It's been a year
Please go to Notional Slurry for my actual blog.
I'll be moving these posts over there sometime soon.
I'll be moving these posts over there sometime soon.
posted by Vaguery | 9:04:00 AM
Fragments And Exercises
A few simple questions. Some are mathematical, statistical, probabilistic; some involve complex systems, some are complex systems. Some have answers. • Difficulty varies. You can answer some with a pencil on a napkin. Simple computer programs or spreadsheets will manage the rest. • In the end, maybe we all learn. The thing I try for when jotting these down is extensibility; I think they all lead down branching paths towards usefulness and fun. Or maybe madness and ruin.
- It's been a year
- Whack that sucker hard enough, it'll break
- Before you die of tetanus
- via Notional Slurry
February 12, 2006
February 15, 2005
Whack that sucker hard enough, it'll break
"Oy! Ya fiddling lickspittle doodler, where the fuck have ya got to?"
"Here, My Lord, in the second greenhouse. I was just sketching these fern gametophytes. They're the first I've encountered since we started propagating this cultivar."
"Yeh, right. So.... Which, the dirt?"
"No, Sire, these small green organisms. Their presence indicates that the ferns we brought have--"
"Yer sketchin mold? Jesus. I made a fuckin Michelangelo of mold, I did. The very Leonardo of little tiny flecks of cabbage, are ya?"
"Yes, Sire. As you say."
"You know, if I didn't know better, I'd swear yer weren't my clone at all.... Agh, no mind. Right. So: you know this Martian inter-- embargo thingie? With the War?"
"Interdiction, Lord?"
"Yeh, right, Interdiction. So I'm thinkin -- what a load of crap that is, eh? I mean, how do you even interdite a whole planet, eh? I mean, here it is, not a mere city or nothin mind you, it's a fuckin great gravity well, right in plain sight in the middle of the fuckin Solar System. Big red thing, I seen it dozens a times. I mean, you just lob shite at it, right, an it just lands there, bang, you know? What're they thinkin? I mean, sure, there's re-entry to contend with, but Jesus, they did those probe thingies, dint they?"
"Ports, Sire."
"Wot?"
"'Ports.' Technically you were correct when you mentioned an embargo. The Authorities are maintaining a presence at all the ports of entry, so as to prohibit sanctioned trades and communications entering or leaving the settlements. The word 'Interdiction' is a slight misnomer."
"Well, why they call it that then? Fuckin half-arsed weasel word, innit? You mean embargo, you say 'embargo' is what I say. Might as well call it 'Mommie' an --"
"Ahem. Yes, Sire. I believe it was a reference to the Solar Interdiction."
"Oh, right. Stuck in their minds, I spose, eh? The Icebags pissed em off enough, spoiled their sense of interstellar adventure an Grand Destiny of Man an Made in His Image an all when they found out the lords of this particular corner of Everything's Arse are fuckin great Oorters with invisible rocks fer brains. And so the fussy little monkey bigots spend their dwindlin bluster all runnin around mutterin the damned word all the time. 'Interdiction interdict interdite' this an that and that over there. Gets in yer subconscious, you get a pancultural religiopolitical shock like that, an you mutter mutter mutter to yourself all day long. Start to use words inappropriately. Sign of imminent institutional breakdown, I say."
"Indeed."
"Or ongoing, eh? With the War, eh? That'd count.... You know, I'd flat forgot about the Icebags' Cube. Heh. Funny thing, that, how you can be thinkin about somethin real hard, an miss the big.... Innat interesting? Not that it has any fuckin thing to do with what I been tryin to say all along, oh, no. So. Right. We saw that big darkmass a few days back, right? Headed what? About 40 degrees up out towards Saturn ways, wunnit? Whizzin along about point three or so?"
"Yes, Sire. Would you like to pursue it now? At the time--"
"See, I'm thinkin they're gonna wanna have some, you know, big Bad Motherfucker bang-er-up bombs an shit on Mars, someday soon, yeh?"
"One would imagine both sides would, yes. Those, or food. Possibly both."
"Right, an ships, right? Need darkmass for ships' drives, at least for all those gleamy military needles so they look good in the parade. An, I shall remind you, not to mention this great wobbly filigree of silvery self-fabricated shite we call home, eh? How long to catch it?"
"Hmmm. I believe it would have wrapped by now, or would before we reached it. I can calculate the distance to the Weeuruunie Interdiction Boundary easily; the last surveys of both the boundaries and the darkmasses made before the recent hostilities broke out were very accurate. If the darkmass has wrapped, we will surely be able to catch it without any trouble on the other side."
"Well, that'd be a good part aways back across the Cube, wouldn't it? I'm not fuckin made of money. Or fuel."
"One good-sized harvest of darkmass would more than pay for the effort, Sire."
"Yeh, right! An that is why I said, to you, not five minutes back, 'Oy, where the fuck are ya?' Yer damn lettuce threw me off, that an all your natterin. Here: We're gonna go get all of em."
"..."
"Yeh, heheh, you'd better close your gapin mouth, laddie mine, before you catch a gymnosperm or somethin in it, because see, I got a plan. And didst I not make thee from the flesh of my flesh, or words to that effect? Dost thou not trustest me who made thee in mine image, O my Sunny Jim? Heh. See this? Right here? This in here is original brains, they are. Yours a mere Derivative Work."
"My Lord, there are seven mapped darkmasses within the confines of the Interdiction Cube! The expeditions have been planned and in one case executed--"
"Ach, dammit, did I forget to grow yer balls when I made ya? O, ye of little balls. Where's yer sense of... well, greed boy? I mean, come on, you know very well what I paid for that wee 14 million tons of darkmass down there -- three hundred thousand souls, it was, I made myself. And there's a fuckin War on, an a biggie at that, an big war is as a matter of well-known fact big money, QED. First interplanetary war in fuckin history, an who knows how grand an blowy-uppy it'll get, an here we are out in the middle of the Upper Armpit of Nowhere, noteworthily I shall remind you not in the blowy-uppy part of the fuckin War, an we have a fast ship, complete with fuel an partially synthetic brains an little fuckin cabbage leaves growin the orangery. An I might add a unique ship what's uniquely ship-shape enough that it does not need to deal with acceleration in any ways whatsoever, nosiree, but merely communes with the quantum informational thingie for permission and changes frames of reference as if by magic. Come on, there's just seven of em."
"Sire. There are... well, I suppose it..."
