“You Majored in WHAT?”

I’m in my twentieth year of teaching mostly science and mathematics, so it is understandable that most people are surprised to learn that I majored in, of all things, history.

It’s true. I focused on Western Europe, especially modern France, for my B.A., and post-WWII Greater China for my M.A. My pre-certification education classes, including student teaching, were taken between these two degree programs.

Student teaching in social studies did not go well, for the simple reason that I explain things by reducing them to equations. For some reason, this didn’t work so well in the humanities, so I took lots of science and math classes, and worked in a university physics department, while working on my history M.A. degree, so that I could job-hunt in earnest, a year later, able to teach physics and chemistry. As it ended up, I taught both my first year, along with geometry, physical science, and both 9th and 12th grade religion. Yes, six preps: for an annual salary of US$16,074.

History to mathematics? How does one make that leap? In my mind, this explains how:

  • History is actually the story of society over time, so it’s really sociology.
  • Sociology involves the analysis on groups of human minds in interaction. Therefore, sociology is actually psychology.
  • Psychology is the study of the mind, but the mind is the function of the brain, one of the organs of the human body. Psychology, therefore, is really biology.
  • Biological organisms are complex mixtures of interacting chemicals, and, for this reason, biology is actually chemistry.
  • Chemistry, of course, breaks down to the interactions of electrons and nuclei, governed by only a few physical laws. Chemistry, therefore, is really physics.
  • As anyone who has studied it knows, physics often involves more mathematics than mathematics itself.

…And that at least starts to explain how someone with two history degrees ended up with both a career, and an obsession, way over on the mathematical side of academia.

Buckminsterfullerene Molecular Models: Three Different Versions

Buckminsterfullerene, a molecule made of 60 carbon atoms, and having the shape of a truncated icosahedron, is easily modeled with Stella 4d: Polyhedron Navigator (see http://www.software3d.com/Stella.php to try or buy this program). The first image shows the”ball and stick” version used by chemists who want the bonds between atoms to be visible.

Trunc Icosa
The second model is intermediate between the ball-and-stick version, and the space-filling version, which follows it.

Trunc Icosa2

Here’s the “closely packed” space-filling version, taken to an extreme.

Trunc Icosa3

Which version better reflects reality depends on the certainty level you want for molecular orbitals. A sphere representing 99% certainty would be larger than one for 95% certainty.

Do Not Drink the Twenty Proof Gasoline!


We’ve all seen labels like this, stuck to gasoline pumps. While filling up my car’s gas tank earlier today, I felt compelled to take a picture of this familiar label — because I suddenly realized that what this small sign actually means is that the alcohol content of the gasoline being sold (in an area where liquor sales are illegal, no less) might be as much as twenty proof.

Twenty proof gasoline. Twenty proof gasoline! One never thinks of it this way, but it is both mathematically and chemically accurate. There are many different alcohols, but the one people drink for purposes of intoxication, and the one found in this gasoline, are the exact same molecule: C2H5OH. I then realized that the people who design these labels are being sneaky with the wording on purpose, for they don’t put “contains alcohol,” or anything like that, on these stickers found on gas pumps all over the place.

The reason for use of the official, less-familiar chemical term “ethanol” then became both obvious, and horrifying, all at once. Gas pumps must be labeled this way because there are people out there who are so incredibly stupid that they would actually drink gasoline if they knew it contained, well, booze.

What’s more, there is an unwritten assumption in play here, and I think (or at least hope) it is a valid one: anyone sufficiently educated to know that “ethanol” and the “the alcohol people drink to get drunk” are synonyms is also, presumably, smart enough to know better than to drink gasoline. Drinking gasoline would, of course, be dangerous in the extreme. Even inhaling gasoline fumes is hazardous, but drinking the stuff would be far worse. Consuming enough of this ethanol-containing gasoline to actually get drunk would, in fact, very likely be fatal, due to the mixture of toxic hydrocarbons present, in addition to the alcohol. The most toxic component of gasoline with which I am familiar is benzene, a potent carcinogen. Benzene is really nasty stuff, if it somehow makes it into a human body.

