Public Schools in the United States Should Rename the “Free Lunch”

Posted on 8 August 2014 by RobertLovesPi

If you live in the USA, you are probably familiar with the phrase “free lunch,” or “free and reduced lunch,” as used in a public-school context. For those outside the USA, though, an explanation of what that phrase means, in practice, may be helpful, before I explain why a different name for such lunches should be used.

The term “free and reduced lunch” originated with a federal program which pays for school lunches, as well as breakfasts, with money collected from taxpayers — for students whose families might otherwise be unable afford these meals. The program’s eligibility requirements take into account both family income and size. There’s a problem with it, though: the inaccuracy of the wording used, especially the troublesome word “free.” The acronym above, “TANSTAAFL,” is familiar to millions, from the works of Robert A. Heinlein (science fiction author), Milton Friedman (Nobel-Prize-winning economist), and others. It stands for the informally-worded phrase, “There ain’t no such thing as a free lunch,” which gets to the heart of the problem with the terminology we use when discussing school lunches. (Incidentally, I have seen an economics textbook use the phrase “TINSTAAFL,” in its place, to change “ain’t no” to “is no.” I do not use this version, though, for I am unwilling to correct the grammar of a Nobel laureate.)

The principle that “free lunches” simply do not exist is an important concept in both physics and economics, as well as other fields. In physics, we usually call it the Law of Conservation of Mass and Energy, or the First Law of Thermodynamics. This physical law has numerous applications, and has been key to many important discoveries. Learning to understand it, deeply, is an essential step in the education of anyone learning physics. Those who teach the subject, as I have in many past years, have an even more difficult task: helping students reach the point where they can independently apply the TANSTAAFL principle to numerous different situations, in order to solve problems, and conduct investigations in the laboratory. It is a fundamental statement of how the universe works: one cannot get something for nothing.

TANSTAAFL applies equally well in economics, where it is related to such things as the fact that everything has a cost, and those costs, while they can be shifted, cannot be made to simply disappear. It is also related to the principle that intervention by governments in the economy always carries costs. For example, Congress could, hypothetically, raise the federal minimum wage to $10 per hour — but the cost of doing so would be increased unemployment, especially for those who now have low-paying jobs. Another possible cost of a minimum-wake hike this large would be a sudden spike in the rate of inflation, which would be harmful to almost everyone.

To understand what people have discovered about the fundamental nature of physical reality, physics must be studied. To understand what is known about social reality in the modern world, economics must be studied. Both subjects are important, and understanding the TANSTAAFL principle is vital in both fields. Unfortunately, gaining that understanding has been made more difficult, for those educated in the United States, simply because of repeated and early exposure to the term “free lunch,” from childhood through high school graduation. How can we effectively teach high school and college students that there are no free lunches, when they have already been told, incessantly, for many years, that such things do exist? The answer is that, in many cases, we actually can’t — until we have first helped our students unlearn this previously-learned falsehood, for it stands in the way of the understanding they need. It isn’t a sound educational practice to do anything which makes it necessary for our students to unlearn untrue statements.

I am not advocating abolition, nor even reduction, of this federal program, which provides essential assistance for many families who need the help. Because I am an American taxpayer, in fact, I directly participate in funding this program, and do not object to doing so. I do take issue, however, with this program teaching students, especially young, impressionable children in elementary school, something which is untrue.

We need to correct this, and the solution is simple: call these school lunches what they actually are. They aren’t free, for we, the taxpayers, pay for them. Nothing is free. We should immediately replace the phrase “free and reduced lunch” with the phrase “taxpayer-subsidized lunch.” The second phrase is accurate. It tells the truth, but the first phrase does the opposite. No valid reason exists to try to hide this truth.

A 240-Atom Fullerene, and Related Polyhedra

Posted on 5 August 2014 by RobertLovesPi

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 favorites 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.

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 hexagons each have their own color.

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.

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.)

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.