"Ahh, see, there's the kind of enthusiasm I like to hear from the fruit of my -- well my cheek as I recall. An -- mind you, mind you! -- as I might have mentioned there's a fuckin War on, see, an so nobody else will be lookin for em. They're all busily ready-aim-firing-oh-no-wait-not-yet with their bombs an asteroids an shit, an noisily diplomatin an televisin at each other, an fundin black labs an gold retrieval missions an shit. Hell, boy, nobody else will even have a fuckin clue where these darkmasses have got to, once that last survey's got a bit obsolete an they go whizzin an wrappin around the Cube a few times! Heheh. Seven, you say?"
"Yes. As I recall the survey was accurate to within one meter, and one centimeter per second velocity. That was about three months ago, so the uncertainty clouds will have spread somewhat by now. And they'd all have wrapped one or more times in that period, adding more uncertainty...."
"Well, we're not a passive observer here, you know. We are, do not forget, in a fuckin awesome space ship. In a pinch we can even make it go 'whoosh' an fart hot hydrogen out its arse an there you are, dealing with yer uncertainties proactively by chasin stuff. Or so I am told, when I am not resigned to merely sittin in the middle of fuckin nowhere consumed with a passion for growin intelligent flower gardens. Or if you prefer we might consider, oh I dunno, maybe usin the fuckin lattice drive we grew last month, instead of just ponderin its quantum splendor an all."
"Indeed, point taken. That said, Sire, we do have a limited amount of fuel, as you mentioned. Each use of the lattice drive to shift our frame of reference will consume a substantial portion of the darkmass we carry. We should plan a course that minimizes the amount of velocity we need to reach their vicinities. Or the most we can reach. It may not be feasible to expect to get all seven."
"Ay, ay -- thank you very much young Master Exposition, but I'll have no half-assed clone talk, you. All seven or nothin."
"Well, the Cube is 11,323,344,384 kilometers across, so--"
"That would be natterin again. Do you always think out loud like that? I might could fix that, you know. Hey, hang on, hang on! People were tryin to figure out how Pluto gets in an out, but nothin else can, right? I mean, why ain't Pluto jumpin all over the fuckin sky every time it crosses the Cube boundary, when every other piece of stuff in the fuckin box just wraps over to the other side unchanged an all, just blip an there you are with the shadows all upside-down of a sudden, eleven billion-an-more klicks from where you started?"
"Something about dark matter, I'm told, Sire, though that seems like a hedge. I suspect the Weeuruunie wrote an exception into the matter filter for Pluto when they constructed the Interdiction Cube, so that early astronomers would not become overexcited if they noticed the planetoid jumping."
"Heh. As if. They was keepin the dinosaurs in, an no mistake."
"Indeed. And those were the astronomers that were meant not to be excited. At any rate, the Cube is of known size, and the position and velocities of the seven darkmasses are known to great accuracy, and of course we know our own position very well. It should boil down to a simple dynamic programming problem."
"Yeh, well, we got what? just below fifteen million tons of darkmass an hydrogen fuel on board. So no needless sightseein. We give a wee tap on the gas, we coast, we nab, we prod again, got it? Extended burns will leave us sittin out here twiddlin our thumbs up our asses an eatin gynocarps an shit. No offense, little moldies."
"Gametophytes, Lord, but only the sporophyte stage of this cultivar have organs of hearing.... There is one problem."
"Yeh. The fuckin War, eh?"
"Yes, Sire. We will need to avoid a substantial region around the core of the system, unless we pass through at relatively high speed. More than one millicee, for example. I suspect we will be better served if we use the Interdiction Boundary to wrap across the Solar System instantly. It does increase the complexity of the tour planning somewhat."
"Right. So hop on it, boyo. I'll wake up the ship an the others, an then let's go nab us all the darkmass anybody's ever not seen, before they get wise to us."
"I will be done here momentarily, Sire."
"An did I ever tell you ya sound like Mister fuckin Spock? I didn't build that in, did I? You do it on purpose, don't ya, as an actin out thing. Heh. No, wait, I know, you talk like that Vizier fellow in that old movie, innit? Not sure that's entirely appropriate for a fourteen year old boy, now, is it?"
"No, Sire. Not at all."
"Heh. Smartass. But always remember -- Derived Work."
The Weeuruunie Interdiction Cube is 11 323 344 384 kilometers on each edge, measured to within one centimeter, and is as near as can be determined perfect cube with center coincident with the center of mass of the Sun, and oriented so that one particular corner is aligned permanently with the center of mass of Neptune. Any object more complicated than a formaldehyde molecule (except for some reason the planet Pluto and its moon) attempting to pass across the boundary in any direction from either side is instantaneously transported to the opposite face at the exact equivalent position, without loss of velocity or any noticeable inertial effect. The easiest way to visualize this is to recall the late 20th century game Asteroids, in which objects disappeared off the right side and reappeared on the left instantly; topologically, an Asteroids screen maps the Euclidean space of the screen onto the surface of a torus, while the Weeuruunie Interdiction Cube maps the Solar System onto the "surface" of a higher-dimensional torus. As far as matter is concerned the boundary is equivalent to continuous space; you could in theory stick your arm across it and goose somebody eleven billion kilometers away, with no ill effects beyond a slap, though you would be unable to see where your arm had gone -- the boundary implements an intimidating suite of information-based filters... well, let's just say it's "magic" and make this shorter.
Note: The wrapping property of the Interdiction Cube means the shortest distance to a given point from your location might lie in any of 27 possible directions.
By common convention, spatial coordinates inside the Cube are referred to relative to the Neptune corner, with that described as (0.0, 0.0, 0.0) and the opposite corner as (1.0, 1.0, 1.0). The particular orientation of axes is unimportant for the problems here, but suffice to say they are determined by long negotiations by a series of squabbling multinational bodies, and subject to change on short notice.
The seven known darkmasses are not conventional matter, though they are affected by the Interdiction Cube in both the typical and an unusual manner. Unaffected by gravity, they only move in straight lines at constant velocity with respect to the Cube, though the particulars of their size, vectors of motion and speed differ from darkmass to darkmass. Little is known about their composition, which is one reason it is referred to by physicists as "eccentric matter", even though it forms the basis of the new lattice drive technology and has numerous other engineering and military applications. It is very dense, though practically invisible, and does not create or experience gravitational fields. Which is a big part of why it's called "eccentric."
The most modern ships (such as yours, and some military vessels) which rely on darkmass-fueled lattice drives are also -- as a side effect -- unaffected by external gravitation. If sufficiently advanced (as is yours) they can change velocity within one second by transforming darkmass into conventional matter and expelling it... well, let's just call it "magic" and make this shorter, and will travel in a straight line (also relative to the Cube) until the velocity is actively changed again.
