1992.

Most of the discoveries mentioned above are planets that are either too large or too close to their sun to possess liquid water, much

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How fast is the universe expanding? New results deepen the controversy

A large cluster galaxy in the center acts as a gravitational lens, splitting the light from a more distant supernova into four yellow images (arrows)

The standard model of physics has reigned supreme since the 1970s, successfully describing experimental physical reality in a vast array of experimental tests. Among other things, the standard model predicted the existence of a particle, now known as the Higgs boson, underlying the phenomenon of mass. This particle was experimentally discovered in 2012, nearly 50 years after it was first predicted.

Yet physicists have known for many years that the standard model cannot be the

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Protein folding via machine learning may spawn medical advances

Complex of bacteria-infecting viral proteins modeled in CASP 13

Introduction

In an advance that may presage a dramatic new era of pharmaceuticals and medicine, DeepMind (a subsidiary of Alphabet, Google’s parent company) recently applied their machine learning software to the challenging problem of protein folding, with remarkable success. In the wake of this success, DeepMind and other private companies are racing to further extend these capabilities and apply them to real-world biology and medicine.

The protein folding problem

Protein folding is the name for the physical process in which a protein chain, defined by a linear sequence of amino

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Homo Deus: A brief history of tomorrow

Homo Deus

In his new book Homo Deus, Israeli scholar Yuval Noah Harari has published one of the most thoughtful and far-reaching analyses of humanity’s present and future. Building on his earlier Sapiens, Harari argues that although humanity has made enormous progress across in the past few centuries, the future of our society, and even of our species, is uncertain.

Harari begins with a reprise of human history, from prehistoric times to the present. He then observes that although religious beliefs are much more nuanced and sophisticated than in the past, human society still relies heavily on the narratives

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Mathematicians prove result tied to the Riemann hypothesis

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Ken Ono, Emory University

Don Zagier, Max Planck Institute

Michael Griffin, BYU

Larry Rolen, Vanderbilt University

Introduction

Four mathematicians, Michael Griffin of Brigham Young University, Ken Ono of Emory University (now at University of Virginia), Larry Rolen of Vanderbilt University and Don Zagier of the Max Planck Institute, have proven a significant result that is thought to be on the roadmap to a proof of the most celebrated of unsolved mathematical conjecture, namely the Riemann hypothesis. First, here is some background:

The Riemann hypothesis

The Riemann hypothesis was first posed by the German

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Computational tools help create new living organism

A colony of the new Syn61 bacteria; credit: BBC

Creating life

In a remarkable development with far-reaching consequences, researchers at the Cambridge Laboratory of Molecular Biology have used a computer program to rewrite the DNA of the well-known bacteria Escherichia coli (more commonly known as “E. coli”) to produce a functioning, reproducing species that is far more complex than any previous similar synthetic biology effort.

Venter’s 2010 project

This effort has its roots in a project spearheaded by J. Craig Venter, the well-known maverick biomedical researcher known for the “shotgun” approach to genome sequencing pioneered by his team at

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Researchers use “magic functions” to prove sphere-packing results

Optimal stacking of oranges.

The sphere-packing problem

The Kepler conjecture is the assertion that the simple scheme of stacking oranges typically seen in a supermarket has the highest possible average density, namely pi/(3 sqrt(2)) = 0.740480489…, for any possible arrangement, regular or irregular. It is named after 17th-century astronomer Johannes Kepler, who first proposed that planets orbited in elliptical paths around the sun.

Hales’ proof of the Kepler conjecture

In the early 1990s, Thomas Hales, following an approach first suggested by Laszlo Fejes Toth in 1953, determined that the maximum density of all possible arrangements could be obtained

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Pi, climate change denial and creationism

Introduction

Right off, it may not sound like pi, climate change denial and young-Earth creationism have much in common. In fact, there is an important connection. Here is some background.

Credit: Michele Vallisneri, NASA JPL

Computing pi

Pi = 3.1415926535…, namely the ratio between the circumference of a circle and its diameter, has fascinated not only mathematicians and scientists but the public at large for centuries. Archimedes (c.287–212 BCE) was the first to present a scheme for calculating pi as a limit of perimeters of inscribed and circumscribed polygons, as illustrated briefly in the graphic to the right (see

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