Sunday, December 16, 2012
Some Rejoiceables
The U-Bend is re-organizing their whole public interface, with the (unintended?) effect that it's hard to tell what will and won't work, so I'll just link to this, one of the best performance captures of one of my favourites. Oh, and Gaudete!
Thursday, December 13, 2012
It did rain pennies from heaven
... and might again.
Reported here; in my own abstract: any metals denser than basalts and that can be mined fell from solar orbit onto the Earth long after its crust formed and mostly settled-down. And remember that there's still plenty of stuff out there!
I thought this might interest some of my theologically-minded fellow web-loggers.
the triviafer
Reported here; in my own abstract: any metals denser than basalts and that can be mined fell from solar orbit onto the Earth long after its crust formed and mostly settled-down. And remember that there's still plenty of stuff out there!
I thought this might interest some of my theologically-minded fellow web-loggers.
the triviafer
Saturday, December 8, 2012
Poco a poco diminuendo...
(Incidentally, if things seem to be slowing down, I suppose I might blame my recent foray into tumblng, but as that's been less than a week it doesn't really answer. No, I shan't post a link here. Send a message to /dev/null ... not your own, of course, but to the address on that pseudonym's profile page. But it's teribly mathy, so... so I don't know what.)
Update Whee! it works nicely now! Thanks, screencountry!/Update
Since everyone now agrees that the particular warranty wasn't worth the electrons it jiggled and juggled, I don't mind telling everyone (anonymously, at least) that my shiny laptop screen became cracked after a thoroughly-preventable fall of about four feet onto concrete, and that I really should have thought to zip up the carrying bag, but on the other hand I don't see the practical use of a clamshell design like that. Perhaps I shall add gores later. Anyways, the endpoint-purveyor offered to charge me more than 33% more than the original price for it to replace the screen, which offer I declined with some irritation. This was after the manufacturer-designated service centre took four days to notice it had been sent to them (though they were expecting it) and a further week to decide that no, they wouldn't do anything helpful, not even offer to charge me more than &c. So, it turns out there's a friendly business with warehouses in Seattle and in Vancouver that sell ... surplus? excess? anyways, replacement lcd screens. If the thing I'm expecting next wednesday works, I might just tell you all about it. But really I mustn't drop the thing again, that was the worst, and this sort of rigor mallearum is expensive, and would be again even knowing the short-cut at the end. It's a nifty thing, but the screen was the only broken thing I found, after the drop, so things might have gone much worse. I didn't even stub a toe!
Well, your patience during this little ramble has been most appreciated, believe you me.
God bless you all!
Update Whee! it works nicely now! Thanks, screencountry!/Update
Since everyone now agrees that the particular warranty wasn't worth the electrons it jiggled and juggled, I don't mind telling everyone (anonymously, at least) that my shiny laptop screen became cracked after a thoroughly-preventable fall of about four feet onto concrete, and that I really should have thought to zip up the carrying bag, but on the other hand I don't see the practical use of a clamshell design like that. Perhaps I shall add gores later. Anyways, the endpoint-purveyor offered to charge me more than 33% more than the original price for it to replace the screen, which offer I declined with some irritation. This was after the manufacturer-designated service centre took four days to notice it had been sent to them (though they were expecting it) and a further week to decide that no, they wouldn't do anything helpful, not even offer to charge me more than &c. So, it turns out there's a friendly business with warehouses in Seattle and in Vancouver that sell ... surplus? excess? anyways, replacement lcd screens. If the thing I'm expecting next wednesday works, I might just tell you all about it. But really I mustn't drop the thing again, that was the worst, and this sort of rigor mallearum is expensive, and would be again even knowing the short-cut at the end. It's a nifty thing, but the screen was the only broken thing I found, after the drop, so things might have gone much worse. I didn't even stub a toe!
Well, your patience during this little ramble has been most appreciated, believe you me.
God bless you all!
Tuesday, November 27, 2012
More topological reflections of arithmetic
For this part of the story I want a really-pretty diagram that even the mathjax still doesn't understand yet how to draw, so I've prepared a dead picture of it.
The doubled-arrows say what kinds of squares they cross --- what sort of equations. The whole thing is basically a fancy elaboration of the idea $ \Phi + X = Z + Y $ and $ \Psi Z = C \Phi $ and $ A Z = X C $ and $ A Y = X B $ and $ B \Phi = Y \Psi $; and it says that if what it means by those are true, then (that's the dotted double arrow) also what it means by $ \Psi + A = C + B $ is true. Note, however, that not all arithmetically true things are topologically true; in arithmetic we have: if $ A B = 1 $ and $ C B = 1 $, then $A = C$; but this isn't true in figures: one would need the right diagram to make it work.
The best known (if not best-known) generality described by the above diagram is known as “Mather's Second Cube Theorem”, and it's at the heart of a lovely little construction I mentioned earlier, which you get with $ \Psi = * $ . There's essentially only one way for a space to “be $ * $”, so these situations are rather special; on the other hand, as long as $ \Phi $ all comes in one piece, there's exactly one way to fit any $A,B,C$ as in the diagram over any compatible $X, Y, Z$. What it meant for the earlier story is that one doesn't need to worry about $ \Phi$ , because it might as well be $ * $. Making that change doesn't even change the ... well, I can't quite say what it didn't change, because I haven't spelled it, here. But when you can simplify the problem without changing the solution, you're making progress!