Whenever the lattice drive is activated, determine the amount of darkmass expended using this algorithm: First, calculate the magnitude of the change in velocity vectors. For example, from rest v=(0,0,0), a jump to v=(0.0000001, 0.0000001, 0) is an absolute change of Sqrt[2]*10-7. Multiply this by the mass being moved, in tons, and that is the percentage of darkmass consumed by the maneuver. So for example a ship weighing 40 million tons of which 12 million tons were darkmass that undertook the previous maneuver would lose 5.65% of the darkmass it was carrying, or 678823 tons, instantaneously. This percentage is independent of the absolute amount of darkmatter on hand (except in terms of its mass), so small ships carrying small amounts of darkmass can be very maneuverable.
Remember, it's eccentric matter.
That said, a speed of 0.05 of the speed of light is the absolute speed limit for lattice-driven ships inside the Interdiction Cube, for reasons that remain obscure, though it is thought possible that "wild" darkmasses could travel faster.
Ships can harvest fuel from a "wild" darkmass by first intercepting it, then matching velocities (thus requiring a minimum of two uses of the lattice drive). 1000 tons of darkmass can be taken on board per second.
The seven known darkmasses' masses, positions, and velocities as of 2 678 400 seconds ago exactly are
Question #1: Ignoring uncertainty in the above measurements, and relativistic effects, where are each of the seven darkmasses now (2 678 400 seconds later)?
Question #2: The ship is located exactly at Cube coordinates (0.82, 0.86, 0.44), and has at present a relative velocity of (0,0,0), and has 14 000 000 tons of darkmass fuel on board for the lattice drive. Again ignoring relativistic effects and uncertainties, which of the seven darkmasses is the best to visit first, if the point is to maximize the average rate at which fuel is acquired (or lost) over the entire duration of the operation?
Question #3: What is the best route that visits and completely harvests all seven darkmasses, but which completely avoids entering the spherical region centered on the sun, of radius 400 000 000 km (The Warzone)? By "best" we again mean the one that maximizes the average rate at which darkmass is harvested over the duration of the operation, as well as the total amount of darkmass on board at the end?.
Question #You Thought I Completely Forgot About the Title, Didn't You?:
"There has been a complication, Sire."
"Fuckin fuck! What?! I was just puttin more memory into this philodendron here. Don't sneak up on me like that! Dammit. Sorry, sorry, wee plantie. Waddya mean, 'complication'?"
"Well, our earlier visits to the wild darkmasses have been more... tentative than this trip. When we colocated with the first darkmass...well, it broke."
"Broke? No, wait, let me guess. It 'broke' 'eccentrically', I'll wager."
"Yes, Sire. It seems that exactly half of the darkmass was somehow knocked off by our abrupt arrival, and is now proceeding in the same direction we were originally headed when we colocated with its position, traveling at our previous velocity."
"Like a halfa fuckin billiard ball, eh? But we still have half of what we expected to be waitin for us? Fuckin Oorters, eh? Makin weird shit happen just cause they can. Someday I think I wanna meet one of them. An take him apart. Betcha his brains're worth a mint an a half."
"Yes, Sire. So, from the readings we've taken, we would need to colocate with the darkmass at a relative velocity of no more than 10-9 to avoid this phenomenon. Coinciding with a darkmass at any lower velocity, we should be able to retrieve it entirely; higher, and half will fly off just as fast as we arrived."
"Well, hell, that'd take a pantload more fuel, then, wouldn't it, slowin down like that? Fuck! Agh! But if we don't, we need to chase down the pieces as we make em."
"Indeed, Sire."
"Well... just figger it out. You're a smart lad. Make it fuckin so. An gimme that chip."
So: After completing the acquisition of the half-darkmass you first encountered in your original route, describe a new route to the remaining seven (including the half of the one you thought to take out completely) and harvests every one, in the shortest time, and with the largest remaining amount of darkmass fuel on board, completely avoiding the Warzone, under the new rule that if the velocity of the ship relative to the darkmass upon arrival is more than 10-9 (in lattice coordinates), half of the darkmass is lost in the direction you were moving, at the velocity you were moving.
Bonus question #1: That uncertainty thing. It was mentioned. It turns out to be valid. Indeed, the position of the darkmasses was only mapped to within one meter (they are, as we have pointed out several times, very dense), and their velocity only accurate to within one centimeter per second. First, what are these infinitesimal-sounding distances in proportion to the Cube edge? More important, if the center of each darkmass was somewhere within a sphere of radius 1 meter centered on the reported coordinates, and the velocity accurate to within one centimeter per second in magnitude and 0.01 arcsecond in direction [no, we didn't mention that; clones don't recall everything], how large a region might you have to search to find each darkmass?
"Here, My Lord, in the second greenhouse. I was just sketching these fern gametophytes. They're the first I've encountered since we started propagating this cultivar."
"Yeh, right. So.... Which, the dirt?"
"No, Sire, these small green organisms. Their presence indicates that the ferns we brought have--"
"Yer sketchin mold? Jesus. I made a fuckin Michelangelo of mold, I did. The very Leonardo of little tiny flecks of cabbage, are ya?"
"Yes, Sire. As you say."
"You know, if I didn't know better, I'd swear yer weren't my clone at all.... Agh, no mind. Right. So: you know this Martian inter-- embargo thingie? With the War?"
"Interdiction, Lord?"
"Yeh, right, Interdiction. So I'm thinkin -- what a load of crap that is, eh? I mean, how do you even interdite a whole planet, eh? I mean, here it is, not a mere city or nothin mind you, it's a fuckin great gravity well, right in plain sight in the middle of the fuckin Solar System. Big red thing, I seen it dozens a times. I mean, you just lob shite at it, right, an it just lands there, bang, you know? What're they thinkin? I mean, sure, there's re-entry to contend with, but Jesus, they did those probe thingies, dint they?"
"Ports, Sire."
"Wot?"
"'Ports.' Technically you were correct when you mentioned an embargo. The Authorities are maintaining a presence at all the ports of entry, so as to prohibit sanctioned trades and communications entering or leaving the settlements. The word 'Interdiction' is a slight misnomer."
"Well, why they call it that then? Fuckin half-arsed weasel word, innit? You mean embargo, you say 'embargo' is what I say. Might as well call it 'Mommie' an --"
"Ahem. Yes, Sire. I believe it was a reference to the Solar Interdiction."
"Oh, right. Stuck in their minds, I spose, eh? The Icebags pissed em off enough, spoiled their sense of interstellar adventure an Grand Destiny of Man an Made in His Image an all when they found out the lords of this particular corner of Everything's Arse are fuckin great Oorters with invisible rocks fer brains. And so the fussy little monkey bigots spend their dwindlin bluster all runnin around mutterin the damned word all the time. 'Interdiction interdict interdite' this an that and that over there. Gets in yer subconscious, you get a pancultural religiopolitical shock like that, an you mutter mutter mutter to yourself all day long. Start to use words inappropriately. Sign of imminent institutional breakdown, I say."