On “Digging to China”

Posted on 4 August 2014 by RobertLovesPi

When I was a little kid, my sister and I dug a big hole, in our front yard, and simply called it “the digging-hole.” It looked a lot like the hole shown above, except for the fact that, during daylight hours, our digging-hole usually included two small, dirt-covered, determined children, armed with plastic shovels. We tried, for years, to dig that hole as deep as possible. My personal goal, of course, was the Earth’s molten core, not India, and certainly not China.

Why do Americans so often talk about digging a hole straight down to China, anyway? Even if the Earth were solid all the way through its interior, digging straight down, from almost anywhere in the contiguous 48 states of the USA, would not put you in China, nor even India (which is, at least, closer to being correct than is China), but at the bottom of the Southern Indian Ocean. Salty water would suddenly rush into your newly-dug tunnel, killing you instantly, as soon as you got close to enough to the other side for the extreme water-pressure there to finish your digging project for you. The only exceptions to this watery doom would be coming out of the tunnel on one of the islands in that ocean, which would require great precision to hit deliberately.

Also, the fact that China and the USA are both Northern-hemisphere nations easily rules China out as the hypothetical “solid-earth” destination for Americans who dig straight down, and all the way through. If you could go through the center of the earth from North of the equator, you’d have to end up South of the equator. Isn’t that obvious? Don’t people look at globes?

“Evolution is just a theory.” Please STOP saying this!

Posted on 4 August 2014 by RobertLovesPi

Why?

Well, just to get started, these three things are also “just” theories:

1. Germs are the cause of many diseases.
2. Everything you have ever touched is made of atoms.
3. The spinning earth doesn’t fling us into outer space because of gravity.

Would any reasonable person actually think the phrase “just a theory” makes sense for any of these three things? Use of this phrase, for evolution, the Big Bang, or anything else, indicates one thing: the person talking does not understand the meaning of the word “theory.” Theories are the best science has to offer, and science is the foundation of modern civilization. These theories are based on the repeated testing of hypotheses, using experiment, to explain what we observe — so they are evidence-based explanations, not mere guesses, as the annoying phrase “just a theory” implies.

Evolution is every bit as well-established a theory as the three examples cited above. All theories are subject to further testing, which is an important self-correcting mechanism in science. No theory is beyond revision or replacement, if new experimental evidence calls for it. However, that fact doesn’t make any particular theory invalid — it simply helps explain why science works. It also works just as well whether people believe in it, or approve of it, or agree with it — or not.

If you want to disprove the theory of evolution, just find a fossilized rabbit in a one-billion-year-old rock, as J.M.S. Haldane famously observed. It will only take one such finding to accomplish your goal, and you can publish your results, and become famous – if you can find such a fossil. For your own safety, though, please do not hold your breath while looking.

Craters and Slopes Near the South Pole of the Moon Adorn the Faces of a Rhombic Enneacontahedron

Posted on 3 August 2014 by RobertLovesPi

The images on the faces of this polyhedron are based on information sent from NASA’s Lunar Reconnaisance Orbiter, as seen at http://lunar.gsfc.nasa.gov/lola/feature-20110705.html and tweeted by @LRO_NASA, which has been happily tweeting about its fifth anniversary in a polar lunar orbit recently. I have no idea whether this is actually an A.I. onboard the LRO, or simply someone at NASA getting paid to have fun on Twitter.

To get these images from near the Lunar South Pole onto the faces of a rhombic enneacontahedron, and then create this rotating image, I used Stella 4d: Polyhedron Navigator. There is no better tool available for polyhedral research. To check this program out for yourself, simply visit www.software3d.com/Stella.php.

Surface Gravitational Field Strengths for Numerous Solar System Objects

Posted on 1 August 2014 by RobertLovesPi

It isn’t difficult to find rankings for the most massive objects in the solar system, rankings of objects in terms of increasing distance from the sun, or rankings of objects by radius. However, ranking objects by surface gravitational field strength is another matter, and is more complicated, for it is affected by both the mass and radius of the object in question, but in different ways. If two objects have different masses, but the same radius, the gravitational field strength will be greater for the more massive object. However, increasing the radius of an object decreases its surface gravitational field strength, in an inverse-square relationship.