So, for the record, do not drink the up-to-twenty-proof gasoline — even though that is an accurate way to describe it.

My Aqua Regia Story

This is my twentieth year teaching, but only the first year when I have not taught at least one class in chemistry, and I miss it. One of my fondest memories of chemistry lab involves the one time I experimented with aqua regia — a mixture of acids which, unlike any single acid, can dissolve both gold and platinum, the “noble metals.” I had read a story of a scientist’s gold Nobel Prize being protected from the Nazis by dissolving it in aqua regia, and then recovering the gold from solution after World War II had ended. Having read about this, I wanted to try it myself, and also thought it would make an excellent lab for classroom use — if I could figure out how to recover the gold, and also learn what precautions would be needed to allow high school students to perform this experiment safely. For sensible and obvious reasons, I conducted a “trial run” without students present, but with another chemistry teacher nearby, since aqua regia, and the gases it produces when dissolving gold, are quite dangerous. Someone else has put a video on YouTube, showing aqua regia dissolving gold, so you can see something much like what I saw, simply by watching this video.

First, I obtained one-tenth of a troy ounce of gold, which cost about $80 at the time. I had read about the extreme malleability of gold, one of the softest metals, and wanted to see evidence for it for myself — so, before I prepared the the aqua regia, I used a hammer to try flattening the gold sample into a thin sheet. That didn’t work, but it didn’t take long for me to figure out why — I had accidentally bought gold coin-alloy, which is 10% copper, not pure gold. Since this alloy is far less malleable than pure gold, my attempt to flatten it had failed, but I also knew this would not pose a problem for my primary experiment — the one involving aqua regia. Also, I didn’t have another spare $80 handy, to purchase another 1/10 troy ounce of pure gold, so I proceeded to make, for the first time in my life, a small amount of aqua regia — Latin for “royal water.”

Unlike what is shown in the video above, I prepared the acid-mixture first, before adding the gold, using a slightly-different recipe:  the traditional 1:4 ratio, by volume, of concentrated nitric acid to concentrated hydrochloric acid. Both these acids look (superficially) like water, but the mixture instantly turned yellow, and started fuming, even before anything was added to it. Wearing full protective gear, I watched it for a few minutes — and then, using tongs held by gloved hands, lowered my hammer-bashed sample of gold into the fuming, yellow mixture of concentrated acids.

It worked. It was a fascinating reaction, and a lot of fun to watch. At approximately the same time that the last of my gold sample dissolved, something occurred to me:  I had failed to research how to recover the dissolved gold from the resulting solution! No problem, I thought — I can figure this out. (I am seldom accused of lacking self-confidence, even when I’m wrong.)

My first idea was to use a single-replacement reaction. Many times, I have had students extract pure silver from a solution of silver nitrate by adding a more-active metal, such as copper. The copper dissolves, replacing the silver in the silver nitrate solution, and silver powder forms, as a precipitate, on the surface of the copper. Thinking that a similar process could be used to precipitate out the gold from my gold / aqua regia mixure, I simply added come copper to the reaction beaker. The corrosive properties of my aqua regia sample had not yet been exhausted, though, and so the remaining aqua regia simply “ate” the copper. The result was a mess — I had only succeeded in turning an already-complicated problem into an even-more-complicated problem, by adding more chemicals to the mixture. More attempts to turn the gold ions back into solid gold dust, using other chemicals, followed, but all of them failed. Finally, I used a strong base, sodium hydroxide, to neutralize the still-acidic mixture, and then, disgusted by my failure to recover the gold, found a way to safely dispose of the mixture, and did so.