"Indeed."
"Or ongoing, eh? With the War, eh? That'd count.... You know, I'd flat forgot about the Icebags' Cube. Heh. Funny thing, that, how you can be thinkin about somethin real hard, an miss the big.... Innat interesting? Not that it has any fuckin thing to do with what I been tryin to say all along, oh, no. So. Right. We saw that big darkmass a few days back, right? Headed what? About 40 degrees up out towards Saturn ways, wunnit? Whizzin along about point three or so?"
"Yes, Sire. Would you like to pursue it now? At the time--"
"See, I'm thinkin they're gonna wanna have some, you know, big Bad Motherfucker bang-er-up bombs an shit on Mars, someday soon, yeh?"
"One would imagine both sides would, yes. Those, or food. Possibly both."
"Right, an ships, right? Need darkmass for ships' drives, at least for all those gleamy military needles so they look good in the parade. An, I shall remind you, not to mention this great wobbly filigree of silvery self-fabricated shite we call home, eh? How long to catch it?"
"Hmmm. I believe it would have wrapped by now, or would before we reached it. I can calculate the distance to the Weeuruunie Interdiction Boundary easily; the last surveys of both the boundaries and the darkmasses made before the recent hostilities broke out were very accurate. If the darkmass has wrapped, we will surely be able to catch it without any trouble on the other side."
"Well, that'd be a good part aways back across the Cube, wouldn't it? I'm not fuckin made of money. Or fuel."
"One good-sized harvest of darkmass would more than pay for the effort, Sire."
"Yeh, right! An that is why I said, to you, not five minutes back, 'Oy, where the fuck are ya?' Yer damn lettuce threw me off, that an all your natterin. Here: We're gonna go get all of em."
"..."
"Yeh, heheh, you'd better close your gapin mouth, laddie mine, before you catch a gymnosperm or somethin in it, because see, I got a plan. And didst I not make thee from the flesh of my flesh, or words to that effect? Dost thou not trustest me who made thee in mine image, O my Sunny Jim? Heh. See this? Right here? This in here is original brains, they are. Yours a mere Derivative Work."
"My Lord, there are seven mapped darkmasses within the confines of the Interdiction Cube! The expeditions have been planned and in one case executed--"
"Ach, dammit, did I forget to grow yer balls when I made ya? O, ye of little balls. Where's yer sense of... well, greed boy? I mean, come on, you know very well what I paid for that wee 14 million tons of darkmass down there -- three hundred thousand souls, it was, I made myself. And there's a fuckin War on, an a biggie at that, an big war is as a matter of well-known fact big money, QED. First interplanetary war in fuckin history, an who knows how grand an blowy-uppy it'll get, an here we are out in the middle of the Upper Armpit of Nowhere, noteworthily I shall remind you not in the blowy-uppy part of the fuckin War, an we have a fast ship, complete with fuel an partially synthetic brains an little fuckin cabbage leaves growin the orangery. An I might add a unique ship what's uniquely ship-shape enough that it does not need to deal with acceleration in any ways whatsoever, nosiree, but merely communes with the quantum informational thingie for permission and changes frames of reference as if by magic. Come on, there's just seven of em."
"Sire. There are... well, I suppose it..."
"Ahh, see, there's the kind of enthusiasm I like to hear from the fruit of my -- well my cheek as I recall. An -- mind you, mind you! -- as I might have mentioned there's a fuckin War on, see, an so nobody else will be lookin for em. They're all busily ready-aim-firing-oh-no-wait-not-yet with their bombs an asteroids an shit, an noisily diplomatin an televisin at each other, an fundin black labs an gold retrieval missions an shit. Hell, boy, nobody else will even have a fuckin clue where these darkmasses have got to, once that last survey's got a bit obsolete an they go whizzin an wrappin around the Cube a few times! Heheh. Seven, you say?"
"Yes. As I recall the survey was accurate to within one meter, and one centimeter per second velocity. That was about three months ago, so the uncertainty clouds will have spread somewhat by now. And they'd all have wrapped one or more times in that period, adding more uncertainty...."
"Well, we're not a passive observer here, you know. We are, do not forget, in a fuckin awesome space ship. In a pinch we can even make it go 'whoosh' an fart hot hydrogen out its arse an there you are, dealing with yer uncertainties proactively by chasin stuff. Or so I am told, when I am not resigned to merely sittin in the middle of fuckin nowhere consumed with a passion for growin intelligent flower gardens. Or if you prefer we might consider, oh I dunno, maybe usin the fuckin lattice drive we grew last month, instead of just ponderin its quantum splendor an all."
"Indeed, point taken. That said, Sire, we do have a limited amount of fuel, as you mentioned. Each use of the lattice drive to shift our frame of reference will consume a substantial portion of the darkmass we carry. We should plan a course that minimizes the amount of velocity we need to reach their vicinities. Or the most we can reach. It may not be feasible to expect to get all seven."
"Ay, ay -- thank you very much young Master Exposition, but I'll have no half-assed clone talk, you. All seven or nothin."
"Well, the Cube is 11,323,344,384 kilometers across, so--"
"That would be natterin again. Do you always think out loud like that? I might could fix that, you know. Hey, hang on, hang on! People were tryin to figure out how Pluto gets in an out, but nothin else can, right? I mean, why ain't Pluto jumpin all over the fuckin sky every time it crosses the Cube boundary, when every other piece of stuff in the fuckin box just wraps over to the other side unchanged an all, just blip an there you are with the shadows all upside-down of a sudden, eleven billion-an-more klicks from where you started?"
"Something about dark matter, I'm told, Sire, though that seems like a hedge. I suspect the Weeuruunie wrote an exception into the matter filter for Pluto when they constructed the Interdiction Cube, so that early astronomers would not become overexcited if they noticed the planetoid jumping."
"Heh. As if. They was keepin the dinosaurs in, an no mistake."
"Indeed. And those were the astronomers that were meant not to be excited. At any rate, the Cube is of known size, and the position and velocities of the seven darkmasses are known to great accuracy, and of course we know our own position very well. It should boil down to a simple dynamic programming problem."
"Yeh, well, we got what? just below fifteen million tons of darkmass an hydrogen fuel on board. So no needless sightseein. We give a wee tap on the gas, we coast, we nab, we prod again, got it? Extended burns will leave us sittin out here twiddlin our thumbs up our asses an eatin gynocarps an shit. No offense, little moldies."
