Powers of Ten

Powers of Ten,
by Philip and Phylis Morrison,
and the Office of Charles and Ray Eames.

One of my favorite books is Powers of Ten by Philip Morrison, Phylis Morrison, and the Office of Charles and Ray Eames. The authors describe their book in a section called “Advice to the Reader.”

The core of this book is the scenes on the forty-two right-hand pages that follow. By themselves, they present a visual model of our current knowledge of the universe, showing along one straight line both the large and the small. Each image stands against a black background, a little reminiscent of a darkened theater. Across from every black-framed page is a page of text and picture, a pause at each step along the journey to examine detail, evidence, or the history of knowledge.

The step from one scene to its neighbor is always made a tenfold change: The edge of each square represents a length ten times longer or shorter than that of its two neighbors. The small central square frames the scene next inward.

Stephen Jay Gould said of the book “The effect is stunning and teaches more about the size of things than any turgid treatise could.”

Powers of Ten was based on an earlier film by the Office of Charles and Ray Eames of the same title, which can be viewed on YouTube. The film is based on an earlier book, Cosmic View: The Universe in Forty Jumps, by the Dutch educator Kees Boeke.

Powers if Ten.

https://www.youtube.com/watch?v=55Gpm1Q0abk

As I discussed in the October 12, 2007 entry in this blog, Russ Hobbie and I added a section on Distances and Sizes to the 4th edition of Intermediate Physics for Medicine and Biology, motivated in part by Powers of Ten. We find the ability to imagine the relative sizes of biological objects to be crucial for understanding life.

You don't have to buy Powers of Ten to enjoy it (although your money will be well spent if you do). At a website based on the book you can view many of the pictures and find other interesting items. For instance, thumbnail pictures at each power of ten have been collected into a poster, so you can view the different scales of the universe all at once. And Nickelodeon Magazine has even turned these pictures into a child's “Powers of Ten Game.”

Charles and Ray Eames were best known not as scientists or science educators, but as designers. Just recently, the United States Postal Service issues a series of stamps highlighting their work. Philip Morrison was a well-known and respected MIT physicist.

Distances and Sizes

One of the additions that Russ Hobbie and I made to the 4th edition of Intermediate Physics for Medicine and Biology is an initial section in Chapter 1 about Distances and Sizes.

In biology and medicine, we study objects that span a wide range of sizes: from giant redwood trees to individual molecules. Therefore, we begin with a brief discussion of length scales.

The Machinery of Life,
by David Goodsell.

We then present two illustrations. Figure 1.1 shows objects from a few microns to a few hundred microns in size, including a paramecium, an alveolus, a cardiac cell, red blood cells, and E. coli. Figure 1.2 contains objects from a few to a few hundred nanometers, including HIV, hemoglobin, a cell membrane, DNA, and glucose. Many interesting and important biological structures were left out of these figures.

I admit that our figures are not nearly as well drawn as, say, David Goodsell’s artwork in The Machinery of Life. But, I enjoy creating such drawings, even if I am artistically challenged. So, below are two new illustrations, patterned after Figs. 1.1 and 1.2. Think of them as supplementary figures for readers of this blog.

FIGURE 1.1½. Objects ranging in size from 1 mm down to 1 μm.
(a) Human hair, (b) human egg, or ovum, (c) sperm,
(d) large myelinated nerve axon, (e) skeletal muscle fiber,
(f) capillary, (g) yeast, and (h) mitochondria.

FIGURE 1.2½. Objects ranging in size from 1 μm down to 1 nm.
(a) Ribosomes, (b) nucleosomes, (c) tobacco mosaic virus,
(d) antibodies, and (e) ATP.

Powers of Ten.

When you combine these figures with those in IPMB, you get a nice overview of the important biological objects at these spatial scales. Two things you do not get are a sense of their dynamic behavior (e.g., Brownian motion) at the microscopic scale, and an appreciation for the atomic nature of all objects (you could not detect single atoms in Fig. 1.2½, but they lurk just below the surface; ATP consists of just 47 atoms).

If you like this sort of thing, you will love browsing through The Machinery of Life or Powers of Ten.

