Luminous horned goat head surrounded by stars with one glowing red eye

A Mote in Capricorn’s Eye

On By rolcottIn Astronomy, Crazy Theories, UncategorizedLeave a comment

<bzzt, bzzt> “You again, Moire. What is it this time?”

“Better get over to my office, Walt. I’ve got some charts you’ll want to see, but you probably don’t want them spread all over.”

“Fifteen minutes. No, make that twenty. Bye.” <click>

Twenty minutes later on the dot, his cane taps my office doorpost. “Let’s see those charts.”

“Let me cast this video from Old Reliable to my wall screen. Watch it while I give you the background.”

Data from JPL’s Small Body Database

“I’ll need background to even read this thing.”

“I figured. Here’s a sketch for vocabulary. The big oval is a comet orbit, though they’re usually a lot skinnier than that, roughly ten times longer than wide. ‘Perihelion’ is Greek for ‘close to the Sun’. The word for its opposite point is ‘aphelion’. Your maybe‑incoming chunk of neutronium would be a quarter the Sun’s mass but far too small to see. Even so, I thought its gravity field could be enough to generate a signal made of long‑period comets kicked inward from the Oort cloud.”

“Ah, that’s why your chart’s doing aphelions. A comet at aphelion far from the Sun would be out in the Oort cloud and vulnerable to being kicked. Side‑to‑side in the chart is east‑west; up top is the Pole Star then down to the southern sky. So what you’re showing me is where in the sky each long‑term comet would be when it’s farthest from the Sun. The markers are pretty much everywhere each decade. What’s with the orange fringes?”

“They’re bar charts to summarize relative comet count in each 10‑degree slice of latitude or longitude. The biggest bar in each longitude set represents 4 comets; the biggest latitude bars represent 6. If comets are coming in randomly around the Equator, the summaries along the bottom would be essentially flat, like it shows for the 50 years starting in 1950.”

“This is the kind of map that stretches Greenland relative to Africa, isn’t it?”

“Mm-hm. The latitude lines are drawn parallel to each other although they actually converge on an spherical surface.”

“So I can correct for that by mentally squeezing distances between latitudes … I’d expect relatively few markers close to the poles compared to what’s near the Equator. Right, that’s roughly the latitude summary pattern given the small sample sizes. But what’s going on after 2010?”

“You noticed that red-dot cluster centered above the Equator at about –75°. It’s in the constellation Capricorn. I asked Old Reliable to select the group between –100° and –50° and chart where and when each comet was at its most recent aphelion. Here’s the result. By the way, the slope estimate is at the 75% confidence level. You don’t often see data as tight as that.”

“Whoa! Around 1100 A.D. something from beyond 200 a.u. in the general direction of Capricorn started knocking long‑period comets down from the Oort cloud. Every century that thing’s gotten 17 a.u. closer which gives us … about 400 years before planetary chaos breaks out. What’s an ‘a.u.’?

“Astronomical unit, essentially the radius of Earth’s orbit — 140 million kilometers. But here’s a plot of the same variables including all the long‑period comets reported since 1950. The red dots aren’t special; they behave like the general population. Feeling better?”

“Absolutely not. This looks like the entire Oort cloud is collapsing on us!”

“Relax. You’re looking at an observational artifact — a subtle one, the kind that scientists hate but they get their jollies from uncovering one. Especially if it’s someone else’s.”

“How is this an artifact? The dates and distances are right there.”

“The dates hide something deeper. We can’t see these guys at aphelion — they’re dark and far away. Our telescopes can only pick them up when they’re within 5 a.u. of the Sun, 10 a.u. at the most. Five a.u. is less than 3% of a 200‑a.u. orbit. Within a few percent, observation dates are perihelion dates in disguise. The chart’s horizontal axis is really perihelion‑to‑aphelion cycle time. We’ve known since Kepler’s time that more energetic comets fly farther and take longer to return.”

“Hmph. So your red dots are probably just a random eddy in the Oort cloud. Bye.”

~ Rich Olcott

Luminous horned goat head surrounded by stars with one glowing red eye

Neutronium Chill

On By rolcottIn Astronomy, Astronomy: Stellar Science, Crazy Theories, UncategorizedLeave a comment

<chirp, chirp> I recognize the caller’s number. “Afternoon, Walt. Weren’t you going to trash that phone?”

“Afternoon, Moire. I decided to keep it, just for you.”

“Aww.”

“Don’t get cute. My people have a follow-on question for you. The neutronium chunk that might or might not rearrange our Solar System — how much lead time do we have between when we detect it and when things go sour?”

“So many unknowns. The object’s size, its speed, what signal we’re monitoring, even what you mean by ‘go sour’.”

“Yes, yes, very uncertain. Make some assumptions.”