Gravitational field strength is measured in N/kg, which are equivalent to m/s², the units for acceleration. The terms “gravitational field strength” and “acceleration due to gravity,” both of which are symbolized “g,” are actually synonymous. I prefer “gravitational field strength” because referring to acceleration, when discussing the weight of a stationary object on the surface of a planet, can cause confusion.

Use of the numbers given below is easy: given the mass of a thing (an imaginary astronaut, for example), in kilograms, simply multiply this figure by the given gravitational field strength, and you’ll have the weight of the thing, in newtons, on the surface of that planet (or other solar system object). If, for some odd reason, you want the weight in the popular non-metric unit known as the “pound,” simply divide the weight (in newtons) by 4.45, and then change the units to pounds.

How is surface gravitational field strength determined? To explain that, a diagram is helpful.

The large green circle represents a planet, or some other solar system object, and the blue thing on its surface, which I’ll call object x, can be pretty much anything on the solar system object’s surface. There are two formulas for F_g, the force of gravity pulling the planet and the thing on its surface toward each other. One is simply F_g= m_xg, a form of Newton’s Second Law of Motion, where “g” is the gravitational field strength, and m_xis the mass of the object at the surface. The other formula is more complicated: F_g= (Gm_xm_p)/r². This is Newton’s Law of Universal Gravitation, where “G” (not to be confused with “g”) is the universal gravitational constant, 6.67259 x 10^-11Nm²/kg², and m_pand r are the mass and radius of the planet (or other solar system object). Because they each equal F_g, the expressions m_xg and (Gm_xm_p)/r² can be set equal to each other, yielding the equation m_xg = (Gm_xm_p)/r², which becomes g = (Gm_p)/r² after m_xis cancelled. The mass of the object on the surface is not needed — “g” is simply a function of m_pand r.

There is a problem, however, with the idea of “surface” gravitational field strength — and that is the fact that the five largest objects in the solar system, the sun and the gas giants, all lack visible solid surfaces. One cannot stand on Jupiter — if you tried, you’d simply fall inside the planet. Therefore, for Jupiter, picture a solid platform floating at the top of the visible clouds there, and place the test object on this solid platform. Under those conditions, multiplying the test object’s mass by the Jovian value of “g” will, indeed, yield the weight of the object there, as it could be measured by placing it on a bathroom scale, at rest on the floating platform. For the other gas giants, as well as the sun, the idea is the same.

The objects included in the list below are the sun, all eight major planets, all dwarf planets (and dwarf planet candidates) with known values of “g,” all major satellites, some minor satellites, and a few of the largest asteroids. Many more objects exist, of course, but most have values for “g” which are not yet known.

Here are the top five:

Sun/Sol, 274.0 N/kg

Jupiter, 24.79 N/kg

Neptune, 11.15 N/kg

Saturn, 10.44 N/kg

Earth/Terra, 9.806 65 N/kg

The top five, alone, make me glad I undertook this project, for I did not realize, before doing this, that our planet has the highest surface gravitational field strength of any object in the solar system with a visible solid surface.

The next five include the rest of the major planets, plus one Jovian moon.

Venus, 8.87 N/kg

Uranus, 8.69 N/kg

Mars, 3.711 N/kg

Mercury, 3.7 N/kg

Io, 1.796 N/kg

The third set of five are all planetary moons, starting with earth’s own moon. The others are Jovian moons, except for Titan, which orbits Saturn.

Moon/Luna, 1.622 N/kg

Ganymede, 1.428 N/kg

Titan, 1.352 N/kg

Europa, 1.314 N/kg

Callisto, 1.235 N/kg

The fourth set of five begins with the largest dwarf planet, Eris, and includes two other dwarf planets as well.

Eris, 0.827 N/kg (dwarf planet)

Triton, 0.779 N/kg (Neptune’s largest moon)

Pluto, 0.658 N/kg (dwarf planet)

Haumea, 0.63 N/kg (dwarf planet)

Titania, 0.38 N/kg (largest moon of Uranus)

The fifth set of five includes the remaining dwarf planets with known values of “g.”