There are lots more nifty diagrams involved, probably the prettiest being
--- the $\gg$ here is doing what the $\Downarrow$ was above, never mind. Anyways, I was going to write a fun little paper about it, but then I found this fine gem, which I think describes about the same argument, but better! This happens to me a lot, as you might imagine. One day...
Anyway, if anyone wants more, just ask me!
The doubled-arrows say what kinds of squares they cross --- what sort of equations. The whole thing is basically a fancy elaboration of the idea $ \Phi + X = Z + Y $ and $ \Psi Z = C \Phi $ and $ A Z = X C $ and $ A Y = X B $ and $ B \Phi = Y \Psi $; and it says that if what it means by those are true, then (that's the dotted double arrow) also what it means by $ \Psi + A = C + B $ is true. Note, however, that not all arithmetically true things are topologically true; in arithmetic we have: if $ A B = 1 $ and $ C B = 1 $, then $A = C$; but this isn't true in figures: one would need the right diagram to make it work.
The best known (if not best-known) generality described by the above diagram is known as “Mather's Second Cube Theorem”, and it's at the heart of a lovely little construction I mentioned earlier, which you get with $ \Psi = * $ . There's essentially only one way for a space to “be $ * $”, so these situations are rather special; on the other hand, as long as $ \Phi $ all comes in one piece, there's exactly one way to fit any $A,B,C$ as in the diagram over any compatible $X, Y, Z$. What it meant for the earlier story is that one doesn't need to worry about $ \Phi$ , because it might as well be $ * $. Making that change doesn't even change the ... well, I can't quite say what it didn't change, because I haven't spelled it, here. But when you can simplify the problem without changing the solution, you're making progress!
There are lots more nifty diagrams involved, probably the prettiest being
--- the $\gg$ here is doing what the $\Downarrow$ was above, never mind. Anyways, I was going to write a fun little paper about it, but then I found this fine gem, which I think describes about the same argument, but better! This happens to me a lot, as you might imagine. One day...
Anyway, if anyone wants more, just ask me!
Monday, November 26, 2012
A little less about that dullest of things
Dear Ludo,
After that long interruption, I had thought of returning to the earlier theme, but I find that Stilwell has been writing more articulately than I could a decent approximation to an Economics of Charity; I hesitate at some of his bolder sloganisms --- I hesitate at most slogans, for I am a timid creature by temperament. Particularly, it seems wrong to say that a true fiat issue is “not backed” and even by nature “unbackable”: rather, the backing of a true issue is its tradability, or under another aspect, the actual wealth of its market, or again, the productivity and honesty of its citizens.
Modulo such concerns, which are either natural or naturally bunk (I am not qualified to guess), I think I can recommend a cautious reading of his notes.
That is all.
from the armchair
After that long interruption, I had thought of returning to the earlier theme, but I find that Stilwell has been writing more articulately than I could a decent approximation to an Economics of Charity; I hesitate at some of his bolder sloganisms --- I hesitate at most slogans, for I am a timid creature by temperament. Particularly, it seems wrong to say that a true fiat issue is “not backed” and even by nature “unbackable”: rather, the backing of a true issue is its tradability, or under another aspect, the actual wealth of its market, or again, the productivity and honesty of its citizens.
Modulo such concerns, which are either natural or naturally bunk (I am not qualified to guess), I think I can recommend a cautious reading of his notes.
That is all.
from the armchair
Sunday, November 11, 2012
Ignosce me
Non veterescentur nobiscum consenescentis;
Senectudino non delassantur neque damnabuntur aevis.
Occidens Solis, ac Oriens, ei commemorabimus.
It isn't really one of mine; the sense is of a verse they recite in these parts every Armistice Day, and the individual stem words are mostly out of Whitaker's. But we keep trying!
Thursday, November 8, 2012
Well, that was ...
... that was something of an else.
Dear Mr. Returning,
I had limping hopes two years back that an hostile lower chamber might have called their king to account for transmuting and muting "creed and trust", in the Primordial Fix, to "quodcumquelatry", without consulting them or getting it officially rewritten. Yea, limpid limping hopes, but alas, it seems they did not much care.
In any case, we'll (or... they will... I'm a remote-dwelling foreigner, afterall) have four more Epiphanies and Easters and Christmases under your presiding, maybe you'll notice them? I'm afraid your managers won't let you notice them, properly; it'd be such a relief, though, if you could escape the Lower Downs for a little bit, and just soak-up the splendour and fear of some things "above your pay grade", or consider some of the things others built before anyone here was born, but are still using today. It's a sad thing for the princes, though, that princes are not by their handlers allowed that much contemplation, but only entertainment and diversion.
Anyway, as you boldly march forward with eyes firmly fixed on anything but where you're steppeing or standing, may I recommend a little book about a genuine (and successful!) search for common ground? Here's sufficient excerpt to find which one:
an extern
Dear Mr. Returning,
I had limping hopes two years back that an hostile lower chamber might have called their king to account for transmuting and muting "creed and trust", in the Primordial Fix, to "quodcumquelatry", without consulting them or getting it officially rewritten. Yea, limpid limping hopes, but alas, it seems they did not much care.