“Big assignment. I’ll start with size. The smallest neutron star we’ve ever seen is about 1.3 solar masses, but Oppenheimer’s calculations say smaller chunks could be stable. Suppose it’s near the theoretical minimum, say about a quarter of a solar mass, maybe 20‑25 kilometers in diameter.”

“A whole neutron star’s about that diameter. This chunk would be a small cast‑off.”

“No, the cast‑off could be wider than its parent. It’s only a fraction of the star’s mass, so there’d be less gravity squeezing its component particles together. Anyway, it’s a reasonable size for calculating the telescopic detection limit.”

“Walk me through that.”

“Sure. Our sharpest telescopes, the Hubble and the Webb, both use detectors that span about about 40 milliarcseconds per pixel‑width.”

” ‘Milli-‘ I get; that’s one one‑thousandth. ‘Arcseconds’?”

“Works like a clock. Sixty arcseconds to an arcminute, sixty arcminutes to a degree. While I’m at it, 360 degrees is full circle is 2π radians. Put it all together, one of those pixels covers about 0.2 microradian across the sky. Now visualize a triangle. One side spans the chunk; the angle opposite it is at Hubble’s detector.”

“A long, pointy triangle.”

“Pointy enough that the angle in radians is essentially equal to its sine and we can do trigonometry. Your question boils down to, how close does the chunk have to be so its light spans a Hubble pixel?” <keyboard clatter> “You’re not going to like this. A 20‑kilometer object casting a one‑pixel figure can’t be more than 180 million kilometers away. That’s already well inside the Solar System, closer to us than the Asteroid Belt. Just a one‑pixel ‘image,’ not a portrait, from our best telescope for visible light.”

“How about another part of the spectrum?”

“I wouldn’t bet on it. The chunk probably radiated X‑rays early on, but it’s had millions of years in interstellar space to cool off.”

“How is it millions? The Crab Nebula hosts a neutron star and its supernova looks pretty random to me. The Nebula could have flung a chunk at us and it’s only 6500 lightyears away.”

“The fastest interstellar traveler we’ve seen so far is 3I/Atlas, got here from who knows where with a base velocity of 58 kilometers per second. I’ll give you a factor of two — say the Crab flung your chunk at 120 km/sec. That’s still 2500 times slower than lightspeed. At that rate it’d take 16 million years to arrive here. The Crab supernova’s light got here only 970‑some years ago so no chance for an IR‑bright Crab chunk any time soon.”

“Neutrinos?”

“The IceCube Neutrino Observatory detects a million cosmic‑ray muons for every neutrino it sees. Tough odds.”

“Telescopic surveillance isn’t an option. Observing neutrinos is a crapshoot. How about gravitational messing with planet orbits?”

“Something subtler would appear first. Imagine a spherical shell where the inward pull of the Sun’s gravity just equals the outward pull of everything outside.”

“That’d be the outer edge of the Oort Cloud.”

Data from
wikipedia.org/wiki/List_of_long-period_comets

“Pretty much. The chunk will carry its own gravity sphere. Where the two spheres intersect, the opposing gravity fields fight. Some Oort objects fly outward, many drop inward to become long‑period comets. The chunk‑detection signal would be a drastic rise in long‑period comet sightings. Typical long‑period comets take hundreds of years to get down to where we can see them. Nonetheless, we’ve had an upward spike recently.”

“Because of improved technology?”

“Or improved financing. Congress is interested in finding Earth‑threatening asteroids. Look for asteroids, maybe find comets.”

“Lead time in centuries, opening bell may have rung. Bye”

Author’s conception of a neutronium fragment
being expelled from the Crab Nebula.

~ Rich Olcott

A Three Burdens Story

On By rolcottIn Astronomy: Stellar Science, Crazy Theories, Physics: Quantum Mechanics, UncategorizedLeave a comment

<bzzzt, bzzzt> “Who is this and how did you get this number?”

“Morning, Walt. It’s Sy Moire. Does hitting ‘Reply’ to your entry on my phone log count as spycraft?”

“Drat, another burner phone to trash. Oh well. Why the call?”

“Some history for your neutronium question, may make a difference.”

“Good difference or bad difference?”

“Depends on how you slant it, I suppose. The good news is that detecting that Earth‑threatening lump of neutronium would be easier than I thought. The bad news is that detecting it would be easier than I thought.”

“Forward and backward in the same statement. Unravel that.”

“It all goes back to Oppenheimer—”

“J Robert, the guy who ran the Manhattan Project?”

“The same, except before that he did theoretical Physics, not project management. Probably the most prominent physicist of the 1930s. He was active and influential in everything from molecular quantum mechanics and quantum field theory to astrophysics. He knew pretty much everyone who counted; everyone else knew of him. That made him a natural choice as a recruiter for the Project. I think many people were surprised when he turned out to be a good administrator.”