In retrospect, I think I know where I messed up — I should have neutralized the remaining acids in the mixture with sodium hydroxide first, before adding copper to cause the gold to precipitate out, in a no-longer-acidic solution of ions with much less hydronium present. That, I think, will work, and I do intend to try it sometime — after doing more research first, to increase my level of certainty, and also after waiting for the current price of gold to drop to less-expensive levels. Right now, after all, a tenth of a troy ounce of gold costs roughly $120, not a mere $80.

As for the lost $80, I’m not upset about that anymore. I definitely learned things while doing this, and now view the $80 spent as simply the cost of tuition for an educational experience.

The “Destabilized” Element, Bismuth, Plus Others Which May Join It Soon

There is a chemical element, bismuth, which many people — even chemists — think has at least one stable isotope. However, the truth, discovered in 2003 (but still not well-known), is that it has no stable isotopes, but does have one with an extremely long half-life — so long that it, and other isotopes with similarly-long half-lives, are often deemed “effectively stable.” Bismuth is shown in green on the table, and its “effectively stable” isotope, bismuth-209,  has a half-life of at least 1.9 x 1019 years. For comparison, it has “only” been ~1.38 x 1010 years since the Big Bang. Bismuth-209’s half-life is, therefore, over a billion times longer than the total amount of time which has existed, so far.


In addition, the yellow boxes indicate elements which have only radioactive and “observationally stable” isotopes. “Observationally stable” means that radioactivity (in some cases, even the spontaneous-fission variety), with an extremely long half-life, is predicted, or at least thought to be possible, but no actual decay has yet been observed — so the yellow elements’ perhaps-stable, perhaps-not isotopes are “on watch.” The red boxes, by contrast, are for elements which have been long-known to have no stable isotopes.

None of this takes into consideration the unresolved issue of hypothesized long-term proton decay. If protons turn out to be unstable, all atoms likely are as well, unless simply having them exist in atoms somehow stabilizes them, as is the case for neutrons, which decay in isolation, but do not in stable nuclei. This is an area of uncertainty — another way of saying that this is something which needs further study.

The Eleven Oddball Symbols on the Periodic Table of the Elements

periodic table oddballs

Most symbols for elements on the periodic table are easy to learn, such as those for carbon, oxygen, and nitrogen:  C, O, and N. There are eleven “oddballs,” though, because their symbols originated in other languages (Latin, mostly), and do not match their English names. Here’s a list of them, by atomic number, with an explanation for each.

11. Na stands for sodium because this element used to be called natrium.

19. K stands for potassium, for this element’s name used to be kalium.

26. Fe stands for iron because this element was formerly named ferrum.

29. Cu stands for copper because it used to be called cuprum.

47. Ag’s (silver’s) old name was argentum.

50. Sn’s (tin’s) name used to be stannum.

51. Antimony’s symbol, Sb, came from its former name, stibium.

74. Tungsten, with the symbol W, was once called wolfram. In some parts of the world, it still goes by that name, in fact.

79. Gold (Au) was called aurum in past centuries.

80. Mercury’s (Hg’s) old name is impossible (for me, anyway) to say five times, quickly:  hydrargyrum.

82. Lead (Pb) was once called plumbum because plumbers used it to weight the lower end of plumb-lines.

I think learning things is easier, with longer retention, if one knows the reasons behind the facts, rather than simply attempting rote memorization.

A 240-Atom Fullerene, and Related Polyhedra

The most well-known fullerene has the shape of a truncated icosahedron, best-known outside the world of geometry as the “futbol” / “football” / “soccer ball” shape — twenty hexagons and twelve pentagons, all regular. The formula for this molecule is C60. However, there are also many other fullerenes, both larger and smaller. One of my favorite is C240, simply because I sometimes make class projects out of building fullerene models with Zome (available at www.zometool.com), and the 240-atom fullerene is the largest one which can be built using Zome. Here’s what it looks like, as molecular models are traditionally colored.