In any case, we'll (or... they will... I'm a remote-dwelling foreigner, afterall) have four more Epiphanies and Easters and Christmases under your presiding, maybe you'll notice them? I'm afraid your managers won't let you notice them, properly; it'd be such a relief, though, if you could escape the Lower Downs for a little bit, and just soak-up the splendour and fear of some things "above your pay grade", or consider some of the things others built before anyone here was born, but are still using today. It's a sad thing for the princes, though, that princes are not by their handlers allowed that much contemplation, but only entertainment and diversion.
Anyway, as you boldly march forward with eyes firmly fixed on anything but where you're steppeing or standing, may I recommend a little book about a genuine (and successful!) search for common ground? Here's sufficient excerpt to find which one:
You hold that your heretics and sceptics have helped the world forward and handed on a lamp of progress. I deny it. Nothing is plainer from real history than that each of your heretics invented a complete cosmos of his own which the next heretic smashed entirely to pieces. Who knows now exactly what Nestorius taught? Who cares?
an extern
Sunday, October 28, 2012
Anticipating the Commemoration of This Friday
A longish excerpt (I can never seem to trim them at all... ), from a rather sad thread from a novel by one of the most sympathetic writers Her Britannic Majesty Victoria might have read in the first printing.
Thus Jo. This isn't quite the first time we've seen this little land behind the iron gate, nor is it the last we will see of it; but it's probably the best showing the sorry spot has. We may gather from some details that this particular "berryin' ground" is rather overused, in a pauperish way, for its size. It is, in way, a geographic distillation of all the sadness that grows about the venerable institution not at all far off, where the learned gentlemen discuss every side of "equity" that will do no one any good untill there's nothing left to be equitable about.
For locus focus
a reader
By many devious ways, reeking with offence of many kinds, they come to the little tunnel of a court, and to the gas-lamp (lighted now), and to the iron gate.
"He was put there," says Jo, holding to the bars and looking in.
"Where? Oh, what a scene of horror!"
"There!" says Jo, pointing. "Over yinder. Among them piles of bones, and close to that there kitchin winder! They put him wery nigh the top. They was obliged to stamp upon it to git it in. I could unkiver it for you with my broom if the gate was open. That's why they locks it, I s'pose," giving it a shake. "It's always locked. Look at the rat!" cries Jo, excited. "Hi! Look! There he goes! Ho! Into the ground!"
The servant shrinks into a corner, into a corner of that hideous archway, with its deadly stains contaminating her dress; and putting out her two hands and passionately telling him to keep away from her, for he is loathsome to her, so remains for some moments. Jo stands staring and is still staring when she recovers herself.
"Is this place of abomination consecrated ground?"
"I don't know nothink of consequential ground," says Jo, still staring.
"Is it blessed?"
"Which?" says Jo, in the last degree amazed.
"Is it blessed?"
"I'm blest if I know," says Jo, staring more than ever; "but I shouldn't think it warn't. Blest?" repeats Jo, something troubled in his mind. "It an't done it much good if it is. Blest? I should think it was t'othered myself. But I don't know nothink!"
Bleak House, Charles Dickens
Thus Jo. This isn't quite the first time we've seen this little land behind the iron gate, nor is it the last we will see of it; but it's probably the best showing the sorry spot has. We may gather from some details that this particular "berryin' ground" is rather overused, in a pauperish way, for its size. It is, in way, a geographic distillation of all the sadness that grows about the venerable institution not at all far off, where the learned gentlemen discuss every side of "equity" that will do no one any good untill there's nothing left to be equitable about.
For locus focus
a reader
Friday, September 28, 2012
I was sort-of daydreaming...
so I don't quite recall how much there is...
but, well, Here is a lovely recording
what struck me most about it is the drums. They can particularly be heard around the final Amen, but there's somewhat more.
Here's another I just can't get out of my head at the moment
I blame Robert Reilly.
Over the summer, I was contemplating an arrangement of things like the following: start with a special square
$$ \begin{array}{ccc}
A & \to & B \\
\downarrow & \ulcorner & \downarrow \\
C & \to & P
\end{array}
$$
There is then some delightful nonsense which guarantees the following special squares
$$
\begin{array}{cccccc}
& E & \to & F & \to & *\\
& \downarrow & \lrcorner & \downarrow & \lrcorner & \downarrow \\
& A & \to & B & \to & P \\
E & \to & G & \to & * \\
\downarrow & \lrcorner & \downarrow & \lrcorner & \downarrow \\
A & \to & C & \to & P \\
& & & E & \to & F \\
& & & \downarrow & \ulcorner & \downarrow \\
& & & G & \to & *
\end{array}$$
That last one is the really-special one, because of where the $*$ sits in it. It makes $E$ rather simpler than it otherwise might have been... But that's a story for another day.
but, well, Here is a lovely recording
Here's another I just can't get out of my head at the moment
I blame Robert Reilly.