The Intermediate Physics for Medicine and Biology Tourist

Over the Christmas break I was browsing through the Guidebook for the Scientific Traveler: Visiting Physics and Chemistry Sites Across America, and it got me to wondering what sites a reader of the 4th edition of Intermediate Physics for Medicine and Biology might want to visit. Apparently having too much time on my hands, I devised a trip through the United States for our readers. (Perhaps I’ll prepare an international edition later.) The trip starts and ends in Rochester, Michigan, where I work, but the path consists of a large circle and you can begin anywhere. I have not visited all these places, but I know enough about them to suspect you would enjoy them all. Tell me if I have forgotten any important sites. Happy travels!

• Oakland University in Rochester, Michigan. OU is home to Intermediate Physics for Medicine and Biology (IPMB) coauthor Brad Roth, in the Department of Physics. Here Roth collaborated with Russ Hobbie to prepare the 4th edition of IPMB.
• The University of Chicago in Chicago, Illinois. The elementary charge (the magnitude of the charge of an electron, mentioned in IPMB in Chapter 3 and many times later) was first measured accurately by Robert Millikan at the University of Chicago using his famous oil drop experiment. The American Physical Society has an initiative to present commemorative plaques at important sites in the history of physics. Be sure to visit the Millikan plaque. You can see the original equipment used by Millikan at Chicago’s wonderful Museum of Science and Industry. Chapter 1 of IPMB cites the book Powers of Ten by Phillip and Phylis Morrison and the office of C. and R. Eames. The book is centered on a couple picnicking in Chicago, near Soldier Field and the Shedd Aquarium. Be sure to stop there, with your copy of Powers of Ten in hand.
• Morrison, Illinois. Robert Millikan was born in Morrison, and a downtown park in this small town about 120 miles west of Chicago bears his name (although there is no sign or marker to indicate it). IPMB coauthor Brad Roth grew up in Morrison.
• University of Minnesota, in Minneapolis, Minnesota. IPMB’s author Russ Hobbie worked at the University of Minnesota for years, and remains an emeritus faculty member in the Department of Physics and Astronomy. Stop by a visit him in nearby Saint Paul. While in Minneapolis, be sure to visit the Bakken Museum, perhaps the only museum in the country dedicated entirely to electricity and magnetism, and especially bioelectricity and biomagnetism, as discussed in Chapters 6–9 of IPMB. Earl Bakken was one of the founders of the medical device company Medtronic. Stop by at Medtronic's nearby Mounds View Bakken Education Center.
• University of California Berkeley, in Berkeley, California. The cyclotron, crucial for nuclear medicine (see Chapter 17 of IPMB), was invented by Ernest Lawrence at UC Berkeley. See the APS plaque commemorating this invention. Material from a cyclotron in Lawrence’s lab led to the discovery of technetium, an element with no stable isotopes that is widely used in nuclear medicine imaging and is discussed at length in Chapter 17 of IPMB. While in the San Francisco area, visit Stanford University where Felix Bloch performed his pioneering experiments in nuclear magnetic resonance (Chapter 18, IPMB), and where Mark Denny has his Biomechanics Laboratory (Denny’s book Air and Water is cited often in IPMB). Don’t forget to visit the Exploratorium.
• California Institute of Technology, in Pasadena, California. Carl Anderson discovered positrons while working at CalTech (see the APS plaque for Anderson’s discovery). Positrons are used in positron emission tomography (PET) imaging (Chapter 17, IPMB). Also from Cal Tech in Richard Feynman, whose Lectures on Physics are cited in IPMB.
• Washington University in St Louis, in St Louis, Missouri. Arthur Compton performed his groundbreaking experiments on Compton Scattering (Chapter 15, IPMB) at Washington University. See the APS plaque commemorating his work. Their biomedical engineering department now is home to many leading researchers in cardiac electrophysiology, including post doc Debbie Janks, a reader and often a commenter on the IPMB facebook group and blog.
• Vanderbilt University, in Nashville, Tennessee. IPMB coauthor Brad Roth attended graduate school at Vanderbilt, working with John Wikswo in the Department of Physics and Astronomy. There, they measured the magnetic field of a single nerve axon, as described in Chapter 8 of IPMB. IPMB author Russ Hobbie was a Visiting Professor at Vanderbilt in 1999. Max Delbruck, an early biological physicist who contributed to our understanding of genetics, performed many of his Nobel Prize winning experiments at Vanderbilt.
• Duke University, in Durham, North Carolina. Duke’s Department of Biomedical Engineering has been the home of many leaders in bioelectricity and cardiac electrophysiology, including Robert Plonsey, whose books are often cited often in IPMB. The Duke Biology Department is home to Steven Vogel, author of Life in Moving Fluids, another book cited in IPMB. Be sure to find the statue of former Duke physiologist Knut Schmidt-Nielsen studying a camel, which graces the Duke campus.
• National Institutes of Health, in Bethesda, Maryland. No tour of biomedical facilities in the United States would be complete without stopping at the NIH campus in Bethesda. Be sure to visit the Stetton Museum of Medical Research in Building 10: the Warren Grant Magnuson Clinical Center. Stop by Building 13 and see where IPMB coauthor Brad Roth worked on transcranial magnetic stimulation (IPMB, Chapter 8) and where his friend Peter Basser invented MRI Diffusion Tensor Imaging (IPMB, Chapter 18) (Peter’s office is still there; stop by and say hi). You could spend a week visiting all the historic medical research sites in the Washington DC area.
• Yale University, in New Haven, Connecticut. Visit Yale and walk the path of the early American physicist Josiah Williard Gibbs, whose work on chemical thermodynamics is discussed in Chapter 3 of IPMB, including the Gibbs Free Energy. See the APS plaque commemorating Gibbs’ work.
• Framingham, Massachusetts. Visit the town that contributed more to uncovering the diseases of the heart than any other, through the Framingham Heart Study. Framingham is one of the few locations mentioned explicitly in IPMB, in Chapter 2.
• Harvard University, in Cambridge, Massachusetts. Edward Purcell performed his early experiments on nuclear magnetic resonance at Harvard, which resulted in the Nobel Prize. He is also author of a beloved paper cited in IPMB, “Life at Low Reynolds Number.” Visit the site of the Harvard cyclotron, where IPMB author Russ Hobbie was a graduate student, and where Allan Cormack worked on the mathematical methods underlying computed tomography (IPMB, Chapter 16). Visit the nearby Massachusetts Institute of Technology Museum, containing a collection of artifacts related to science and technology (IPMB author Russ Hobbie obtained his undergraduate degree from MIT). While near Boston, visit the Museum of Science, especially their Theater of Electricity.
• Woods Hole Marine Biological Laboratories, in Woods Hole, Massachusetts. At Woods Hole, Kenneth Cole developed the voltage clamp method (Chapter 6, IPMB), which played an important role in the discovery of how nerves conduct action potentials. Stop by the Visitors Center and take a tour.
• Oakland University, in Rochester Michigan. Back to the starting point. Be sure to stop by Brad Roth’s office (166 Hannah Hall) and see his collection of all four editions of IPMB sitting on his bookshelf.