"Gametophytes, Lord, but only the sporophyte stage of this cultivar have organs of hearing.... There is one problem."
"Yeh. The fuckin War, eh?"
"Yes, Sire. We will need to avoid a substantial region around the core of the system, unless we pass through at relatively high speed. More than one millicee, for example. I suspect we will be better served if we use the Interdiction Boundary to wrap across the Solar System instantly. It does increase the complexity of the tour planning somewhat."
"Right. So hop on it, boyo. I'll wake up the ship an the others, an then let's go nab us all the darkmass anybody's ever not seen, before they get wise to us."
"I will be done here momentarily, Sire."
"An did I ever tell you ya sound like Mister fuckin Spock? I didn't build that in, did I? You do it on purpose, don't ya, as an actin out thing. Heh. No, wait, I know, you talk like that Vizier fellow in that old movie, innit? Not sure that's entirely appropriate for a fourteen year old boy, now, is it?"
"No, Sire. Not at all."
"Heh. Smartass. But always remember -- Derived Work."
The Weeuruunie Interdiction Cube is 11 323 344 384 kilometers on each edge, measured to within one centimeter, and is as near as can be determined perfect cube with center coincident with the center of mass of the Sun, and oriented so that one particular corner is aligned permanently with the center of mass of Neptune. Any object more complicated than a formaldehyde molecule (except for some reason the planet Pluto and its moon) attempting to pass across the boundary in any direction from either side is instantaneously transported to the opposite face at the exact equivalent position, without loss of velocity or any noticeable inertial effect. The easiest way to visualize this is to recall the late 20th century game Asteroids, in which objects disappeared off the right side and reappeared on the left instantly; topologically, an Asteroids screen maps the Euclidean space of the screen onto the surface of a torus, while the Weeuruunie Interdiction Cube maps the Solar System onto the "surface" of a higher-dimensional torus. As far as matter is concerned the boundary is equivalent to continuous space; you could in theory stick your arm across it and goose somebody eleven billion kilometers away, with no ill effects beyond a slap, though you would be unable to see where your arm had gone -- the boundary implements an intimidating suite of information-based filters... well, let's just say it's "magic" and make this shorter.
Note: The wrapping property of the Interdiction Cube means the shortest distance to a given point from your location might lie in any of 27 possible directions.
By common convention, spatial coordinates inside the Cube are referred to relative to the Neptune corner, with that described as (0.0, 0.0, 0.0) and the opposite corner as (1.0, 1.0, 1.0). The particular orientation of axes is unimportant for the problems here, but suffice to say they are determined by long negotiations by a series of squabbling multinational bodies, and subject to change on short notice.
The seven known darkmasses are not conventional matter, though they are affected by the Interdiction Cube in both the typical and an unusual manner. Unaffected by gravity, they only move in straight lines at constant velocity with respect to the Cube, though the particulars of their size, vectors of motion and speed differ from darkmass to darkmass. Little is known about their composition, which is one reason it is referred to by physicists as "eccentric matter", even though it forms the basis of the new lattice drive technology and has numerous other engineering and military applications. It is very dense, though practically invisible, and does not create or experience gravitational fields. Which is a big part of why it's called "eccentric."
The most modern ships (such as yours, and some military vessels) which rely on darkmass-fueled lattice drives are also -- as a side effect -- unaffected by external gravitation. If sufficiently advanced (as is yours) they can change velocity within one second by transforming darkmass into conventional matter and expelling it... well, let's just call it "magic" and make this shorter, and will travel in a straight line (also relative to the Cube) until the velocity is actively changed again.
Whenever the lattice drive is activated, determine the amount of darkmass expended using this algorithm: First, calculate the magnitude of the change in velocity vectors. For example, from rest v=(0,0,0), a jump to v=(0.0000001, 0.0000001, 0) is an absolute change of Sqrt[2]*10-7. Multiply this by the mass being moved, in tons, and that is the percentage of darkmass consumed by the maneuver. So for example a ship weighing 40 million tons of which 12 million tons were darkmass that undertook the previous maneuver would lose 5.65% of the darkmass it was carrying, or 678823 tons, instantaneously. This percentage is independent of the absolute amount of darkmatter on hand (except in terms of its mass), so small ships carrying small amounts of darkmass can be very maneuverable.
Remember, it's eccentric matter.
That said, a speed of 0.05 of the speed of light is the absolute speed limit for lattice-driven ships inside the Interdiction Cube, for reasons that remain obscure, though it is thought possible that "wild" darkmasses could travel faster.
Ships can harvest fuel from a "wild" darkmass by first intercepting it, then matching velocities (thus requiring a minimum of two uses of the lattice drive). 1000 tons of darkmass can be taken on board per second.
The seven known darkmasses' masses, positions, and velocities as of 2 678 400 seconds ago exactly are
| mass | position (relative to cube axes) | velocity (relative to cube axes) |
| 10 000 000 tons | (0.1, 0.1, 0.1) | (0.00000011, -0.00000022, 0.00000006) |
| 5 000 000 tons | (0.5, 0.2, 0.9) | (-0.00000015, -0.00000017, 0.00000037) |
| 20 000 000 tons | (0.7, 0.9, 0.3) | (0.00000100, -0.00000039, -0.00000012) |
| 10 000 000 tons | (0.2, 0.8, 0.7) | (-0.00000034, 0.00000028, -0.00000021) |
| 60 000 000 tons | (0.5, 0.6, 0.5) | (-0.00000021, -0.00000030, 0.00000035) |
| 5 000 000 tons | (0.7, 0.1, 0.6) | (0.00000018, 0.00000013, 0.00000021) |
| 40 000 000 tons | (0.8, 0.8, 0.1) | (0.00000011, -0.00000020, 0.00000020) |
Question #1: Ignoring uncertainty in the above measurements, and relativistic effects, where are each of the seven darkmasses now (2 678 400 seconds later)?
Question #2: The ship is located exactly at Cube coordinates (0.82, 0.86, 0.44), and has at present a relative velocity of (0,0,0), and has 14 000 000 tons of darkmass fuel on board for the lattice drive. Again ignoring relativistic effects and uncertainties, which of the seven darkmasses is the best to visit first, if the point is to maximize the average rate at which fuel is acquired (or lost) over the entire duration of the operation?
Question #3: What is the best route that visits and completely harvests all seven darkmasses, but which completely avoids entering the spherical region centered on the sun, of radius 400 000 000 km (The Warzone)? By "best" we again mean the one that maximizes the average rate at which darkmass is harvested over the duration of the operation, as well as the total amount of darkmass on board at the end?.
Question #You Thought I Completely Forgot About the Title, Didn't You?:
"There has been a complication, Sire."
"Fuckin fuck! What?! I was just puttin more memory into this philodendron here. Don't sneak up on me like that! Dammit. Sorry, sorry, wee plantie. Waddya mean, 'complication'?"
"Well, our earlier visits to the wild darkmasses have been more... tentative than this trip. When we colocated with the first darkmass...well, it broke."
"Broke? No, wait, let me guess. It 'broke' 'eccentrically', I'll wager."
"Yes, Sire. It seems that exactly half of the darkmass was somehow knocked off by our abrupt arrival, and is now proceeding in the same direction we were originally headed when we colocated with its position, traveling at our previous velocity."
"Like a halfa fuckin billiard ball, eh? But we still have half of what we expected to be waitin for us? Fuckin Oorters, eh? Makin weird shit happen just cause they can. Someday I think I wanna meet one of them. An take him apart. Betcha his brains're worth a mint an a half."
"Yes, Sire. So, from the readings we've taken, we would need to colocate with the darkmass at a relative velocity of no more than 10-9 to avoid this phenomenon. Coinciding with a darkmass at any lower velocity, we should be able to retrieve it entirely; higher, and half will fly off just as fast as we arrived."
"Well, hell, that'd take a pantload more fuel, then, wouldn't it, slowin down like that? Fuck! Agh! But if we don't, we need to chase down the pieces as we make em."
"Indeed, Sire."
"Well... just figger it out. You're a smart lad. Make it fuckin so. An gimme that chip."
So: After completing the acquisition of the half-darkmass you first encountered in your original route, describe a new route to the remaining seven (including the half of the one you thought to take out completely) and harvests every one, in the shortest time, and with the largest remaining amount of darkmass fuel on board, completely avoiding the Warzone, under the new rule that if the velocity of the ship relative to the darkmass upon arrival is more than 10-9 (in lattice coordinates), half of the darkmass is lost in the direction you were moving, at the velocity you were moving.
Bonus question #1: That uncertainty thing. It was mentioned. It turns out to be valid. Indeed, the position of the darkmasses was only mapped to within one meter (they are, as we have pointed out several times, very dense), and their velocity only accurate to within one centimeter per second. First, what are these infinitesimal-sounding distances in proportion to the Cube edge? More important, if the center of each darkmass was somewhere within a sphere of radius 1 meter centered on the reported coordinates, and the velocity accurate to within one centimeter per second in magnitude and 0.01 arcsecond in direction [no, we didn't mention that; clones don't recall everything], how large a region might you have to search to find each darkmass?
posted by Vaguery | 8:37:00 PM
February 11, 2005
Before you die of tetanus
File under: geometry, numerical optimization, multiobjective optimization
Inspired by: This post at Notional Slurry
You are at the summer cottage. Alas, the weather is gray and chill and blustery, and there is nothing to do. The cottage, you will recall, is devoid of civilized amenities such as computers and televisions, having been created solely for the infuriation of young persons.
Grandmama points out, however, that there is the Rainy Day Cupboard. What is in it, you ask? Several dried husks of decorative indian corn, nibbled by mice; a number of desiccated stingray egg cases, stinking of distant seas and Clorox; an edition of Scrabble missing twelve letter tiles; five jigsaw puzzles; a television antenna (what?); and a mysterious frame.
When you encounter this frame, you initially mistake it for a picture frame: it is of sturdy wood, and rectangular and open like a picture frame. But unlike most such mundane objects this one is fashioned from a dark and glistening wood, and has many ornately fashioned little sharp nails or pins sticking out of it, which have apparently collected lint and dust and rust through the years -- ouch!
After Grandmama nurses your finger with a magic Band-Aid, you ask her about the frame, which possesses a strange allure for you now that it has tasted of your very life's blood. Grandmama tells you that many, many years ago, a thing called string art was very popular, before being suppressed by right-thinking people. What you have found in the Rainy Day Cupboard is, in fact, a partially-completed String Art project. Would you like to finish it, she asks?
How would you do that?
Well, she says, one supposes one would tie string between the nails. Ahh, of course. String art! You warily approach the apparatus, and your well-trained little eye notes that there are 100 of the pins sticking up from one of the members of the frame, and 100 pins sticking up from the opposite edge. It is an amazing piece of folk craftsmanship you realize, because the pins are placed with absolute precision at equal distance along the frame. Whoever this long-dead master pin-setter was, he must surely have been a leader among string artisans!
(As noted, you have an eye for these things. You have watched Find many times with Mummy. It is fun to try to identify the Keno Brothers. One, you have learned, is actually a rather well-designed CGI program....)
As Grandmama hands you some of her sewing thread to play with, she notes with a sudden intake of breath the increasingly painful swelling afflicting your Band-Aided finger. A look of concern passes over her face, and she pats you on the head and tells you to play nice -- but carefully -- while she goes and makes a call on the telephone to Mummy. Mummy is digging for old bones in Uzbekistan, you know, so she will be very cross at being awakened.
Outside, the gray day appears to be getting rather stormy, and whitecaps are crashing on the strand outside the cottage window....
So, you think: Let us label the 100 pins on the left member of this mysterious frame, starting at the top, as L1 through L100. Similarly, we shall label the pins on the right member as R1 at the top, and R100 at the bottom. How cunning your fevered mind is becoming! Now. If I were to tie a string tightly to pin L1, and stretch it ever so tightly across the frame to R1 and tie it there, then... well, I'd have connected two pins. And I could continue in that ways until I had tied each of Li to Ri in turn. It would look something like a harp, I imagine. And I could strum it, tra-la!
A frown passes over your face, as the wind picks up a bit, and the lights flash briefly off and on. You absentmindedly rub your forearm, where a dull ache is encroaching.
Now, that wouldn't be very artistic would it? Grandmama said it was string art, so I shall make the lines cross! Yes! So, if I were instead to connect pin L1 to R100, and L2 to R99, and so on, then every string would cross. But... well, the master pin-setter has done such a wonderful job that there would in fact be only one crossing point, in the very center. While conceptually interesting to those who prefer a minimalist aesthetic, I would like a bit more challenge.
You frown a bit. Rain pelts the windows, and Grandmama is speaking quickly and rather loudly to somebody on the telephone now.
I have it! I shall connect the pins to one another in such a way as to maximize the number of crossings. I shall connect each of 100 strings from one bare left pin to one bare right pin, and count how many crossings there are. I shall be happiest if the number the maximum possible!
Sweat gleams upon your little brow as you set to work. Above, you hear the stormcock squawking as it pivots madly on the roof....