Over the summer, I was contemplating an arrangement of things like the following: start with a special square
$$ \begin{array}{ccc}
A & \to & B \\
\downarrow & \ulcorner & \downarrow \\
C & \to & P
\end{array}
$$
There is then some delightful nonsense which guarantees the following special squares
$$
\begin{array}{cccccc}
& E & \to & F & \to & *\\
& \downarrow & \lrcorner & \downarrow & \lrcorner & \downarrow \\
& A & \to & B & \to & P \\
E & \to & G & \to & * \\
\downarrow & \lrcorner & \downarrow & \lrcorner & \downarrow \\
A & \to & C & \to & P \\
& & & E & \to & F \\
& & & \downarrow & \ulcorner & \downarrow \\
& & & G & \to & *
\end{array}$$
That last one is the really-special one, because of where the $*$ sits in it. It makes $E$ rather simpler than it otherwise might have been... But that's a story for another day.
Monday, September 24, 2012
A microcosm of true liberty
Dear Carmen,
For your consideration.
It is not often that the music director at our parish asks us to attempt polyphony --- understandably, for it is rather a lot more work for him, and there aren't really that many of us singing for him.
But in consequence it is not often that (and hence it is notable when) I get to enjoy this particular curious experience: when a singer is able to sing his part near-automatically, he can actually hear the whole music around him even as he is singing it. I am convinced that this --- to hear the whole even while attentively making part of it --- is the true purpose of polyphony.1 It is, I would like to say, a microcosm of true liberty.
What I mean by this: what each voice has to do is well-delineated, and it is necessary that each attend to both the governor and the other voices so far as not to overwhelm or misclash with them; yet at the same time, assignment of parts is done with each singer's particular voice in mind, and singers assigned to the same part do not, must not sing identically --- it is usually necessary to breathe at inopportune moments, and so we support eachother, taking turns. Within the stringently prescribed form, there are deeps to navigate and intelligence to be applied. But, when each knows well-enough how he is to sing his part, he isn't trapped within it, but by the whole is lifted above his own line, and... it's hard to find words for it. If folk could live like that, it should have been a happy shire indeed.
Anyways, that's what I did this Sunday between about 10:30 and 1:30.
vox clamantis in urbis
1) This is not to say this is the true purpose of the art of polyphonic music or its performance --- that were God's Glory; but the particular means chosen by polyphony as opposed to plainchant, for instance.
For your consideration.
It is not often that the music director at our parish asks us to attempt polyphony --- understandably, for it is rather a lot more work for him, and there aren't really that many of us singing for him.
But in consequence it is not often that (and hence it is notable when) I get to enjoy this particular curious experience: when a singer is able to sing his part near-automatically, he can actually hear the whole music around him even as he is singing it. I am convinced that this --- to hear the whole even while attentively making part of it --- is the true purpose of polyphony.1 It is, I would like to say, a microcosm of true liberty.
What I mean by this: what each voice has to do is well-delineated, and it is necessary that each attend to both the governor and the other voices so far as not to overwhelm or misclash with them; yet at the same time, assignment of parts is done with each singer's particular voice in mind, and singers assigned to the same part do not, must not sing identically --- it is usually necessary to breathe at inopportune moments, and so we support eachother, taking turns. Within the stringently prescribed form, there are deeps to navigate and intelligence to be applied. But, when each knows well-enough how he is to sing his part, he isn't trapped within it, but by the whole is lifted above his own line, and... it's hard to find words for it. If folk could live like that, it should have been a happy shire indeed.
Anyways, that's what I did this Sunday between about 10:30 and 1:30.
vox clamantis in urbis
1) This is not to say this is the true purpose of the art of polyphonic music or its performance --- that were God's Glory; but the particular means chosen by polyphony as opposed to plainchant, for instance.
Thursday, September 20, 2012
Concerning Hobbits
Dear Ploughman,
Today is (if you have time for it) an excellent day for a party (or dinner, anyways) of special magnificence! I see it noted that tomorrow is "Second Breakfast" day, in honour of LXXV Publication Anniversary of the first installment of The Red Book. Hooray! This year also saw (or... will have seen?) the LVIII Publication Anniversary of the second installment, and together they make... er... CXXXIII, 133, a number nowhere to be known as one grass, of people or otherwise.
Alas that it will be Ember Friday; even more sorrowful, the Baggins' Combined Birthday will be Ember Saturday. Thus I remark with some admiration the Hobbit Custom of not assigning a Week Day to the Overlithe (two of them in leapyears); for by this simple concession to the regularity of Nature in defiance of Numerology, they have assured us that 22nd of September (SR) is always the Thursday (SR) between the Autumnal Ember Wednesday and Friday, because the 14th (SR) is always Wednesday (SR) in the week before. And so the purchase of provisions can quite sensibly fall to nearly nothing on the subsequent two days, and it did not matter much.
Now, to be sure, there are other fascinating emergent rhythms that this custom would tend to mute --- for instance, the cycling among the Joyful, Sorrowful and Glorious Mysteries that each fixed feast progresses through. I can't remember whose 'blog wrote something to remark on that, but indeed it is something else to contemplate the Crucifixion on the very feast of the Annunciation (a.k.a. New Year's Day, in Gondor).
In Any Case, 'tis Today the Penultimate Thursday of September, and so Happy Birthday, Bilbo and Frodo!
having never cooked rabbit
Today is (if you have time for it) an excellent day for a party (or dinner, anyways) of special magnificence! I see it noted that tomorrow is "Second Breakfast" day, in honour of LXXV Publication Anniversary of the first installment of The Red Book. Hooray! This year also saw (or... will have seen?) the LVIII Publication Anniversary of the second installment, and together they make... er... CXXXIII, 133, a number nowhere to be known as one grass, of people or otherwise.