“Good for him and us. Why drag him into this?”

“He showed how neutron stars and black holes could happen. We’d known since 1915 that black holes were implicit in Einstein’s equations. Einstein himself didn’t think the Universe would allow matter to be that dense. In 1939 Oppenheimer showed how a star’s collapse could compress matter enough for the job. Actually, black holes were the easy part.”

“How could that be easy?”

“A star’s outer layers are supported by thermal pressure from the gas inside, right? When there’s not enough fuel to supply the heat, those outer layers come smashing inward like a roof when you knock out a load‑bearing wall. A big‑enough star would have plenty of gravitational potential energy to crush its core into black‑hole mode. Oppenheimer’s real challenge was explaining why the fall stopped partway for a smaller star. He used a tip from Fermi.”

“Enrico Fermi, the guy who built the nuclear reactor under the football field?”

“The same, except it wasn’t under the field, it was under the bleacher seats, but that’s another story. Anyway, Fermi had applied Pauli’s exclusion principle—”

“Out from the weeds, please.”

“Sorry. A couple of years prior, Fermi had worked out how a gas of quantum particles would behave under compression.”

“We’re talking electron or neutron gases?”

“Those among others. Squeeze a sparse quantum gas and measure the pressure it exerts at each decrease in volume. Pressure times volume is energy. For normal gases, pressure change times volume change at constant temperature is essentially constant. With quantum gases there’s a quantum‑limit pressure where the ΔP×ΔV energy goes nuts regardless of temperature. You need to input drastic amounts of energy to achieve even a small volume decrease. Technically, the limit’s called degeneracy pressure but I’d avoid that term in your report.”

“Because negative electron repels negative electron, naturally.”

“Nope, this is completely quantum, nothing to do with charge. It applies to neutrons just like it does to electrons although the limits are different. Try to cram too many quantum particles into a given volume and something has to give.”

“I see where you’re going. Does that energy limit depend on particle mass?”

“Indeed. Heavier particles have a higher limit.”

“Let me guess. White dwarf stars collapse to the electron limit, neutron stars to the neutron limit, pressure crushing beyond that makes a black hole. That’s what Oppenheimer calculated?”

“Mm‑hm. Better yet, he calculated minimum mass for each object type. That’s where the good‑news/bad‑news comes in. You need a lot of gravitational pressure to maintain a stable neutronium object. Below about a tenth of a solar mass, BLOOEY! If your nasty nova were to fling a too‑small chunk of neutronium at us, the chunk would disintegrate long before it could get this far.”

“So the good‑bad news is, a mass of neutronium could only get near us if it was more than a tenth of the Sun’s mass. Anything that heavy would make noticeable changes to planetary orbits long before we could see it. Bye.” <click>

~ Rich Olcott

Abstract cosmic composition with colorful swirling ribbons, organic forms, and stars

A Cosmic Trattoria

On By rolcottIn Astronomy: Multi-messenger, Crazy Theories, UncategorizedLeave a comment

Walt is not amused. ” ‘Not likely but probably’ — what kind of answer is that, Moire? You going for Pratchett’s ‘million-to-one chances crop up nine times out of ten’?”

“You asked me two questions, I gave you two answers. I won’t even bill you for one of them.”

“You know my opinion about lagniappes. We’ll pay for both, but only if we get value.”

“Your first question was, could a nova have flung a chunk of neutronium that’ll hit our Solar System?” <starts tapping on Old Reliable’s screen> “I assume your people are worried about protecting Earth.”

“I neither confirm nor deny that assumption.”

“Either way, by geometry alone the likelihood’s negligible. All our System’s planets and most of the satellites and asteroids and comets are within a few degrees of the Plane of the Ecliptic. Think of everything from Neptune’s orbit inward as a thin flat disk. From above, Earth’s cross‑section area is almost a trillionth of the disk’s area. Suppose your nova is 10 times as far away as our nearest star. Worst case is the nova hanging directly above the Ecliptic. Our disk would be a circular patch in the nova’s sky. If the nova throws a neutronium chunk, it can choose randomly from 20 billion patches the same size as ours; Earth is only a trillionth of one of them. Best case is the nova’s located edge-on to our disk; the odds of a hit would be far smaller. A neutronium chunk striking Earth is a framazillion times less likely than a hit from one of our own near‑Earth asteroids.”

“‘Framazillion.’ I’m going to quote you on that.”

“No extra charge. Now for your second question, whether we could detect a neutronium interloper. Yes, if we’re looking in the right direction with the right detectors.”

“Needs a full-sky survey, obviously, because we have no clue where it’s coming from. What kind of detectors? It’s got to be warmer than empty space. Infrared?”