C240 fullerene 2

This polyhedron still has twelve pentagons, like its smaller “cousin,” the truncated icosahedron, but far more hexagons. What’s more, these hexagons do not have exactly the same shape. If this is re-colored in the traditional style of a polyhedron, rather than a molecule, it looks like this. In this image, also, the different shapes of hexagon each have their own color.

C240 fullerene 1

Like other polyhedra, a compound can be made from this polyhedron and its dual. In this case, the dual’s faces are shown, below, as red triangles. The original fullerene-shape is in purple for the pentagonal faces, and orange for the hexagons.

C240 compound with dual

In the base/dual compound above, it can be difficult to tell exactly what this dual is, but that can be clarified by removing the original fullerene. What’s left is called a geodesic sphere — or, quite informally, a ball made of many triangles. The larger a fullerene is, the more hexagonal rings/faces it will have, and the more triangles will be found on the geodesic sphere which is its dual. For the 240-atom fullerene shown repeatedly, above, here is the dual, by itself, with different colors indicating slightly different triangle-shapes. (An exception is the yellow and green triangles, which are congruent, but have different colors for aesthetic reasons.)

C240 dual

I made these four rotating images using Stella 4d:  Polyhedron Navigator. To try this program for yourself, simply visit www.software3d.com/Stella.php. At that site, there is a free trial download available.


The Unintentional Bomb: A True Story

picric acid

Nineteen years ago, I began my teaching career at a small, private Arkansas high school. One of the classes I taught was Chemistry, and my principal happened to be a former chemistry teacher, himself.  We were both new to the school, and knew that there was a high turnover rate there for teachers in that field. They’d had perhaps eight teachers for that class in the previous five years. I stayed there six years, teaching chemistry every year.

The new principal saw the need for upgraded laboratory facilities, and we got them, including a new, larger chemical stockroom. The old stockroom was a nightmare, and the chemicals needed to be transferred to their new home. This was a massive undertaking, for many of my predecessors had ordered chemicals, not taking the time to inventory the stockroom to see if the school already had what they needed. Even worse, the chemicals were stored in approximate alphabetical order.

Experienced chemists and chemistry teachers know how scary the phrase “alphabetical order” is, in this context. For reasons of safety, chemicals need to be stored by families, using a shelving pattern that keeps incompatible chemicals far apart. I was not an experienced teacher of anything at this point, but the principal showed me the classification scheme he’d used before, himself. It’s the one recommended by Flinn Scientific, and you can see it at http://www.flinnsci.com/store/Scripts/prodView.asp?idproduct=16069. At his direction, over a couple of weeks, I took the chemicals from the old storage area to the new one, de-alphabetizing them into a much safer arrangement, onto category-labelled shelves. In the process, of course, I saw every laboratory chemical that school had, recognizing many (jar after jar of liquid mercury, for example) as highly dangerous, and making certain proper precautions were taken with such substances. If I didn’t recognize a chemical well enough to categorize it (sulfates together, halides together, etc.), I looked it up, in order to find its place. I wouldn’t even open a container with an unfamiliar chemical in it, until researching it. As it turned out, my caution with unfamiliar chemicals literally saved my life.

There are hundreds of different acids, and I doubt anyone knows them all. When I encountered a hand-labeled jar reading “picric acid,” I had never heard of that chemical, the structure for which is shown above. When I looked it up, I learned picric acid is safe if it is all in solution with water, but is a shock-sensitive explosive in solid form. I examined the liquid carefully, without actually touching the container. Sure enough, solid crystals had already started to form, over the years, as some of the water in the container slowly evaporated, and escaped.

Great, I thought, sarcastically — a shock-sensitive explosive. I then kept reading the hazard alerts, and noticed that they stated that picric acid should never be stored in any container with a metallic lid, because that invites the formation of explosive metal picrates which can be detonated simply by the friction caused by an attempt to open the lid. The picric acid I was dealing with, of course, not only had the dangerous solid crystals — it also had a metal lid, and a partially corroded one at that.