Alas that it will be Ember Friday; even more sorrowful, the Baggins' Combined Birthday will be Ember Saturday. Thus I remark with some admiration the Hobbit Custom of not assigning a Week Day to the Overlithe (two of them in leapyears); for by this simple concession to the regularity of Nature in defiance of Numerology, they have assured us that 22nd of September (SR) is always the Thursday (SR) between the Autumnal Ember Wednesday and Friday, because the 14th (SR) is always Wednesday (SR) in the week before. And so the purchase of provisions can quite sensibly fall to nearly nothing on the subsequent two days, and it did not matter much.
Now, to be sure, there are other fascinating emergent rhythms that this custom would tend to mute --- for instance, the cycling among the Joyful, Sorrowful and Glorious Mysteries that each fixed feast progresses through. I can't remember whose 'blog wrote something to remark on that, but indeed it is something else to contemplate the Crucifixion on the very feast of the Annunciation (a.k.a. New Year's Day, in Gondor).
In Any Case, 'tis Today the Penultimate Thursday of September, and so Happy Birthday, Bilbo and Frodo!
having never cooked rabbit
Saturday, September 15, 2012
Have you noticed the titles diverging from subject matter?
cat >& /dev/null << eof
It's sad, really. Somehow, I just can't get those films out of my head, to the point that in my latest re-read, I'm constantly interrupting the author to remark "... see, that's another thing they totally didn't get in the movies..."
What am I supposed to do, now? Maybe I'll have to find my copy of Leaf by Niggle, again...
eof
It's sad, really. Somehow, I just can't get those films out of my head, to the point that in my latest re-read, I'm constantly interrupting the author to remark "... see, that's another thing they totally didn't get in the movies..."
What am I supposed to do, now? Maybe I'll have to find my copy of Leaf by Niggle, again...
eof
Wednesday, September 12, 2012
Vultum tuum deprecabuntur omnes divites plebis
On this (Extraordinary Calendar) Feast of The Holy Name of Mary,
$$\begin{array}{rl}
\mathrm{XXXVI} & \mbox{Solid ground underfoot}\\
\mathrm{XXXVII} & \href{http://introibo.fr}{\mbox{Chant and such-like}}\\
\mathrm{XXXVIII} & \mbox{Cream}\\
\mathrm{XXXIX} & \mbox{Work, temporal or }\href{http://wdtprs.com/blog/2012/09/benedict-xvi-there-is-no-such-thing-as-a-superfluous-or-useless-prayer/}{\mbox{spiritual}} \\
\mathrm{XL} & \mbox{Roman numerals, as it happens...}
\end{array}
$$
Happy feast day. Try the croissants!
$$\begin{array}{rl}
\mathrm{XXXVI} & \mbox{Solid ground underfoot}\\
\mathrm{XXXVII} & \href{http://introibo.fr}{\mbox{Chant and such-like}}\\
\mathrm{XXXVIII} & \mbox{Cream}\\
\mathrm{XXXIX} & \mbox{Work, temporal or }\href{http://wdtprs.com/blog/2012/09/benedict-xvi-there-is-no-such-thing-as-a-superfluous-or-useless-prayer/}{\mbox{spiritual}} \\
\mathrm{XL} & \mbox{Roman numerals, as it happens...}
\end{array}
$$
Happy feast day. Try the croissants!
Monday, September 10, 2012
What an odd day!
Dear DNS,
(And others...) For about three hours, I found I couldn't get to the websites of LifeSite News, Crisis Magazine, Dappled Things, Alt-Catholicah (a recent discovery --- some nifty things, there), or Catholic Vote. Perhaps these sites are all (it's beyond my knowledge) hosted in the same rack of some server host, and they had some trouble? They're all apparently back, now --- they even all returned about the same time, but I had an interesting little while musing on why this might happen.
But I don't want to cast asperities around, nor imaginative suspicions, so... folks, be grateful for your internet: it's slightly more fragile than we like to think, sometimes. There was a boast going around suggesting that the thing was designed to "withstand a nuclear holocaust". What a silly idea! The internet might just survive a "nuclear holocaust" in the same sense that the biosphere would --- in patches. There is no more guarantee that your internet service would persist than there is that you would persist to enjoy it.
It's a beautiful night after a beautiful sunset after a beautiful day; there is chocolate, and whipped cream. The strawberries, alas, were a fleeting joy.
a subscriber
(And others...) For about three hours, I found I couldn't get to the websites of LifeSite News, Crisis Magazine, Dappled Things, Alt-Catholicah (a recent discovery --- some nifty things, there), or Catholic Vote. Perhaps these sites are all (it's beyond my knowledge) hosted in the same rack of some server host, and they had some trouble? They're all apparently back, now --- they even all returned about the same time, but I had an interesting little while musing on why this might happen.
But I don't want to cast asperities around, nor imaginative suspicions, so... folks, be grateful for your internet: it's slightly more fragile than we like to think, sometimes. There was a boast going around suggesting that the thing was designed to "withstand a nuclear holocaust". What a silly idea! The internet might just survive a "nuclear holocaust" in the same sense that the biosphere would --- in patches. There is no more guarantee that your internet service would persist than there is that you would persist to enjoy it.