“Mm-hm. The chunk’s free electrons would shine with black‑body radiation.at all wavelengths but IR’s your best choice.”

“Free electrons? I thought the thing was made of neutrons.”

“Sorry, that’s pop‑sci simplification. The neutrons in a neutronium object are stabilized by gravitational compression, but the gravity field tapers off near the surface. Deep inside you’ve got neutron soup, possibly superfluid, but the particles further up aren’t crowded quite as tightly and have ways of freezing up. There’s a slushy lower crust of nuclear pasta, then an atomic upper crust of iron and its transition‑metal relatives plus some hydrogen and helium. Finally there’s a free‑electron ‘atmosphere’.”

“Wait — nuclear pasta?”

“Technical term, believe it or not. Liquid water and ice make a slush, right, one ingredient occurring in two phases. Water has about 20 different phases—”

“Ice-nine, in Vonnegut’s book.”

“That’s a plot device, I’m talking real ice here. You get different ices at different temperatures and pressures because the microscopic‑level components re‑arrange. We think neutronium does the same. Near the top of a neutron star’s lower crust, the pressure’s low enough that neutrons and protons can clump together as monster nuclei with a high neutron/proton ratio. Theorists depict them as ‘gnocci‘ (or ‘meatballs’) floating in the uncrystallized neutron phase. Down below where the pressure’s higher, the gnocci phase breaks up to form long thin strands of ‘spaghetti.’ Even further down you get sheets of ‘lasagna’ and so on. Physicists can have fun, too. Anyway, IR detectors would be best because the star’s gravity field redshifts UV and visible black‑body emissions to longer wavelengths. But your problem is, that’s not neutronium‑specific.”

“Right, any warm rock would emit IR. Is there something else?”

“Yes, there is, and we already have the right observatories in place and operating.”

“What do they detect?”

“Neutrinos, the lightest particles we know of, about a millionth the mass of an electron. Every time a proton absorbs an electron and converts to a neutron, neutrinos are emitted. That happens a lot in neutronium; a chunk of it ought to be a bright neutrino source. We’ve got a dozen neutrino telescopes in operation all over the world. If they spot something, you’ll hear about it.”

“Bye.”

“Don’t mention it.”

Abstract cosmic composition with colorful swirling ribbons, organic forms, and stars

~ Rich Olcott

Rugged asteroid surrounded by glowing debris in a colorful nebula

Neutral Under Pressure

On By rolcottIn Astronomy: Stellar Science, Crazy Theories, Physics, UncategorizedLeave a comment

The Acme Building’s elevator takes me to the 12th floor with no hiccups, for a change. I unlock my office door and there’s “Walt,” standing in the middle of the room brandishing his cane with what looks like a fat cell phone plugged into it somehow. “The place is clean, Mr Moire. No listeners in the elevator shaft, corridor or office space, and I put temporary foam over your windows to block laser‑bounce sound pickup.”

“That’s good to know, Walt. Should I give you credit for fixing the elevator?”

“Not publicly. Found a dark web relay routine freeloading elevator’s controller computer. Disabled it so the machine can do its real job.”

“Nice work. But why so surreptitious? You know I don’t do military consults. And why come up to my office instead of meeting at Eddie‘s or the park as usual?”

“Tradecraft, Mr Moire — never use the same location too often. Espionage is a thing, military or not, and it doesn’t recognize national boundaries. My people have their secrets, too.”

“Naturally. What are your people curious about this time?”

“Neutronium.”

“Neutronium?”

“Yes, what can you tell me about it?”

“It’s really dense. You’ve heard the line that atoms are mostly empty space. An atomic nucleus is maybe a few femtometers across, but it floats in a ball of space that’s a thousand to ten thousand times wider. The space isn’t empty, of course — it’s occupied by spread‑out electronic charge. Neutronium doesn’t have that not‑quite‑empty space. A neutronium ball with the Sun’s mass would be a dozen kilometers across.”

“Mashed‑together charge like a metal. Would neutronium be shiny?”

“Possibly. I’ve never seen any.”

“Are there neutronium ores you could mine for?”

“Not on Earth, guaranteed. Except for the noble metals like gold, ore atoms are isolated from each other, dispersed in minerals like oxides or sulfides. Neutronium by its nature would have to be in a massive isolated deposit.”

“By its nature?”

“An isolated neutron lives about 15 minutes before it spits out an electron and becomes a proton. A neutron near enough to another nuclear particle is stabilized by the weak nuclear force, but that’s ineffective beyond a few femtometers. Either a neutron’s in an atomic nucleus or it’s mashed up against other neutrons in neutronium or it’s dead.”