The Machinery of Life

The Machinery of Life,
by David Goodsell.

In the very first section of the 5th edition of Intermediate Physics for Medicine and Biology (Sec. 1.1), Russ Hobbie and I discuss “Distances and Sizes.”

In biology and medicine, we study objects that span a wide range of sizes: from giant redwood trees to individual molecules. Therefore, we begin with a brief discussion of length scales.

At the end of this section, we conclude

To examine the relative sizes of objects in more detail, see Morrison et al. (1994) or Goodsell (2009).

I have talked about the book Powers of Ten by Morrison et al. previously in this blog. I have also mentioned David Goodsell’s book The Machinery of Life several times, but until today I have never devoted an entire blog entry to it.

In the 4th edition of IPMB, Russ and I cited the first edition of The Machinery of Life (1998), and that is the edition that sits on my bookshelf. When preparing the 5th edition, we updated our references, so we now cite the second edition of Goodsell's book (2009). Is there much difference between the two? Yes! Like when Dorothy left Kansas to enter Oz, the first edition is all black and white but the second edition is in glorious color. And what a difference color makes in a book that is first and foremost visual. The second edition of The Machinery of Life is stunningly beautiful. It is not just a colorized version of the first edition; it is a whole new book. Goodsell writes in the preface

I created the illustrations in this book to help bridge this gulf and allow us to see the molecular structure of cells, if not directly, then in an artistic rendition. I have included two types of illustrations with this goal in mind: watercolor paintings which magnify a small portion of a living cell by one million times, showing the arrangement of molecules inside, and computer-generated pictures, which show the atomic details of individual molecules. In this second edition of The Machinery of Life, these illustrations are presented in full color, and they incorporate many of the exciting scientific advances of the 15 years since the first edition.