It's a beautiful night after a beautiful sunset after a beautiful day; there is chocolate, and whipped cream. The strawberries, alas, were a fleeting joy.
a subscriber
Friday, September 7, 2012
I seem to have gone all batty.
Dear Gadfly,
Of course, this is part, or signal, of the problem:
For she is absolutely right. The strict correctness, from a strategic point of view, that democratic elections anywhere these days demand the winners ... reserve their mind? ... engage in verbal misdirection? ... well, exaggerate and waffle are two current words for it, anyways --- the correctness of this proposition is irrefutable. And what's more is that just about everyone so expects the candidates on offer to speak at variance with their eventual behaviour, that it almost achieves the innocivity of the English "good open lie system" we've all been reading about lately. You have, haven't you?
The main difficulty I can see in this hopeful view of the matter is that people do seem to believe enough of what their candidates suggest. One can tell this by the offense taken and outcry returned when eventual behaviour is other than (even honestly so) that predicted on campaign.1 There are, of course many reasons why a person might predict incorrectly what he means to do for the coming seven years (or whatever). Maybe he forgets. Maybe forceful unforeseen contingencies arise. Maybe he independently changes his mind. Bl. John Henry did, and then had to answer books with books on how he wasn't actually a deceitful scoundrel --- and he wasn't even holding or seeking public office! And, yes, sometimes it's he-really-lied-to-us.
One of the nifty things about a healthy dictatorial monarchy is that, occasionally, it is possible that subordinate officials are appointed on their actual merits; consider St. Thomas More's term as Chancelor of England. I sometimes dream that a similar thing may be possible, and even in a more stable way, in a democratic state, but it would demand something of all citizens. In particular, if a people really wishes for true self government, enough of them must first learn to govern themselves. It's one thing to be your own master; it's quite another to achieve self-mastery. This, more than the love for other-life-than-mine, is what seems to me most lacking, in the republics, in the Commonwealth, anywhere in the West as much as in the East.
For all the laws against frivolous divorce will do no good if folk still make frivolous marriages --- or neglect the forms of solemn marriage altogether. All the laws specifically against the taking of life-in-utero will do no good if folk still hate the ordered progression from love to regeneration. It matters not that those places where the law has already degraded are democracies; their peoples have rebelled against God, why should they not rebel against man?
But let us not close gloomily. The duty of a Christian is what it ever was, and he ought to judge as he ever has judged; nor are good and evil one thing among Christians and another among Saracens and Zulus (though what they know of it may vary). Let we who can and will live upright and noble lives; let those who love and marry live in exemplary fashion --- like B.A. and Seraphic; like John and Sheila and Marko and Michael; I had news recently of a poet friend's engagement, and have nothing but good hopes for that alliance, too. Let those who are single live joyous and upright lives (and sometimes help Save The Storks and such-like) --- I do my meager best, and plenty of others do very well, too.
Let the World see how happy we can be, and perhaps they might believe it, too!
sperens
1) Now, if I were a scholar of politics I'd pull out some references to news articles on such-like things, but I'm not such an historian, so you'll just have to rely on your own anecdotal recollections.
Of course, this is part, or signal, of the problem:
Peoples, please! He's a politician! A successful politician! You think he tells us the truth? You think he thinks he can get elected by telling us the truth?
For she is absolutely right. The strict correctness, from a strategic point of view, that democratic elections anywhere these days demand the winners ... reserve their mind? ... engage in verbal misdirection? ... well, exaggerate and waffle are two current words for it, anyways --- the correctness of this proposition is irrefutable. And what's more is that just about everyone so expects the candidates on offer to speak at variance with their eventual behaviour, that it almost achieves the innocivity of the English "good open lie system" we've all been reading about lately. You have, haven't you?
The main difficulty I can see in this hopeful view of the matter is that people do seem to believe enough of what their candidates suggest. One can tell this by the offense taken and outcry returned when eventual behaviour is other than (even honestly so) that predicted on campaign.1 There are, of course many reasons why a person might predict incorrectly what he means to do for the coming seven years (or whatever). Maybe he forgets. Maybe forceful unforeseen contingencies arise. Maybe he independently changes his mind. Bl. John Henry did, and then had to answer books with books on how he wasn't actually a deceitful scoundrel --- and he wasn't even holding or seeking public office! And, yes, sometimes it's he-really-lied-to-us.
One of the nifty things about a healthy dictatorial monarchy is that, occasionally, it is possible that subordinate officials are appointed on their actual merits; consider St. Thomas More's term as Chancelor of England. I sometimes dream that a similar thing may be possible, and even in a more stable way, in a democratic state, but it would demand something of all citizens. In particular, if a people really wishes for true self government, enough of them must first learn to govern themselves. It's one thing to be your own master; it's quite another to achieve self-mastery. This, more than the love for other-life-than-mine, is what seems to me most lacking, in the republics, in the Commonwealth, anywhere in the West as much as in the East.
For all the laws against frivolous divorce will do no good if folk still make frivolous marriages --- or neglect the forms of solemn marriage altogether. All the laws specifically against the taking of life-in-utero will do no good if folk still hate the ordered progression from love to regeneration. It matters not that those places where the law has already degraded are democracies; their peoples have rebelled against God, why should they not rebel against man?