I never so much as touched that lid. Very carefully, I gently carried this container to the new stockroom, gave it a shelf all by itself, and didn’t so much as give it a nasty look, for the rest of the time I taught there. Leaving it alone, with me being the only person with access to that room, was the safest thing I could think of to do, as long as I was teaching there. For six school years, since it was carefully undisturbed, the picric acid behaved itself — and then, seeking a higher salary, I found a job for the following Fall, teaching at a public school. I knew I would not be able to leave this private school, though, without dealing with this picric acid problem once and for all, along with other dangerous chemicals the school did not need. I could have simply turned my keys in, and left, but that would have risked a potentially-fatal explosion in that school in future years, for I could not safely assume the next chemistry teacher would be familiar with, nor research, picric acid. My conscience would not permit that.

The school year being over, I went to see the school’s new principal. Unlike his predecessor, the new principal had never taught chemistry, but he’d been on the faculty, before his promotion, for longer than I had been there, and so we knew each other well. When I went into his office, with my keys, for end-of-the-year checkout, and calmly told him that there were many serious toxins and an unexploded bomb down the hall, he knew immediately that I wasn’t joking. With his permission, I kept my keys into the beginning of the Summer, getting things ready for professional chemical-disposal experts to come in and remove the dangerous materials. Before long, four cardboard boxes had been filled with dangerous chemicals the school did not need, slated for disposal — and that’s after I had already disposed of most things that needed to go, if I had the knowledge, and means, to dispose of them properly.

The first group of professionals who were called in, for help, were from the local fire department. They took some of the chemicals away, without charge, but only the ones that they knew how to deal with safely. The principal and I were informed that, for the remaining chemicals (down to one box now, in which was the picric acid), a professional “hazmat” team would need to be called in, and it wouldn’t be cheap.

It wasn’t. The bill from the hazmat team exceeded US$2000. They took away three or four kilograms of mercury, as well as a lot of other nasty stuff, but also told us, with apologies, that they weren’t taking the picric acid, it being too dangerous for a “mere” professional hazmat team. To get rid of that, we were told, we’d need to call in the bomb squad from the state’s capital city, Little Rock.

I had heard the phrase “bomb squad” in movies, and on TV, but not in real life. Judging from the look on his face, the same can be said for the principal. As it happened, I wasn’t in town on the day the bomb squad came to school, but I did hear numerous first-hand accounts of what transpired, when I came back the following day to turn in my keys.

One of many surprises reported to me by these witnesses is that the FBI arrived with the bomb squad, asking questions and interviewing people. Apparently there wasn’t supposed to be any picric acid in Arkansas schools, for a statewide sweep had been made to gather it all up, and dispose of it, in the 1970s. My guess, and that’s all it is, is that this very old bottle had been overlooked because of it being in a private, rather than a public, school. If the FBI wants to contact me now to ask me questions about this stuff, I’ll answer them, but, at the time, I didn’t mind a bit that I missed out on the interrogation-portion of these events. After the FBI had finished their on-site investigation, the bomb squad began their work.

This K-12 school has a very large campus, with multiple buildings, and my classroom was at one corner of it. The disposal site they chose — the nearest area sufficiently remote from people and buildings — was far behind the gymnasium, at least half a kilometer away, at the opposite corner of the campus. As it was described to me, two bomb squad guys put on what I call “moon suits,” wrapped the picric acid bottle up, with a lot of padding, and placed this padded bundle on a stretcher.  They then walked the stretcher, with its deadly cargo, around and between buildings, across railroad tracks and a street, around the gymnasium, and back into an empty lot, where a deep hole was dug. One of the guys in moon suits then put the picric acid container at the bottom of the hole, along with a stick of dynamite, the idea being to use the smaller dynamite explosion to trigger the much larger explosion of the picric acid.

