Human race 2017 Kardashev scale

🎺Breaking news!🎺

After several years of being a Type 0.727 civilization (going back to 2014), humanity has ticked over in the third decimal place to become a Type 0.728 civilization (source data on page 8 here under “Total World”; calculation here).

Ten years previously in 2007 the score was 0.722. Slow but steady progress to… whatever it is that we’re developing towards.

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Book Review: An Imaginary Tale

Image result for an imaginary tale

An Imaginary Tale: The Story of √-1 by Paul Nahin

I’ve had this book for years, and have read the first few chapters several times but never actually finished it for some reason. I think I got bogged down in the geometry of Chapter 3, even though it’s not that difficult. I even read and reviewed its direct sequel a few years ago. Better late than never I suppose.

This book is about complex numbers, and especially the unit imaginary number i. The first few chapters are about its history, and how mathematicians were suspicious of such quantities for a long time (hell, they were suspicious of negative numbers for centuries, too). Interestingly, their origin traces more to solutions to cubic equations rather than quadratic equations, where they seem, to a modern eye, to more naturally arise.

Then the book jumps around to an assortment of different topics out of chronological order. For instance, there’s a section dealing with solutions to a particular type of electronic circuit, which comes before Euler’s foundational work. At times Nahin will prove some theorem, and at other times he’ll say something to the effect of “To prove this claim isn’t that hard, but this isn’t a math textbook!” Sometimes I wish he did just write a more thorough textbook with his style. So treat it more like “Here’s a variety of interesting things about complex numbers, as selected by the author.”

Which isn’t to say there’s not good stuff being proven. We get for instance Cauchy’s integral formula, which is proved though not to full generality (he only considers a rectangular contour; “not a textbook!”). I remember being blown away by this theorem when I took complex analysis in university, and Nahin expresses similar amazement.

I found some errors in the appendices in this 1998 copy: in Appendix B on p. 133 there are some denominators that are squared which I don’t think should be (it’s a simple matter of FOILing out a quadratic). In the original paper the squares are not present. Also in Appendix E when discussing the Laplace transform, he uses the pound sterling symbol £ rather than the script capital ℒ  which I found somewhat amusing.

Posted in Mathematics, Reviews

Book Review: Special Relativity and Classical Field Theory (The Theoretical Minimum #3)

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Special Relativity and Classical Field Theory: The Theoretical Minimum by Leonard Susskind and Art Friedman

The third book in Susskind’s Theoretical Minimum series (based off the video lectures here) covers special relativity  and classical field theory, specifically electromagnetism (general relativity is the next planned book). It’s assumed the reader has read the first book in the series where things like the action, Lagrangian, and Hamiltonian are developed, but there’s still lots of review and hand-holding. Appendix B is a review of some vector calculus operations.

I thought this book was great, and is filled with a bunch of great nuggets even for somebody with a degree in physics. For example, one longstanding question I’ve had is why the Lagrangian only goes up to the first derivative of position (aside from the “It’s equivalent to Newton’s Second Law” cop-out). Susskind’s satisfying answer: it’s because of locality [1]. And just for fun, Appendix A is all about magnetic monopoles.

Speaking of locality, Susskind lists four fundamental principles that underlie all of physics, and then uses them directly in producing results in electromagnetism. That is he doesn’t just have an interesting philosophical discussion, but translates them into meat-and-potatoes mathematics and then uses them to solve problems. These principles are: the action principle, locality, Lorentz invariance, and gauge invariance.

This is my favorite so far of the Theoretical Minimum series. People who have only seen Maxwell’s equations developed to their vector calculus form that appears on T-shirts (for example, most engineers) will be guided to their “higher” formulation in terms of tensors. The connection between special relativity and electromagnetism is deep, and it’s with that connection that Einstein began (paragraph one!) his famous special relativity paper. With this book the reader will clearly see how once you observe that a current-carrying wire generates a magnetic field, special relativity necessarily follows.

Also, “GROUCHO!” 😄 (see p. 392)

[1] And from Coopersmith I got a satisfying answer to why the Lagrangian depends explicitly on the first derivative: it serves as an infinite set of “internal boundary conditions” to enforce continuity.

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Book Review: Quantum Mechanics (The Theoretical Minimum #2)

QMTM

Quantum Mechanics: The Theoretical Minimum by Leonard Susskind and Art Friedman

This book (based off of these lectures) came out a few years ago, and while I picked it up at launch I never quite finished it and it got kind of lost on the back burner. The third book on special relativity just came out, though, so I dusted this one off and finally completed it. Recall that this quantum mechanics volume is the second in Susskind’s Theoretical Minimum series, after the first book on classical mechanics.

A few years ago at my old university we had David Griffiths as an invited guest speaker. I didn’t take notes so I could be misremembering, but I recall him saying that while the physics education profession had more or less decided on the appropriate structure for a course on electromagnetism, there was still widespread disagreement about how to teach quantum mechanics. Griffiths’ own book starts with the Schrodinger equation on page one, and proceeds from simple potentials like the infinite square well up through the hydrogen atom and beyond, and then moves on to other topics like spin. Longair’s Quantum Concepts in Physics instead takes an historical approach.