But let us not close gloomily. The duty of a Christian is what it ever was, and he ought to judge as he ever has judged; nor are good and evil one thing among Christians and another among Saracens and Zulus (though what they know of it may vary). Let we who can and will live upright and noble lives; let those who love and marry live in exemplary fashion --- like B.A. and Seraphic; like John and Sheila and Marko and Michael; I had news recently of a poet friend's engagement, and have nothing but good hopes for that alliance, too. Let those who are single live joyous and upright lives (and sometimes help Save The Storks and such-like) --- I do my meager best, and plenty of others do very well, too.
Let the World see how happy we can be, and perhaps they might believe it, too!
sperens
1) Now, if I were a scholar of politics I'd pull out some references to news articles on such-like things, but I'm not such an historian, so you'll just have to rely on your own anecdotal recollections.
Tuesday, September 4, 2012
Something Capricious
When I was a wee student, my violin master had the senior orchestra of his students try out this suite --- we had only one cellist, Ludovic the Magnificient, who was not a student of this teacher, but you've got to have a cellist, you know? Suzuki teachers are a sociable lot amongst themselves, and it helps. Anyways, I found this by looking for something that isn't Holst's St. Paul's Suite nor Britten's Simple Symphony that I couldn't remember the name nor composer of --- spectral sequences and extraordinary homology theories have turned some of my memory to comparative mush.
Anyways, I know some of you like this sort of thing.
Ahah! It's by Grieg, the thing I first thought of! But this is very good, too. Happy hunting, if you wish!
Anyways, I know some of you like this sort of thing.
Ahah! It's by Grieg, the thing I first thought of! But this is very good, too. Happy hunting, if you wish!
Sunday, September 2, 2012
A short while later
$$ \begin{array}{rl}
\mathrm{XXXI} & \mbox{The quiet of the middle of nowhere}\\
\mathrm{XXXII} & \mbox{Excellent words, like "corruscating" and "edify"}\\
\mathrm{XXXIII} & \mbox{The way a clear sky turns lemony-yellow in a sunset}\\
\mathrm{XXXIV} & \mbox{Clean water}\\
\mathrm{XXXV} & \mbox{Puzzles!}
\end{array}
$$
\mathrm{XXXI} & \mbox{The quiet of the middle of nowhere}\\
\mathrm{XXXII} & \mbox{Excellent words, like "corruscating" and "edify"}\\
\mathrm{XXXIII} & \mbox{The way a clear sky turns lemony-yellow in a sunset}\\
\mathrm{XXXIV} & \mbox{Clean water}\\
\mathrm{XXXV} & \mbox{Puzzles!}
\end{array}
$$
Thursday, August 30, 2012
An Unfounded Conjecture
Dear Philosophia,
Recall the curious event of Genesis 11:1-9.
Because it is fun to conjecture, I have concocted me a conjecture upon the manner in which God did thwart the city planners of Sennaar:
Recall the curious event of Genesis 11:1-9.
1 And the earth was of one tongue, and of the same speech.
2 And when they removed from the east, they found a plain in the land of Sennaar, and dwelt in it.
3 And each one said to his neighbour: Come, let us make brick, and bake them of stones, and slime instead of mortar.
4 And they said: Come, let us make a city and a tower, the top whereof may reach to heaven: and let us make our name famous before we be scattered abroad into all lands.
5 And the Lord came down to see the city and the tower, which the children of Adam were building.
6 And he said: Behold, it is one people, and all have one tongue: and they have begun to do this, neither will they leave off from their designs, till they accomplish them in deed.
7 Come ye, therefore, let us go down, and there may not understand one another's speech.
8 And so the Lord scattered them from that place into all lands, and they ceased to build the city.
9 And therefore the name thereof was called Babel, because there the language of the whole earth was confounded: and from thence the Lord scattered them abroad upon the face of all countries.
Because it is fun to conjecture, I have concocted me a conjecture upon the manner in which God did thwart the city planners of Sennaar:
Wednesday, August 29, 2012
To Seek Out Prudent Truths
Dear Amavero,
Without pretending to really know what is best, I would at least offer hope that some good thing may be found, "by dint of trying", as St. Jean Vianney put it (in a translation read me by Utrecht).
Without pretending to really know what is best, I would at least offer hope that some good thing may be found, "by dint of trying", as St. Jean Vianney put it (in a translation read me by Utrecht).
Sunday, August 26, 2012
Thursday, August 23, 2012
Say, when was the last time... ?
Oh, hello, there!
I'm not actually here (that is, even less than usual), having written this about two weeks ago; but, well, anyways, today the atmospheric tropical cycle count clicks over the third primorial, a curious number: $2 \times 3 \times 5$. If you were counting on digits you'd need two ordinary humans including one pair of feet. Some of my family seem to be making much of this number; I'm still slightly less than half my mother's age, if only for a couple more weeks. Hobbit-wise, I'm still just in my tweens. Well.
Have a pleasant day, if by any means you can; I know I will!
PS. You know, people make much of how $2$ is the "only even prime"; well, that's not saying much --- it's the first positive even number; it is the standard by which all evens are measured. One might as well say that $3$ is the "only trial prime" and $5$ the "only quicuncial"... but enough!