Oberon, 0.348 N/kg (moon of Uranus)

1 Ceres, 0.28 N/kg (dually classfied: dwarf planet and largest asteroid)

Charon, 0.278 N/kg (largest moon of Pluto)

Ariel, 0.27 N/kg (moon of Uranus)

90482 Orcus, 0.27 N/kg (probable dwarf planet)

The sixth set of five are dominated by Saturnian moons.

Rhea, 0.265 N/kg (Saturnian moon)

4 Vesta, 0.25 N/kg (2^nd largest asteroid)

Dione, 0.233 N/kg (Saturnian moon)

Iapetus, 0.224 N/kg (Saturnian moon)

Umbriel, 0.2 N/kg (moon of Uranus)

The seventh set of five are mostly asteroids.

704 Interamnia, 0.186 N/kg (5^th most massive asteroid)

2 Pallas, 0.18 N/kg (3^rd most massive asteroid)

Tethys, 0.147 N/kg (Saturnian moon)

52 Europa, 0.14 N/kg (7^th most massive asteroid)

3 Juno, 0.12 N/kg (large asteroid, w/~1% of mass of the asteroid belt)

Starting with the eighth group of five, I have much less certainty that something may have been omitted, although I did try to be thorough. My guess is that most future revisions of this list will be necessitated by the discovery of additional dwarf planets. Dwarf planets are hard to find, and there may be hundreds of them awaiting discovery.

Enceladus, 0.114 N/kg (Saturnian moon)

Vanth, 0.11 N/kg (moon of probable dwarf planet 90482 Orcus)

10 Hygiea, 0.091 N/kg (4^th most massive asteroid)

15 Eunomia, 0.08 N/kg (large asteroid, with ~1% of mass of asteroid belt)

Miranda, 0.079 N/kg (moon of Uranus)

Here is the ninth group of five:

Nereid, 0.072 N/kg (Neptunian moon; irregular in shape)

Proteus, 0.07 N/kg (Neptunian moon; irregular in shape)

Mimas, 0.064 N/kg (Saturnian moon / smallest gravitationally-rounded object in the solar system)

Puck, 0.028 N/kg (6^th largest moon of Uranus)

Amalthea, 0.020 N/kg (5^th largest Jovian moon)

Finally, here are “g” values for the two tiny moons of Mars, included because they are nearby, and are the only moons Mars has to offer. A more exhaustive search would reveal many asteroids and minor satellites with “g” values greater than either Martian moon, but smaller than Amalthea, the last solar system object shown in the last set of five.

Phobos, 0.0057 N/kg

Deimos, 0.003 N/kg

The Unintentional Bomb: A True Story

Posted on 31 July 2014 by RobertLovesPi

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.

Pie Chart for Main-Belt Asteroid Masses

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Pie Chart for Main-Belt Asteroid Masses

I looked up enough asteroid masses to use them to make this pie chart. I now have three reactions.

First: oh, that’s why only Ceres is round!

Second: who was stupid enough to name an asteroid Europa? That name is taken!

Third: wow — those small ones sure do make up a lot of the total!

A Simulation of Crystalline Growth Using Polyhedral Augmentation

Posted on 29 June 2014 by RobertLovesPi

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:

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.
Don’t allow one tetrahedral cluster to overlap another one.
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.]

The Sun, On a Trip Through the Electromagnetic Spectrum

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The image above shows the sun’s output of radio waves, which have the longest wavelengths, lowest frequencies, and lowest energies of any part of the electromagnetic spectrum.

This image, above, shows the sun’s microwave output.

Next, infrared:

This next one should be familar. It’s visible light. (Don’t stare at the sun, though.)

Moving on through the spectrum, ultraviolet is next:

After that, x-rays:

And, finally, we arrive at the other side of the spectrum, where the electromagnetic radiation has its shortest wavelengths, and highest frequencies, as well as energy per photon. This is the sun in gamma rays: