## Simple proofs: The impossibility of trisection

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Credit: Vatican Museum

Introduction: Ancient Greek mathematicians developed the methodology of “ruler-and-compass” constructions: if one is given only a ruler (without marks) and a compass, what objects can be constructed as a result of a finite set of operations? While they achieved many successes, three problems confounded their efforts: (1) squaring the circle; (2) trisecting an angle; and (3) duplicating a cube (i.e., constructing a cube whose volume is twice that of a given cube). Indeed, countless mathematicians through the ages have attempted to solve these problems, and countless incorrect “proofs” have

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## Simple proofs: The fundamental theorem of algebra

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Credit: MathIsFun.com

The fundamental theorem of algebra is the assertion that every polynomial with real or complex coefficients has at least one complex root. An immediate extension of this result is that every polynomial of degree $n$ with real or complex coefficients has exactly $n$ complex roots, when counting individually any repeated roots.

This theorem has a long, tortuous history. In 1608, Peter Roth wrote that a polynomial equation of degree $n$ with real coefficients may have $n$ solutions, but offered no proof. Leibniz and Nikolaus Bernoulli both asserted that quartic polynomials of

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## Simple proofs: The irrationality of pi

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Credit: fjordstone.com

Mankind has been fascinated with $\pi$, the ratio between the circumference of a circle and its diameter, for at least 2500 years. Ancient Hebrews used the approximation 3 (see 1 Kings 7:23 and 2 Chron. 4:2). Babylonians used the approximation 3 1/8. Archimedes, in the first rigorous analysis of $\pi$, proved that 3 10/71 < $\pi$ < 3 1/7, by means of a sequence of inscribed and circumscribed triangles. Later scholars in India (where decimal arithmetic was first developed, at least by 300 CE), China and the Middle East computed

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## Simple proofs of great theorems

Euler’s identity Credit: Redbubble.com

Mathematics and beauty

Modern mathematics is one of the most enduring edifices created by humankind, a magnificent form of art and science that all too few have the opportunity of appreciating. The great British mathematician G.H. Hardy wrote, “Beauty is the first test; there is no permanent place in the world for ugly mathematics.” Mathematician-philosopher Bertrand Russell added: “Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music,

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## Does the string theory multiverse really exist?

Credit: Berkeley Center for Cosmological Physics

String theory, fine tuning and the multiverse

String theory is the name for the theory of mathematical physics which proposes that physical reality is based on exceedingly small “strings” and “branes,” embedded in 10- or 11-dimensional space. String theory has been proposed as the long-sought “theory of everything,” because it appears to unite relativity and quantum theory, and also because it is so “beautiful.”

Yet in spite of decades of effort, by thousands of brilliant mathematical physicists, the field has yet to produce specific experimentally testable predictions. What’s more, hopes that string theory

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## The rise of pay-to-publish journals and the decline of peer review

Pi nonsense in peer-reviewed journals

In a previous Math Scholar blog, we lamented the decline of peer review, as evidenced by the surprising number of papers, published in supposedly professional, peer-reviewed journals, claiming that Pi is not the traditional value 3.1415926535…, but instead is some other value. In the 12 months since that blog was published, other papers of this most regrettable genre have appeared.

As a single example of this genre, the author of a 2015 paper, which appeared in the International Journal of Engineering Sciences and Research Technology, states, “The area and circumference of circle has been

## Does beautiful mathematics lead physics astray?

Introduction

In a new book, Lost in Math: How Beauty Leads Physics Astray, Sabine Hossenfelder reflects on her career as a theoretical physicist. She acknowledges that her colleagues have produced “mind-boggling” new theories. But she is deeply troubled by the fact that so much work in theoretical physics today is disconnected from empirical reality, yet is excused because the theories themselves, and the mathematics behind them, are “too beautiful not to be true.”

Hossenfelder notes that there have been numerous instances in the past when scientists’ over-reliance on “beauty” has led it astray. Newton’s clockwork universe, with seemingly self-evident

## Fermi’s paradox and the Copernican principle

Distant galaxies magnified by a gravitational lens

As we have discussed on this forum before (see, for example, previous Math Scholar blog), Fermi’s paradox looms as one of the most profound and puzzling conundrums of science: Given that the universe is presumed to be teeming with intelligent life and technological civilizations, why do we see no evidence of their existence? Although the search for signals and other evidence from extraterrestrial (ET) societies continues (and is accelerating with new facilities and funding), nothing has been found in over 50 years.

Ever since Fermi first declared the paradox in

## Pseudoscience from the political left and right

Pseudoscience through the ages

Projected global mean sea level rise

Through the years, decades and centuries, the world has science has slowly turned back the tide of pseudoscience, with victory after victory against nonsense and ignorance. In the 16th and 17th century, the writings of Copernicus, Galileo, Kepler and Newton overturned the ancient cosmology. The revolting practice of bloodletting was overturned in the late 19th century. Astrology was scientifically defeated in the 18th and 19th century, although, incredibly, it continues to attract faithful adherents even to this day. Young-earth creationism was scientifically overturned by the early 20th century, and now,

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## Chromosomes, DNA and human evolution

History

The Yunis-Prakash diagram comparing the chromosomes of humans, chimpanzees, gorillas and orangutans

Evolution in general and human evolution in particular continue to be bones of contention, so to speak, as evidenced by the ongoing efforts by some groups to prohibit or downplay evolution, or to mandate “equal time” for “intelligent design,” in state and local high school curricula. At of the present date (May 2018), just in the U.S., campaigns are active in Indiana, Louisiana, Oklahoma, South Dakota, Tennessee, Texas and Utah.

Although the role of chromosomes in heredity and evolution was recognized in the 19th century, it was

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