Questions #1: Presuming that Mummy was not so very busy that she has forgotten your booster shots, and that in any case Grandmama has reached the proper authorities before the storm has closed off the causeway, in what pattern do you tie the strings?
Here is a picture of just one way to connect the 100 pins on the left to the 100 pins on the right:
And here is a picture of where the crossings exist in that particular pattern of connections:
Question #2: Having completed the first task appointed you (appointed? By whom? No matter! The work is fascinating!), this business of tying only bare pins to one another begins to chafe. Suppose instead we were released to tie the 100 sacred strings between any pairs of pins on opposite sides. That is, some pins might be left bare, and others be tied to multiple pegs on the other side. So long as no two strings coincide (are tied to the same left and right pins), what can be done to improve the number of total crossings? There must be more crossings!
Question #3: [Programming] As the infection advances -- for it is no normal tetanus, but rather the very Curse of the Penultimate String Artist creeping towards your heart -- you undertake many, many different arrangements of 100 lengths of string. You notice that for arbitrary tying patterns the majority of crossings tend to cluster in the middle of the frame.
Unlike minimalism, this inspires your growing mystical sensibility, and you decide that you would like to add an missing aspect of balance to your work: You still want to try to maximize the overall number of crossings of the strings (you are not allowed to do otherwise by HIM), but at the same time must minimize the difference between the number of crossings at any horizontal point across the frame.
What you will do is this: once a particular arrangement of strings has been tied, you will use a circular beer mat (which is 1/21 the width of the frame) to determine how many crossings are anywhere from the top to the bottom of the frame in the span measured out by the width of the mat. In effect you'll "scan" across a 1/21 portion of the frame's width, and count how many crossings lie in that fraction. You are full of pep and curse, now, so can be very accurate in your measurements so that you may find a pleasantly balanced arrangement of strings before Grandmama's shrieking makes you raise your little glowing eyes from your Great Work.
Here, for example, is what it looks like when you count the number of crossings that lie in every possible beer-mat's-width of the previous example. There are anywhere between about 55 and 165 crossings in the various windows -- far too much variation to please HIM, surely!
Now you have two goals: try to ensure that there is no arrangement that (a) has more crossings and (b) has less variation in the number of crossings across the frame. You may have fewer crossings than what you found in the previous ritual, if the new arrangement is simultaneously better on both goals than all others. How are the strings best arranged?
Question #4: [Programming] Grandmama's crumpled form lies in the corner, the lights are off and lightning flashes all around the cottage... but that is no matter to you. You no longer need to see with mortal eyes in order to complete the Final Gateway Opening Arrangement. The beer mat will still prove useful, regardless of the noisome stains which have splashed it: Again you must maximize the number of crossings. But now you must simultaneously minimize the difference in the number of crossings strictly covered by the circular beer mat! It is 1/21 of the width, and 1/13 of the height of the frame. Now that Grandmama is watching you blandly with her cold eyes, you will have time to take the requisite 31,213 samples of random discs from the (rectangular) frame. If the strings are arranged correctly... THEY will be able to come and play with you! Hurrah! Iä!
Here is what it might look like after you have thrown the beer mat randomly a number of times onto the framework. The green circles indicate the part of the pattern that is covered by the mat on each throw:
Bonus question #1: Suppose that the pin-setter had not been so masterful. How many more crossings (if any) could be achieved if the 100 pins on the right-hand member were located wherever you wanted them to be?
Bonus question #2: Suppose that the frame was a circle, and that the 200 pins were arranged equidistant along the circumference. How many crossings are possible with only one string tied to each pin? With more than one string allowed at a pin?
Bonus question #3: Again, suppose that randomly throwing the circular beer mat (1/21 of the diameter of the circular frame) is the method used to sample the number of crossings covered by 31,213 discs, in order to simultaneously maximize the number of crossings, and even out the spatial distribution of the crossings. What is the best arrangement of the 100 strings?
Bonus question #4: In every case so far where you have been trying to be "even," as well as maximizing the number of crossings, one suspects that there is some arrangement of strings that is maximally even (but not trivially; there have to be some crossings) -- even though this might not maximize the number of crossings. What are the arrangements for the rectangle and the circle, for both single-tied and multi-tied pins, that are the most spatially balanced in two dimensions, which contain one or more crossings? If you can discover this, you may save the world.
Inspired by: This post at Notional Slurry
You are at the summer cottage. Alas, the weather is gray and chill and blustery, and there is nothing to do. The cottage, you will recall, is devoid of civilized amenities such as computers and televisions, having been created solely for the infuriation of young persons.
Grandmama points out, however, that there is the Rainy Day Cupboard. What is in it, you ask? Several dried husks of decorative indian corn, nibbled by mice; a number of desiccated stingray egg cases, stinking of distant seas and Clorox; an edition of Scrabble missing twelve letter tiles; five jigsaw puzzles; a television antenna (what?); and a mysterious frame.
When you encounter this frame, you initially mistake it for a picture frame: it is of sturdy wood, and rectangular and open like a picture frame. But unlike most such mundane objects this one is fashioned from a dark and glistening wood, and has many ornately fashioned little sharp nails or pins sticking out of it, which have apparently collected lint and dust and rust through the years -- ouch!
After Grandmama nurses your finger with a magic Band-Aid, you ask her about the frame, which possesses a strange allure for you now that it has tasted of your very life's blood. Grandmama tells you that many, many years ago, a thing called string art was very popular, before being suppressed by right-thinking people. What you have found in the Rainy Day Cupboard is, in fact, a partially-completed String Art project. Would you like to finish it, she asks?
How would you do that?
Well, she says, one supposes one would tie string between the nails. Ahh, of course. String art! You warily approach the apparatus, and your well-trained little eye notes that there are 100 of the pins sticking up from one of the members of the frame, and 100 pins sticking up from the opposite edge. It is an amazing piece of folk craftsmanship you realize, because the pins are placed with absolute precision at equal distance along the frame. Whoever this long-dead master pin-setter was, he must surely have been a leader among string artisans!
(As noted, you have an eye for these things. You have watched Find many times with Mummy. It is fun to try to identify the Keno Brothers. One, you have learned, is actually a rather well-designed CGI program....)
As Grandmama hands you some of her sewing thread to play with, she notes with a sudden intake of breath the increasingly painful swelling afflicting your Band-Aided finger. A look of concern passes over her face, and she pats you on the head and tells you to play nice -- but carefully -- while she goes and makes a call on the telephone to Mummy. Mummy is digging for old bones in Uzbekistan, you know, so she will be very cross at being awakened.