Susskind and Friedman take a different tack: they start with spin, which doesn’t really have any classical analogue and hence is rather foreign and abstract. However, the mathematics of spin is comparatively very simple (on the order of 2×2 matrices). I’m not sure of this approach. Having take several quantum mechanics courses, it was all fairly old hat to me, but I’m not sure that a beginner wouldn’t feel a bit overwhelmed right off the start. The advantage of beginning with spin is that they are very well placed to then explore issues like entanglement (Chapters 6 and 7). They do a very thorough job developing and motivating the algebra of operators, and having started with spin there’s always ready-made and simple examples to apply the operators to.

While they never solve the infinite square well or the hydrogen atom, the last chapter is devoted to the harmonic oscillator. It’s not as thoroughly developed as in a textbook like Griffiths, since it starts off by declaring but not proving a theorem that the ground state has no nodes, and then the ground state wave function is just given to us. However, they then develop the raising and lowering operators to generate the higher-energy wave functions, whose solutions involve the Hermite polynomials.

In his video lecture series, Susskind had a second course on advanced quantum mechanics, and so I wonder if they’ll eventually publish a similar book in this series. Before that though there’s the third book on special relativity, then presumably we’ll get general relativity, cosmology, and statistical mechanics.

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Book Review: The Lazy Universe

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The Lazy Universe: An Introduction to the Principle of Least Action by Jennifer Coopersmith

There is an interesting phenomenon of popular versions of more technical books. We have Feynman and Hibbs’ Quantum Mechanics and Path Integrals being popularized in the form of Feynman’s QED. David Friedman turned his Price Theory textbook into the wonderful Hidden Order. And now we have an answer to the question: but who will write a “popular” (for non-expert audiences with college-level math training) version of Cornelius Lanczos’ The Variational Principles of Mechanics? Jennifer Coopersmith, that’s who.

I really enjoyed this book. The principle of least action is one of those great gems that bears returning to again and again. One of my pet pursuits is finding more and better explanations for the age-old question “Why is the classical Lagrangian kinetic minus potential energy?” (see here for example for my attempt to understand it a bit better in the case of gravity). Coopersmith devotes Section 6.6 to that very topic.

The Lazy Universe is structured by beginning with some historical material and some mathematical preliminaries (advisory note: you do need to know vector calculus to really profit from this book). Then we march from the principle of virtual work, to D’Alembert’s principle, on to Lagrangian mechanics and then Hamiltonian mechanics. Unlike Susskind’s popular classical mechanics book Coopersmith doesn’t address Poisson brackets. I really like the structure and how she builds from one chapter to the next; I certainly did not sufficiently appreciate the point of virtual work and D’Alembert when I took my classical mechanics class. In the penultimate chapter there’s a survey of how the principle of least action takes its form in other branches of physics.

I’m just not sure who this book is for, exactly. Much of the mathematical nitty-gritty and nearly all the examples are pushed to a multitude of appendices, which makes it annoying to flip back and forth if you want the full story. This was done I think so that less mathematically-adept readers could enjoy the main thread of the argument, but that still requires knowledge of the variational calculus (for example, the Euler-Lagrange equations are not derived like in Susskind’s book). So I would recommend this to people for whom this is their second (or third, etc) pass at analytical mechanics.

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Human race 2016 Kardashev score

BP has released its latest world energy outlook report, and from page 8 of the PDF here we see that total world energy consumption was 13,276.3 Mtoe in 2016. Following the prescription I laid out previously here, we can calculate human civilization’s 2016 Kardashev score to be… 0.727. Which is the same as it was in 2014 and 2015 to the nearest third decimal.

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Year in review 2016

2016 is over, so it’s time to update the end-of-year civilization benchmarks. I’ve dropped the solar power production as energy was already covered by the Kardashev score and overindexing on one particular technology is probably unwise. I’m also not so sure how to think about the ATLAS row, as the energies are changing as the luminosity changes, year on year. ¯\_(ツ)_/¯

2011 2012 2013 2014 2015 2016
Supercomputer (PFLOPs) 10.51 17.59 33.86 33.86 33.86 93.01
Known exoplanets 716 854 1055 1855 2041 3557
ATLAS integ. lumin. (fb-1) 5.46 28.54 28.54 28.54 32.89 71.8
GenBank base pairs (billions) 135.1 148.4 156.2 184.9 203.9 224.9
Kardashev score 0.724 0.725 0.726 0.727 0.727 NA
World population (billions) 7.0 7.1 7.1 7.2 7.3 7.4

What do I usually even write here? I don’t remember. I finished 60 new books this year, only a fraction of which I blog reviewed. I successfully defended my master’s thesis and can now append MSc after my name. You know, if I was a douche. I’ve been undergoing allergy desensitization therapy the past few years, and it paid off big this summer (normally I have hard allergic conjunctivitis from the beginning of May through the end of August). No symptoms, so that’s plus one right there for modern medicine.

For people super bummed about 2016, take the large view, the long view, stop following the news so closely and read or listen to more books, meditate on how bad it could get, listen to some good tunes, and browse some dank memes.

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