“How about a chunk of neutronium buried in one of these ultra‑dense asteroids I’ve read about, like 33 Polyhymnia, or 16 Psyche that we’ve sent a space mission to look at?”

“You have read up, haven’t you? Not even those two asteroids are dense enough to host more than a speck of neutronium. Asteroid mass and volume are separate measurements; you need both to figure mass‑per‑volume density. Each component is a challenging exercise in accurate observation. The first guesstimate for Polyhymnia was that it was more than 10 times denser than iron. That’s why you’ve heard of it. With better data now we think both Polyhymnia and Psyche are at about the density of iron. Higher than most but too low for neutronium by a factor of quintillions.”

“Where else could we go for neutronium?”

“This’ll take some background. The more massive the star, the bigger its diameter and the longer it lives, right?”

Walt gives me a suspicious eye. “You’re fudging, Mr Moire.”

“Nothing gets past you, Walt. More massive stars burn hotter. The heat bulges them out, as you’d expect. The fudge is they exhaust their fuel faster so they die sooner. When the fuel gives out and they go nova, big guys collapse harder than smaller ones can. Their collapse squeezes their residual core tighter. Collapsing a really big star crams its remains into a volume smaller than its Schwartzschild radius. That makes a black hole. Smaller stars can’t go that far. The crush might stop when the residue’s electrons are squeezed into its atomic nuclei. Protons there could capture them and become neutrons. If so, you’ve got neutronium and a neutron star.”

“A nova’s pretty violent, shoots material out randomly?”

“One of the sloppiest, most violent events in the Universe.”

“Could a nova have flung a neutronium fragment through our Solar System like something flung ‘Oumuamua? Could we detect that?”

“Not likely but probably.”

Rugged asteroid surrounded by glowing debris in a colorful nebula

~ Rich Olcott

16830: Not So Special

On By rolcottIn General: Fun Stuff, Mathematics, UncategorizedLeave a comment

Vinnie looks up when I walk into Cal’s coffee shop. “Afternoon, Sy. You’ve got the look of a man deflated. What happened?”

“Dove down a rabbit hole, burned a lot of Old Reliable’s compute time, thought I’d discovered something, but then learned it’s been known since 1830.”

“Big disappointment, huh?”

“Oh, there’s a lot of disappointment in science. You get used to stubborn data blowing away a perfectly good hypothesis. But to be scooped by 200 years — embarrassing! Also frustrating.”

Cal’s pouring coffee. “Spill the story, Sy.”

“It all started with 16830.”

“He got that out of a book by Ken Follett, Cal. Take a number, times itself is a square, times itself again you got a cube. Do that to a thousand numbers in a row, add ’em all up and the total just happens to be the cube of 16830.”

“Vinnie’s right, Cal, except the trick only works if the number you start with is 1134. Start with any other number, one thousand successive cubes don’t add up to any exact cube.”

“You used Old Reliable to check that, right?”

“Naturally. That was the beginning. Once I’d built its program to do that 1000‑cube calculation, I said, ‘Let’s look for other numbers that start a 1000‑cube series that adds up to a cube.’ That program scanned starters up to twenty million. No hits. Well, I did find one other one.”

“Ah-hah!”

“Don’t get excited. Cubing a negative number gives you a negative cube. Cubes from minus‑500 to plus‑500 add up to zero, which is zero cubed.”

“That’s a cheat, Sy. Minus‑500 to plus‑500 is 1001 steps.”

“Good point, Vinnie. My correction stands corrected and you’ve led into my next step. Follett’s novel mentions a four‑member sequence, 113+123+133+143, that adds up to 203. Didn’t take me long to come up with a three‑member chain, 33+43+53=63. Surely there are other windows onto the number line that fit the same pattern if you pick the right starters. I tried a few significant window sizes, like 42 and 100 and 666, no luck. I re‑programmed Old Reliable to try every window from 3 members out to 5000. For each window size, I told it to try every starter from 1 to a million. If it hit an exact‑cube sum the program would print out the starter, the window and the cube‑root of the sum.'”

“Wow. Old Reliable musta been smokin’ when that finished.”

“Pretty much. I did apologize to it. Anyhow, I was happy to see 16830 was in the list, but so were a bunch of other combinations. Then I noticed a pattern in the window sizes. I recognized 1000 as 103, 1331 as 113, and 4096 as 163. Many of the other successful window sizes all the way down to 64=43 were also cubes. Here, I’ve highlighted them in yellow in this screenshot.”

“I thought you only did windows up to 5000.”

“Mm‑hm, but that led to my next experiment. I told Old Reliable to scan perfect‑cube window sizes up to 553. That’s the blue‑highlighted cells. See all the hits?”

“But you’re missing some. Where’s 23=8? Or 33, 63 and 93, for that matter?”