The bomb-squad “astronaut” lit the long fuse on the dynamite, and scrambled out of the hole as quickly as his moon suit would permit. The fuse burned, right up to the dynamite — and then, just as everyone expected a deafening explosion, it fizzled out. They had unknowingly used a stick of dynamite with a defective fuse.

After waiting a while, just to give the dynamite time to, well, change its “mind” about exploding (which didn’t happen), the suited-up bomb squad guy was sent back into the hole, with a second stick of dynamite, which he placed next to the first one. I hope he got paid extra for this, for I would have quit, immediately, rather than re-enter that hole. He, however, did enter, lit the second dynamite stick, and got out in time. This time, the detonation was successful, and the picric acid and both sticks of dynamite were utterly obliterated.

At the time of the explosion, a former student of mine, who had graduated from this same school a few years before, was working in an office building, three or four kilometers away. I got an e-mail from him, and laughed when I read it. Apparently the entire building he was working in had just been shaken by an explosion in the direction of his former school, and he had one question for me:  had I had anything to do with this? I laughed, and replied with an honest answer.

A Simulation of Crystalline Growth Using Polyhedral Augmentation

Crystals and crystalline growth have been studied for centuries because of, at least in part, their symmetry. Crystals are cut in such a way as to increase this symmetry even more, because most people find symmetry attractive. However, where does the original symmetry in a crystal come from? Without it, jewelers who cut gemstones would not exist, for the symmetry of crystalline minerals themselves is what gives such professionals the raw materials with which to work.

To understand anything about how crystals grow, one must look at a bit of chemistry. The growth of crystals:

  • Involves very small pieces:  atoms, molecules, ions, and/or polyatomic ions
  • Involves a small set of simple rules for how these small pieces attach to each other

Why small pieces? That’s easy:  we live in a universe where atoms are tiny, compared to anything we can see. Why is the number of rules for combining parts small, though? Well, in some materials, there are, instead, large numbers of ways that atoms, etc., arrange themselves — and when that happens, the result, on the scale we can see, is simply a mess. Keep the number of ways parts can combine extremely limited, though, and it is more likely that the result will possess the symmetry which is the source of the aesthetic appeal of crystals.

This can be modeled, mathematically, by using polyhedral clusters. For example, I can take a tetrahedron, and them augment each of its four faces with a rhombicosidodecahedron. The result is this tetrahedral cluster:


Next, having chosen my building blocks, I need a set of rules for combining them. I choose, for this example, these three:

  1. Only attach one tetrahedral cluster of rhombicosidodechedra to another at triangular faces — and only use those four triangles, one on each rhombicosidodecahedron, which are at the greatest distance from the cluster’s center.
  2. Don’t allow one tetrahedral cluster to overlap another one.
  3. When you add a tetrahedral cluster in one location, also add others which are in identical locations in the overall, growing cluster.

Using these rules, the first augmentation produces this:


That, in turn, leads to this:


Next, after another round of augmentation:


One more:


In nature, of course, far more steps than this are needed to produce a crystal large enough to be visible. Different crystals, of course, have different shapes and symmetries. How can this simulation-method be altered to model different types of crystalline growth? Simple:  use different polyhedra, and/or change the rules you select as augmentation guidelines, and you’ll get a different result.

[Note:  all of these images were created using Stella 4d: Polyhedron Navigator. This program is available at http://www.software3d.com/Stella.php.]


A Truncated Icosahedron with Sixty Extra Hexagons


A Truncated Icosahedron with Sixty Extra Hexagons

I created this using Stella 4d, which is available (including a free trial download) at http://www.software3d.com/Stella.php. With adjustments in edge lengths to make the bond lengths correct, this would be the shape of a C180 fullerene molecule.

If the thirty-two faces of the truncated icosahedron are hidden, and only the sixty extra hexagons are visible, this polyhedron looks like this:

Dual of Geodesic Trunc Icosa

In “rainbow color mode,” it has an even more interesting appearance:

Dual of Geodesic Trunc Icosa