Outside, the gray day appears to be getting rather stormy, and whitecaps are crashing on the strand outside the cottage window....
So, you think: Let us label the 100 pins on the left member of this mysterious frame, starting at the top, as L1 through L100. Similarly, we shall label the pins on the right member as R1 at the top, and R100 at the bottom. How cunning your fevered mind is becoming! Now. If I were to tie a string tightly to pin L1, and stretch it ever so tightly across the frame to R1 and tie it there, then... well, I'd have connected two pins. And I could continue in that ways until I had tied each of Li to Ri in turn. It would look something like a harp, I imagine. And I could strum it, tra-la!
A frown passes over your face, as the wind picks up a bit, and the lights flash briefly off and on. You absentmindedly rub your forearm, where a dull ache is encroaching.
Now, that wouldn't be very artistic would it? Grandmama said it was string art, so I shall make the lines cross! Yes! So, if I were instead to connect pin L1 to R100, and L2 to R99, and so on, then every string would cross. But... well, the master pin-setter has done such a wonderful job that there would in fact be only one crossing point, in the very center. While conceptually interesting to those who prefer a minimalist aesthetic, I would like a bit more challenge.
You frown a bit. Rain pelts the windows, and Grandmama is speaking quickly and rather loudly to somebody on the telephone now.
I have it! I shall connect the pins to one another in such a way as to maximize the number of crossings. I shall connect each of 100 strings from one bare left pin to one bare right pin, and count how many crossings there are. I shall be happiest if the number the maximum possible!
Sweat gleams upon your little brow as you set to work. Above, you hear the stormcock squawking as it pivots madly on the roof....
Questions #1: Presuming that Mummy was not so very busy that she has forgotten your booster shots, and that in any case Grandmama has reached the proper authorities before the storm has closed off the causeway, in what pattern do you tie the strings?
Here is a picture of just one way to connect the 100 pins on the left to the 100 pins on the right:
And here is a picture of where the crossings exist in that particular pattern of connections:
Question #2: Having completed the first task appointed you (appointed? By whom? No matter! The work is fascinating!), this business of tying only bare pins to one another begins to chafe. Suppose instead we were released to tie the 100 sacred strings between any pairs of pins on opposite sides. That is, some pins might be left bare, and others be tied to multiple pegs on the other side. So long as no two strings coincide (are tied to the same left and right pins), what can be done to improve the number of total crossings? There must be more crossings!
Question #3: [Programming] As the infection advances -- for it is no normal tetanus, but rather the very Curse of the Penultimate String Artist creeping towards your heart -- you undertake many, many different arrangements of 100 lengths of string. You notice that for arbitrary tying patterns the majority of crossings tend to cluster in the middle of the frame.
Unlike minimalism, this inspires your growing mystical sensibility, and you decide that you would like to add an missing aspect of balance to your work: You still want to try to maximize the overall number of crossings of the strings (you are not allowed to do otherwise by HIM), but at the same time must minimize the difference between the number of crossings at any horizontal point across the frame.
What you will do is this: once a particular arrangement of strings has been tied, you will use a circular beer mat (which is 1/21 the width of the frame) to determine how many crossings are anywhere from the top to the bottom of the frame in the span measured out by the width of the mat. In effect you'll "scan" across a 1/21 portion of the frame's width, and count how many crossings lie in that fraction. You are full of pep and curse, now, so can be very accurate in your measurements so that you may find a pleasantly balanced arrangement of strings before Grandmama's shrieking makes you raise your little glowing eyes from your Great Work.
Here, for example, is what it looks like when you count the number of crossings that lie in every possible beer-mat's-width of the previous example. There are anywhere between about 55 and 165 crossings in the various windows -- far too much variation to please HIM, surely!
Now you have two goals: try to ensure that there is no arrangement that (a) has more crossings and (b) has less variation in the number of crossings across the frame. You may have fewer crossings than what you found in the previous ritual, if the new arrangement is simultaneously better on both goals than all others. How are the strings best arranged?
Question #4: [Programming] Grandmama's crumpled form lies in the corner, the lights are off and lightning flashes all around the cottage... but that is no matter to you. You no longer need to see with mortal eyes in order to complete the Final Gateway Opening Arrangement. The beer mat will still prove useful, regardless of the noisome stains which have splashed it: Again you must maximize the number of crossings. But now you must simultaneously minimize the difference in the number of crossings strictly covered by the circular beer mat! It is 1/21 of the width, and 1/13 of the height of the frame. Now that Grandmama is watching you blandly with her cold eyes, you will have time to take the requisite 31,213 samples of random discs from the (rectangular) frame. If the strings are arranged correctly... THEY will be able to come and play with you! Hurrah! Iä!
Here is what it might look like after you have thrown the beer mat randomly a number of times onto the framework. The green circles indicate the part of the pattern that is covered by the mat on each throw:
Bonus question #1: Suppose that the pin-setter had not been so masterful. How many more crossings (if any) could be achieved if the 100 pins on the right-hand member were located wherever you wanted them to be?
Bonus question #2: Suppose that the frame was a circle, and that the 200 pins were arranged equidistant along the circumference. How many crossings are possible with only one string tied to each pin? With more than one string allowed at a pin?
Bonus question #3: Again, suppose that randomly throwing the circular beer mat (1/21 of the diameter of the circular frame) is the method used to sample the number of crossings covered by 31,213 discs, in order to simultaneously maximize the number of crossings, and even out the spatial distribution of the crossings. What is the best arrangement of the 100 strings?
Bonus question #4: In every case so far where you have been trying to be "even," as well as maximizing the number of crossings, one suspects that there is some arrangement of strings that is maximally even (but not trivially; there have to be some crossings) -- even though this might not maximize the number of crossings. What are the arrangements for the rectangle and the circle, for both single-tied and multi-tied pins, that are the most spatially balanced in two dimensions, which contain one or more crossings? If you can discover this, you may save the world.
posted by Vaguery | 8:40:00 PM
via Notional Slurry
I'm starting this Blogger spin-off as a place to try posting the frequent exercises, puzzles, mathematical and computational games I've been developing for years. Notional Slurry should still be considered home base, but has become cluttered of late with a number of different threads and specialties all piled on top of one another.
Maybe here we can have some room.
I'm not entirely convinced that Blogger offers enough features to manage the many figures, diagrams, and such. We'll see. I may just jump back to a WordPress install elsewhere.
Maybe here we can have some room.
I'm not entirely convinced that Blogger offers enough features to manage the many figures, diagrams, and such. We'll see. I may just jump back to a WordPress install elsewhere.
posted by Vaguery | 8:04:00 PM