As with the first edition, I have used several themes to tie the pictures together. One is that of scale. Most of us do not have a good concept of the relative sizes of water molecules, proteins, ribosomes, bacteria, and people. To assist with this understanding, I have drawn the illustrations at a few consistent magnifications. The views showing the interiors of living cells, as in the Frontispiece and scattered through the last half of the book, are all drawn at one million times magnification. Because of this consistent scale, you can flip between pages in these chapters and compare the sizes of DNA, lipid membranes, nuclear pores, and all of the other molecular machinery of living cells. The computer-generated figures of individual molecules are also drawn at a few consistent scales to allow easy comparison.

I have also drawn the illustrations using a consistent style, again to allow easy comparison. A space-filling representation that shows each atom as a sphere is used for all the illustrations of molecules. The shapes of the molecules in the cellular pictures are simplified versions of these space-filling pictures, capturing the overall form of the molecule without showing the location of every atom. The colors, of course, are completely arbitrary since most of these molecules are colorless. I have chosen them to highlight the functional features of the molecules and cellular environments.

I have often wondered how much molecular biology a biological or medical physicist needs to know. I suppose it depends on their research specialty, but in general I believe a physicist who has read and understood The Machinery of Life has most of what you need to begin working at the interface of physics and biology: An understanding of the relative scale of biological objects, an overview of the different types of biological molecules and their structures and functions, and a visual sense of how these molecules fit together to form a cell. To the physicist wanting an introduction to biology on the molecular scale, I recommend starting with The Machinery of Life. That’s why it was included in my ideal bookshelf.

Goodsell fans might enjoy visiting his website: http://mgl.scripps.edu/people/goodsell. There you can download a beautiful poster of different proteins, all drawn to scale. There are many other illustrations and publications. Enjoy!

Listmania! IPMB

Amazon used to have a feature called Listmania! You could make a list of up to 40 books that was visible at Amazon's website. Ten years ago I created a Listmania! list related to Intermediate Physics for Medicine and Biology, reproduced below. Because the list is old, it does not include recent books (such as The Optics of Life) or books that I have discovered recently (such as The First Steps in Seeing). To learn about newer books, search this blog for posts labeled “book review.” Amazon has discontinued Listmania!, but you can still find the lists if you look hard. I miss it.

If you are interested in what I read for pleasure, look here.

Enjoy!

**********************************************************

Intermediate Physics for Medicine and Biology
Bradley J. Roth
The list author says: “Books that are cited by the 4th edition of Intermediate Physics for Medicine and Biology. These are some of the best biological and medical physics books I know of, and are books that have been useful to me during my career.”

Intermediate Physics for Medicine and Biology, 4th edition (Biological and Medical Physics, Biomedical Engineering) All the books listed below are cited in the 4th Edition of Intermediate Physics for Medicine and Biology, written by Russ Hobbie and me.

Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables [Applied Mathematics Series 55] A math handbook that has everything you'll ever need to know.

The 2nd Law: Energy, Chaos, and Form (Scientific American Library Paperback) I love this coffee table book about the second law of thermodynamics.  A painless way to introduce yourself to the subject.

Introduction to Radiological Physics and Radiation Dosimetry Classic in the Medical Physics field.

The Essential Exponential! (For the Future of Our Planet) This book explains why we devoted an entire chapter of Intermediate Physics for Medicine and Biology to the exponential function.

Physics With Illustrative Examples From Medicine and Biology: Mechanics (Biological and Medical Physics, Biomedical Engineering) A classic textbook.

Physics With Illustrative Examples From Medicine and Biology: Electricity and Magnetism (Biological and Medical Physics, Biomedical Engineering) The second edition of the book has much the same content as the first, but the quality of the printing and illustrations is vastly improved.

Physics With Illustrative Examples From Medicine and Biology: Statistical Physics (Biological and Medical Physics, Biomedical Engineering) Benedek and Villars were pioneers in biological and medical physics textbooks.

Random Walks in Biology The best book about the role of diffusion in biology that I know of.

Foundations of Medical Imaging Fine book to study imaging algorithms.

Introduction to Membrane Noise Great book on a little-known topic.

Air and Water One of my favorites. Written by a physiologist with an interest in physics (as opposed to Hobbie and I, who are physicists interested in physiology).

Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles My favorite modern physics textbook.

