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

Zonohedrified Dodeca

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.

Music Video: Murder By Death’s “Those Who Stayed” & “I’m Afraid of Who’s Afraid of Virginia Woolf”

Music: the first two tracks from the Murder By Death album Like the Excorcist, But More Breakdancing. Please visit their website, http://www.murderbydeath.com, to buy this band’s music and merchandise. While you’re there, I recommend checking their concert calendar, to see if they may be playing near you soon. Murder By Death concerts, which I’ve seen six times now, are not to be missed!

Visuals: rotating polyhedra, all with icosidodecahedral symmetry, generated using Stella 4d: Polyhedron Navigator, which you can try for yourself at http://www.software3d.com/Stella.php. The polyhedra shown are, in order of appearance:

  1. The icosahedron
  2. The compound of the icosahedron and its dual, the dodecahedron
  3. The dodecahedron, with all faces the same color
  4. The small stellated dodecahedron, or first stellation of the dodecahedron, in a single color
  5. The small stellated dodecahedron, with only parallel faces having the same color (six-color arrangement)
  6. The great dodecahedron, or second stellation of the dodecahedron, six-color arrangement
  7. The great stellated dodecahedron, or third stellation of the dodecahedron, six-color arrangement
  8. Stellating the dodecahedron a fourth time, to return it to its original form, but in the six-color arrangement this time
  9. The icosidodecahedron, with triangular faces invisible, and pentagonal faces shown using the six-color arrangement
  10. The icosidodecahedron, all faces visible now, and colored by face type
  11. The fourth stellation of the icosidodecahedron (its first stellation is the dodecahedron, the second is the icosahedron, and the third is the compound of the first two, all of which have already been seen)
  12. The fifth stellation of the icosidodecahedron
  13. The convex hull of the fifth stellation of the icosidodecahedron, which is a slightly-truncated icosahedron
  14. The truncated icosahedron which is a true Archimedean Solid, since all its faces are regular
  15. The truncated icosahedron’s second stellation (the first is the already-seen icosahedron)

Heptamandala

heptamandala

On Polyhedral Cages, a Form of Geometrical Art, with Seven Examples

I’ve posted polyhedral cages before — it simply never occurred to me to call such objects by that term. I often tag them as art / geometrical art, rather than mathematics, for they are not true polyhedra, by the generally accepted definition, where edges must involve faces meeting in pairs. Polyhedral cages do not follow this rule, so calling them mathematics causes problems. To do mathematics, after all, is to play games with numbers, and other ideas, according to the rules, with these rules being discovered as we discover new theorems. The rules are respected for one reason alone:  we know they work. If one breaks these rules with, say, a geometric figure, on aesthetic grounds, one is crossing the boundary between mathematics and geometrical art.

The reason for hiding faces of polyhedra is usually aesthetic, not mathematical. I use software called Stella 4d, available at www.software3d.com/Stella.php, to manipulate polyhedra in numerous different ways, trying to discover “new” polyhedra — new, that is, in the sense that these discoveries (not inventions) were never seen before I saw them on my computer screen. When you see a rotating geometrical picture on this blog, such as any of the ones at the bottom of this post, it was created using Stella.

Every now and then, I stumble upon a polyhedron which would look better if selected faces were simply made to disappear — and with Stella, that’s easy. They still exist in the polyhedron, in Stella‘s “mind,” but are rendered invisible in the on-screen image, thus creating the appearance of holes in the polyhedron’s surface. If these holes are regarded as real — “real” in the somewhat confusing sense that there’s nothing where the holes are, holes being absences of what surrounds them — then the former polyhedron is now a polyhedral cage. Here are several examples.  

electric dodecahedron

6xDual of Convex hullhollow rtc chiral varietyZonohedrifiedgfd DodecaZonohedrified DodecaFaceted Cogdfngfvex hull

Dodecahedral Cluster of Cuboctahedra and Icosidodecahedra

Augmented IcosidodDSJFGSca

I made this using Stella 4d:  Polyhedron Navigator, software you may try for yourself at http://www.software3d.com/Stella.php.

Seven Polyhedra with Icosidodecahedral Symmery

Stellated Dual of Cghjonvex hullstellated multiply Penta Hexeconta rainbowConvex hull idConvex hull of a strombic hexacontahedron augmented by 60 more strombic hexacontahedrastellated multiply Penta Hexeconta rainbow 2thingUV

I made all of these using Stella 4d:  Polyhedron Navigator.  You may try this software for yourself at www.software3d.com/Stella.php.

 

Surface Gravitational Field Strengths for Numerous Solar System Objects

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.

gravity

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 Fg, the force of gravity pulling the planet and the thing on its surface toward each other. One is simply Fg= mxg, a form of Newton’s Second Law of Motion, where “g” is the gravitational field strength, and mx is the mass of the object at the surface. The other formula is more complicated:  Fg= (Gmxmp)/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 10-11 Nm²/kg², and mp and r are the mass and radius of the planet (or other solar system object). Because they each equal Fg, the expressions mxg and (Gmxmp)/r² can be set equal to each other, yielding the equation mxg = (Gmxmp)/r², which becomes g = (Gmp)/r² after mis cancelled. The mass of the object on the surface is not needed — “g” is simply a function of mp and 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           (2nd 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          (5th most massive asteroid)

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

Tethys,                 0.147 N/kg          (Saturnian moon)

52 Europa,           0.14 N/kg            (7th 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          (4th 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  (6th largest moon of Uranus)

Amalthea, 0.020 N/kg  (5th 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

A Graph of Infections and Deaths During the First Four Months of the 2014 Ebola Outbreak

Ebola

Source of data for this graph:  http://www.abc.net.au/news/2014-07-31/ebola-timeline-deadliest-outbreak/5639060.

The date I used as “day zero,” March 25, 2014, is the day when the Ministry of Health in Guinea announced an outbreak of ebola was in progress, according to this source: https://en.wikipedia.org/wiki/Ebola_2014. It started earlier, of course, but was not widely known before that date. The last data points shown are for July 27, 2014, the most recent date for which I have the needed information.

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.

Spiraling Stars

spiraling stars