“Good questions. The quick answer is, ‘Because arithmetic,’ but I’m a physicist. I always want to know why. Now that I’d done the calculation I had some samples to ask about. I grabbed a segment from column D and searched it in the Online Encyclopedia of Integer Sequences. Only one hit but it was almost perfect. Even mentioned that the series skips any windows like 63 that are 3‑divisible.”

“So why’s that happen?”

“I had to dig deeper for that. The OEIS just says ‘Here it is,’ leaves explanations to the links. Ben Vitalis posted a collection of formulas in his blog but the real explanation was buried in some hairy algebra in one of Kevin Brown‘s blog posts. Mr Brown’s an old‑school algebraist — used some tricks it took me two days to understand — but he proved that windows divisible by 3 don’t qualify for that family of cube‑making cubes. His post is where I learned about Pagliani and his 1830 book that solved the 1000 case.”

“There’s other families?”

“Small ones. There’s a sparse set where the window size is a perfect square, and a handful of complete mavericks.”

“Could there be others with windows between your blue ones?”

“Maybe, but I’ll let Old Reliable cool off before I look.”

~ Rich Olcott

The 16830 Mandala

On By rolcottIn General: Fun Stuff, Mathematics, UncategorizedLeave a comment

Chasing the number 16830 led me down a rabbit hole, this time in the On-line Encyclopedia of Integer Sequences®, a.k.a. OEIS. You can search it for someone’s name (Euler‘s appears in 8651 entries), a descriptive phrase (“sum of cubes” hits 2679 entries), an integer series (the software’s smart enough to recognize “3,14,15,92,65,35,89,79” as the base‑10 expansion of π) or even just a single integer. After writing A Numbers Game, I had to check where else that particular number has cropped up.

My search on “16830” brought back 48 results. Entry A097811 listed 16830 among “Numbers n such that n3 is the sum of three or more consecutive positive cubes” — that’s how Elspeth won her game, after all. Some of the posts are pretty arcane. For example, I hadn’t known that “digitally balanced” (having equal numbers of each digit) is even a thing, but entry A049355 says 16830 is digitally balanced in base‑4: 1683010 = 100123324. 16830‘s also in the list for entry A136154 “Composites one larger than a prime, with exactly five distinct prime factors.” Sure enough, 16830–1=16829 is indeed prime and 16830 itself is 2×32×5×11×17 (though 3 being in there twice feels like cheating). Who would need to know things like that?

The most eye-worthy fruit of this survey was the series of “stained glass windows” (his words) that Scott R. Shannon had generated as part of his graph theory studies. Graph theory is all about pairwise relationships (you and each of your friends, plus each of them and each of their their friends…).

One of graph theory’s goals is to enumerate and classify all possible pairwise structures. Represent each member of some pair by a dot on a plane (naturally, the mathematicians go for more complicated surfaces but let’s keep things simple). Represent each relationship by a line (straight or curved) between corresponding dots. Then look for patterns.

Incidentally, Euler helped invent graph theory back in 1736. He pointed out that any set of connected lines encloses an area. Count up dots, lines and areas. On a plane, the dot count plus the area count minus the line count will always equal 1. That’s true even for the limiting case of two dots and one line — area count is zero, so 2+0–1=1. Dots plus areas minus lines may sum up to something else on a different surface, but we’re keepin’ it simple, okay?

Shannon diagram for 3 points on the generating circle, from oeis.org/A371374/a371374.jpg

In one Shannon & Co project, they pick a rule for placing dots on a shape, then classify by counting the dots, lines and areas that the rule generates. For example:

  1. Draw a unit circle.
  2. Choose a number n. In this example, n=3.
  3. Mark off n equally-spaced points around the circle.
  4. Draw n more unit circles, each passing through a pair of points on the original circle.

That’s the simplest useful example — n=2 puts two dots 180° apart so a second circle would lie right on top of the first one and that’s boring. Three dots on the circle plus the generated intersection at the center makes 4. Nine areas and twelve arcs (count ’em) between dots: 4+9–12=1 as Euler predicted. To make the picture interesting, Shannon colored each area according to the number of arcs that enclose it: red for 2 arcs, orange for three.

Shannon diagram for 5 points on the generating circle, from oeis.org/A371374/a371374_2.jpg

The 4-dot case is pretty much the same, except the generated circles don’t overlap so there’s no central intersection. Euler’s equation still prevails: 4+9–12=1.

Combinatorial explosion — Shannon’s circle with increasing n
Data from oeis.org/A371373/ and oeis.org/A371374/

Just one more dot on the circle makes a drastic change. The dot count balloons from 4 to 25, connected by 75 arcs that enclose 51 areas (25+51–75=1, Euler wins again). Click the picture or the link and see the ten generated grey circles cut by lots of generated dots. The proliferation of intersections generates more complexity than you’d expect from the first 3 cases.