The Feynman Lectures on Physics (3 Volume Set) (Set v) What physics list could be complete without Feynman?

From Clocks to Chaos Excellent book to learn the biological and medical applications of chaos.

The Machinery of Life Wonderful picture book.  Great way to visualize the relative sizes of biological objects.

Bioelectricity and Biomagnetism Good, thick tome on bioelectricity.

Textbook of Medical Physiology The classic physiology textbook.

Radiobiology for the Radiologist Great place to learn about the biological effects of radiation.

Medical Imaging Physics Standard textbook in medical physics. Hendee is a pioneer in the field.

Ion Channels of Excitable Membranes, Third edition The bible for information about ion channels.

Machines in Our Hearts: The Cardiac Pacemaker, the Implantable Defibrillator, and American Health Care Learn about the history of pacemakers and defibrillators.

The Physics of Radiation Therapy The place to go to learn about radiation therapy.

Bioelectromagnetism: Principles and Applications of Bioelectric and Biomagnetic Fields Fine textbook on bioelectricity.

Powers of Ten (Revised) (Scientific American Library Paperback) Classic work describing how the world looks at different length scales. Required reading by anyone interested in science.

Electric Fields of the Brain: The Neurophysics of EEG, 2nd edition Great way to learn about the physics of the electroencephalogram.

Bioelectricity: A Quantitative Approach Standard textbook for a class in bioelectricity.

Numerical Recipes 3rd edition: The Art of Scientific Computing My go-to book on numerical methods.

Electricity and Magnetism (Berkeley Physics Course, Vol. 2) Best introduction to electricity and magnetism I know. Part of the great Berkeley Physics Course.

Statistical Physics: Berkeley Physics Course, Vol. 5 Great intuitive introduction to statistical mechanics.  Part of the Berkeley Physics Course.

Div, Grad, Curl, and All That: An Informal Text on Vector Calculus (Fourth edition) Need a little review of vector calculus? This is the place to find it.

Scaling: Why is Animal Size so Important? Great book on biological scaling.

How Animals Work Great physiology book. Quirky, but fun.

Nonlinear Dynamics And Chaos: With Applications To Physics, Biology, Chemistry, And Engineering (Studies in Nonlinearity) Best book for a first course in nonlinear dynamics.

Life in Moving Fluids: The Physical Biology of Flow (Princeton Paperbacks) Best book I know of on biological fluid dynamics. Not too mathematical, but full of insight. I recommend all of Vogel's books.

Vital Circuits: On Pumps, Pipes, and the Workings of Circulatory Systems Great for understanding the fluid dynamics of the circulatory system.

Lady Luck: The Theory of Probability (Dover Books on Mathematics) I often find probability theory boring, but not this book. An oldie but goodie.

The Geometry of Biological Time (Interdisciplinary Applied Mathematics) Classic by Art Winfree, who was a leading mathematical biologists.  Be sure to get the 2nd edition.

When Time Breaks Down: The Three-Dimensional Dynamics of Electrochemical Waves and Cardiac Arrhythmias Winfree's classic on the nonlinear dynamics of the heart.

Cardiac Electrophysiology: From Cell to Bedside, 4e Comprehensive reference on cardiac electrophysiology.

Gasiorowicz

Quantum Physics,
by Stephen Gasiorowicz.

One of the standard topics in any modern physics class is blackbody radiation. Indeed, it was the study of blackbody radiation that led to the development of quantum mechanics. In Chapter 14 of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I write

The spectrum of power per unit area emitted by a completely black surface in the wavelength interval between λ and λ + dλ is … a universal function called the blackbody radiation function. …The description of [this] function …by Planck is one of the foundations of quantum mechanics… We can find the total amount of power emitted per unit surface area by integrating¹⁰ Eq. 14.32 [Planck’s blackbody radiation function]…[The result] is the Stefan-Boltzmann law.

As I was reading over this section recently, I was struck by the footnote number ten (present in earlier editions of our book, so I know it was originally written by Russ).

¹⁰This is not a simple integration. See Gasiorowicz (1974, p. 6).

This is embarrassing to admit, but although I am a coauthor on the 4th edition, there are still topics in our book that I am learning about. I always feel a little guilty about this, so recently I decided it is high time to take a look at the book by Stephen Gasiorowicz and see just how difficult this integral really is. The result was fascinating. The integral is not terribly complicated, but it involves a clever trick I would have never thought of. Because math is rather difficult to write in the html of this blog (at least for me), I will explain how to evaluate this integral through a homework problem. When revising our book for the 4th edition, I enjoyed finding “missing steps” in derivations and then creating homework problems to lead the reader through them. For instance, in Problem 24 of Chapter 14, Russ and I asked the reader to “integrate Eq. 14.32 over all wavelengths to obtain the Stephan-Boltzmann law, Eq. 14.33.” Then, we added “You will need the integral [integrated from zero to infinity]

$\int \; x$$3$$/(ex-1)\; dx\; =\; \pi$$4$$/15$ .

Below is a new homework problem related to footnote ten, in which the reader must evaluate the integral given at the end of Problem 24. I base this homework problem on the derivation I found in Gasiorowicz. In our book, we cite the 1974 edition.

Gasiorowicz, S. (1974) Quantum Physics. New York, Wiley.

This is the edition in Kresge library at Oakland University, and is the one I used to create the homework problem. However, I found using amazon.com’s “look inside” feature that this derivation is also in the more recent 3rd edition (2003). In addition, I found the derivation repeated in another of Gasiorowicz’s books, The Structure of Matter.

Problem 24 ½ Evaluate the integral given in Problem 24.
(a) Factor out e^−x, and then use the geometric series 1 + z + z² + z³ + …=1/(1−z) to replace the denominator by an infinite sum.
(b) Make the substitution y = (n+1) x.
(c) Evaluate the resulting integral over y, either by looking it up or (better) by repeated integration by parts.
(d) Make the substitution m=n+1
(e) Use the fact that the sum of 1/m⁴ from 1 to infinity is equal to π⁴/90 to evaluate the integral.

Really, who would have thought to replace 1/(1−z) by an infinite series? Usually, I am desperately trying to do just the opposite: get rid of an infinite series, such as a geometric series, by replacing it with a simple function like 1/(1−z). The last thing I would have wanted to do is to introduce a dreaded infinite sum into the calculation. But it works. I must admit, this is a bit of a cheat. Even if in part (c) you don’t look up the integral, but instead laboriously integrate by parts several times, you still have to pull a rabbit out of the hat in step (e) when you sum up 1/m⁴. Purists will verify this infinite sum by calculating the Fourier series over the range 0 to 2π of the function

f(x) = π⁴/90 – π² x²/12 + π x³/12 – x⁴/48

and then evaluating it at x = 0. (Of course, you know how to calculate a Fourier series, since you have read Chapter 11 of Intermediate Physics for Medicine and Biology). When computing Fourier coefficients, you will need to do a bunch of integrals containing powers of x times cos(nx), but you can do those by—you guessed it—repeated integration by parts. Thus, even if lost on a deserted island without your math handbook or a table of integrals, you should still be able to complete the new homework problem using Gasiorowicz’s method. I’m assuming you know how to do some elementary calculus—integrate by parts and simple integrals of powers, exponentials, and trigonometric functions—without looking it up. (Full disclosure: I found the function f(x) given above by browsing through a table of Fourier series in my math handbook. On that lonely island, you would have to guess f(x), so let’s hope you at least remembered to bring along plenty of scrap paper.)

I checked out the website for Gasiorowicz’s textbook. There is a lot of interesting material there. The book covers many of the familiar topics of modern physics: blackbody radiation, the photoelectric effect, the Bohr model for hydrogen, the uncertainty principle, the Schrodinger equation and more, all the way up to the structure of atoms and molecules. I learned this material from Eisberg and Resnick’s Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (1985), cited several times in Intermediate Physics for Medicine and Biology, when I used their book in my undergraduate modern physics class at the University of Kansas. For an undergraduate quantum mechanics class, I like Griffith’s Introduction to Quantum Mechanics, in part because I have taught from that book. But Gasiorowicz’s book appears to be in the same class as these two. I noticed that Gasiorowicz is from the University of Minnesota, so perhaps Russ knows him.

P.S. Did any of you dear readers notice that Russ and I spelled the name “Stefan” of the “Stefan-Boltzmann law” differently in the text of Chapter 14 and in Problem 24? I asked Google, and it found sites using both spellings, but the all-knowing Wikipedia favors “Stefan”. I’m not 100% certain which is correct (it may have to do with the translation from Slovene to English), but we should at least have been consistent within our book.