“Proliferation” puts it mildly — the thingy count rises as roughly the fourth power of n (see the “Combinatorial explosion” chart).

So about 16830 — that number shows up as the number of arcs in the structure generated by 18 dots on the initial circle. Unfortunately, Shannon didn’t give us that picture, so here’s the one for n=20. Pretty, huh?

Shannon diagram for 20 points on the generating circle, from oeis.org/A371374/a371374_11.jpg

~ Rich Olcott

A Numbers Game

On By rolcottIn General: Fun Stuff, Mathematics, Uncategorized1 Comment

Early morning, but an August morning so it’s hot already. I’m thanking the Acme Building’s air conditioning as I unlock the door to my 12th‑floor office. There’s Vinnie, stretched out on the couch with a book. “What’re you reading there, Vinnie?”

“Morning, Sy. It’s one of Ken Follett‘s books, Pillars of The Earth, about a couple of families battling over a cathedral. One family wants to build it, the other hates whatever the first family wants, and this goes on for generations. Meanwhile, I’m learning more than I want to know about how they made felt in Medieval times.”

“Follett does like to dive down rabbit holes, and he’s proud to show off what he digs up. I just finished another of his books, Code to Zero. It’s a thriller loaded with arcane tidbits. The action’s wrapped around some Soviet plot in the late 1950s to disrupt America’s Explorer satellite program. We get close-ups of rocketship anatomy; we find out why unsymmetrical dimethylhydrazine is a bad bet as a rocket fuel; things like that.”

“Sounds interesting.”

“Yeah, I think you’d like it. Lots of action while the hero’s chasing around trying to figure out what happened to him and why. Spoiler alert — things get frantic toward the end.”

“Thrillers always do, that’s why they’re thrillers.”

“Mm‑hm. But I’m a numbers guy, so what really caught my eye was Follett’s collection of obscure numerical lore. One of the book’s characters, Elspeth, has an ongoing math‑nerd game with another character. Each player names a special number. The other has to explain why that number’s special.”

“Can’t be too hard. Practically every number’s special somehow.”

“True, but Follett found some doozies. For instance, Elspeth plays number 29. It’s special because it’s the root of a whole train of prime numbers: 29, 31, 37, 47,… all united by a common formula. The nth member of the series (counting 29 itself as n=0) is given by 29+2n2. The series breaks at n=29 because 29+2*292=1711, which clearly is divisible by 29.”

“29 days in February, sometimes.”

“That’s not number‑theory special. Another Elspeth gambit involves the abc=a1+b2+c3 formula where each letter is a base‑10 digit on one side and a number raised to a power on the other. She plays 135, special because 11+32+53=1+9+125=135. The other nerd matches her pattern with 11+72+53=1+49+125=175. That got me wondering if any other numbers fit the template. I worked it up on Old Reliable—”

“Of course you would.”

“You know me so well. Turns out that both 518 and 598 qualify, but no others. Elspeth plays with cubes, too. There’s a conversation in the book about 136 and 244 being partners because 13+33+63=244 but 23+43+43=136. Number 8000 is 203 but it’s also 113+123+133+143.”

“How do they even find these things?”

“My guess is heroic mathematicians with time on their hands in a no‑machinery no‑screens environment. Think, these guys were doing arithmetic with an abacus. Four centuries ago Fermat conjectured that numbers of the form 22n+1 are prime. He tried cases n=0 through 4 and they all worked. A century later, Euler showed that 225+1=4 294 967 297 isn’t prime — it’s evenly divisible by 641. Imagine even generating that number on an abacus. Then imagine looking for a zero remainder each time you abacus‑divide by potentially 65000 successive primes. Fortunately Euler struck gold on the 115th test.”

“Rechecked it a couple of times, I bet.”

“Indeed. Those cubes — number theoreticians love sums of powers. There’s one example in Code to Zero that boggles my mind. Up near the book’s frantic finale, Elspeth just happens to know that 168303 is the sum of the 1000 cubes from 11343 through 21333. Why would some hero hit on that combination, and why would they bother to check if the sum is a cube?”

“Dunno, I’m not a numbers theory guy.”

~ Rich Olcott

  • Thanks to Lloyd Boyer for pointing me to Code to Zero and the wonders therein.

Texas Observed

On By rolcottIn General: Fun Stuff, UncategorizedLeave a comment

More map fun this week. Sy and the gang and their science‑y stuff will return in the next post. Unless something else occurs to me.

On a recent trip coming down from Denver to Fort Worth, Texas along I‑25 and US‑287, I was struck by the slew of familiar place names I encountered. Not familiar places, mind you, but names.

About 10 miles east of Childress (in Memphis I had a neighbor family named Childress, wonder if they’ve got kin there) I spotted a forest of cranes half‑a‑mile north of US‑287. Yet another data center under construction. The site will generate heat, air and noise pollution but that’ll be well away from any population centers. However, the construction’s clustered around a slough that feeds Groesbeck Creek. I foresee litigation.

Anyway, along my route there was

  • Des Moines — okay, that was in New Mexico, not Texas, but stay with me.
  • Channing — the one in TX, not the one in Colorado.
  • Memphis — wait, in Texas? I lived in Memphis for 40 years but that was in Tennessee and a lot bigger than this.
  • Hardeman County — waitaminnit, Hardeman County’s in Tennessee, just up I-40 from Memphis
  • Chillicothe — the one in TX, not any of the ones in Illinois, Missouri or Ohio
  • Decatur — one of half‑a‑dozen with that name
  • Fruitland — different from the other dozen scattered across the USA
  • You get the idea.

My theory is that Texas has so much space and so many places that they ran out of names and started using other peoples’ toponyms. As just one example, Texas has its own “Colorado River” masquerading for the real one, which is an outrage but a story for another time. What I wanted to do was stand the name of every Texas city and town against all the rest of the country’s cities and towns, just to quantify the “borrowing.” Yes, towns, too — the population of Chillicothe, TX is only 548 (2020 census) but it counts. My computer’s good, but it’s not good enough to do the matching-up with 1224 Texas names against 19495 USA names broken out by state in a reasonable amount of time. The A.I. model I asked about doing the job said, “You couldn’t afford it.”

So I moved up to the county level. According to this Wikipedia entry, the USA has 3075 counties, including Louisiana’s parishes but excluding Alaska’s boroughs and census areas. The median count per state is about 60. Texas has 254 counties, more than any other state, naturally. That match‑up is doable and summarized in this map (click to expand).

The “Shares” color scale displays the number of out‑of‑state counties that have the same name as a given mapped county. For instance, the L‑marked county near the state’s southeast corner is Liberty County. It rates a “3” because there are Liberty Counties in Florida, Georgia and Montana.

The map’s three top-rated counties (Washington, Jefferson and Franklin) are right in line with the national totals. For some reason, Texas skipped the national number 4, Lincoln, but got back into sync with Jackson. All told, Texas shares 101 county names with other states. On the other hand, it holds 153 names (the blank outlines) all to itself.

Looking at that map raised another hypothesis — that establishing counties started in the relatively urban east where folks were familiar with what county names looked like. “Hey, there are eight other Shelby Counties, we can have one, too.” Only later did edification move westward but people there were willing to put up with names like “Deaf Smith County.”

There’s a way to check that hypothesis. I pulled Creation Date (basically when the state legislature authorized formation of a county government) data from TexasCounties.net and created another map (click to enlarge).

Yep, civilization nucleated in the east and along the state’s northern coast, then took 90 years to spread outward. I guess only a Texan could tell us why Kennedy (southern coast) and Hockley (at the base of the Panhandle) counties waited until 1921 to stand up on their own.

By the way, Elon Musk’s Starbase City is in Cameron County, 100 miles south of Kennedy County (population 350 in 2020). Pure coincidence.

~ Rich Olcott

A Plateful of Plates

On By rolcottIn General: Fun Stuff, Uncategorized2 Comments

Some people collect stamps. I collect “vanity plates,” vehicle tags where someone’s gone to the trouble of selecting certain letters and numbers for the state to hang on their car or truck for public display. I don’t rip off the plates themselves, of course, but I do take a picture when I encounter a choice example (traffic permitting).

What good is a collection if you can’t show it off, right? Here’s a selection from my collection for your inspection:


on a well-used SUV

on an EV

Now for the science-y part, about that map. On a recent trip to Texas I noticed that nearly every car there wore a boring default 3‑letter, 3‑digit mix — far fewer vanity plates than I normally see back home. I wondered if this was a Freakonomics effect. To test that hypothesis I pulled down vehicle registration data from carregistrationfee.com. Registration costs vary widely from state to state and may depend on engine size or horsepower, vehicle class, weight, wheelbase, depreciated value, … you get the idea. Then counties and cities can use their own rules to figure their cut.

In every state, though, there’s an extra fee for customizing your plate with your personal choice of letters and numbers (within limits). The charge for renewing a personalized plate is sometimes less than the first-year price so to equalize matters I calculated three-year totals. The average hit was $114.74, close to the median of $105.00. However, the range was impressive: $29.00 up to $585.00. The map shows where the heaviest levy falls and how far away it is from everyone else:

Three-year total added fee
for a personalized vehicle license.

So, yeah, Freakonomic hypothesis confirmed — what you tax the most you get the least of.

~ Rich Olcott