I'm not actually here (that is, even less than usual), having written this about two weeks ago; but, well, anyways, today the atmospheric tropical cycle count clicks over the third primorial, a curious number: $2 \times 3 \times 5$. If you were counting on digits you'd need two ordinary humans including one pair of feet. Some of my family seem to be making much of this number; I'm still slightly less than half my mother's age, if only for a couple more weeks. Hobbit-wise, I'm still just in my tweens. Well.
Have a pleasant day, if by any means you can; I know I will!
PS. You know, people make much of how $2$ is the "only even prime"; well, that's not saying much --- it's the first positive even number; it is the standard by which all evens are measured. One might as well say that $3$ is the "only trial prime" and $5$ the "only quicuncial"... but enough!
Monday, August 13, 2012
About talking at cross-purposes
Dear Passerby,
One of the weird things about being aHumptydumpty Mathematician is that you very quickly get used to the idea (so quickly that you usually don't think about it) that the meaning of a word --- the signification of a symbol --- is essentially inseparable from its use.
Once we have this hammer in our grasp, we may go hunting for nails; let me pry some up and we'll see if it's in any way handy.
One of the weird things about being a
Once we have this hammer in our grasp, we may go hunting for nails; let me pry some up and we'll see if it's in any way handy.
Sunday, August 12, 2012
Out of Mrs. C's Enchiridion
Let us recall
And so I am reminded that it's much too long since I did one of these.
$$ \begin{array}{rl}
\mathrm{XXVI} &\mbox{All this crazy internet}\\
\mathrm{XXVII} &\mbox{Friends with excellent Common Sense}\\
\mathrm{XXVIII} & \mbox{Small towns just a train-ride away}\\
\mathrm{XXIX} & \mbox{The stars over cornfields}\\
\mathrm{XXX} &\mbox{Birthdays in the family}
\end{array}
$$
Evening--Uncle Gilbert
Here dies another day
During which I have had eyes, ears, hands
And the great world round me;
And with tomorrow begins another.
Why am I allowed two?
And so I am reminded that it's much too long since I did one of these.
$$ \begin{array}{rl}
\mathrm{XXVI} &\mbox{All this crazy internet}\\
\mathrm{XXVII} &\mbox{Friends with excellent Common Sense}\\
\mathrm{XXVIII} & \mbox{Small towns just a train-ride away}\\
\mathrm{XXIX} & \mbox{The stars over cornfields}\\
\mathrm{XXX} &\mbox{Birthdays in the family}
\end{array}
$$
Saturday, August 11, 2012
A House has an inside and an outside and a neighborhood
For the purpose of being well-understood, I was raised (though no-one ever mentioned it) not under the declaration that all men inherit the right to "life, liberty and pursuit of happiness", but rather in a realm which proclaimed "peace, order, and good government". Whether either of us now enjoys any of these goods is a question best left to future historians, perhaps, but never mind.
It would seem there is room for disagreement on the prudence of civil law recognizing the unique and preeminently worthy character of the free and total union of Man and Woman made concrete in the free and total union of one man and one woman coenduring with their common Earthly survival.
I recently highlighted differing opinions on this matter from two good and thoughtful Catholic fathers with whom I respectfully disagree.1 The one asserts that "the state [has to] define marriage and know who is married in order to answer two questions: who owns what and whose kids are whose;" the other affirms his belief "that [one] should not be prohibited from pursuing a legal union with whomever [he?] like, according to the beliefs of another religion or in the eyes of government."
It would seem there is room for disagreement on the prudence of civil law recognizing the unique and preeminently worthy character of the free and total union of Man and Woman made concrete in the free and total union of one man and one woman coenduring with their common Earthly survival.
I recently highlighted differing opinions on this matter from two good and thoughtful Catholic fathers with whom I respectfully disagree.1 The one asserts that "the state [has to] define marriage and know who is married in order to answer two questions: who owns what and whose kids are whose;" the other affirms his belief "that [one] should not be prohibited from pursuing a legal union with whomever [he?] like, according to the beliefs of another religion or in the eyes of government."
Thursday, August 9, 2012
In prelude to some later-to-develope thoughts
Dear Moose,
Compare and contrast, two Catholic men who by their writing and thoughtfulness have much impressed me.
In the North corner, Darwin
In the South, John C.
Er... the cardinal directions cited here bear no relation to anything in the world of real; they're just meant to be different.
b.g. gruff II
Compare and contrast, two Catholic men who by their writing and thoughtfulness have much impressed me.
In the North corner, Darwin
In the South, John C.
Er... the cardinal directions cited here bear no relation to anything in the world of real; they're just meant to be different.
b.g. gruff II
Wednesday, August 1, 2012
Apocrypha Topologica II
Dear Mathematicelli,
If I may proceed,
The Cretan king Minos seems to have hit upon the psychological trick of bewildering his prisoners into imagining a topological obstruction where in fact the obstacle was only metric: that of the Labyrinth (with intimidating monster to keep you distracted). The solution that let Theseus escape the Labyrinth has found renewed popularity of late, and so is worth elaborating. Ariadne's reasoning might conceivably have run thus:
A point worth noting, which may have escaped Ariadne and Theseus in their flight, is that the re-winding and the follow-backwards phases of the solution can be performed in either order if only the string itself is slippery enough --- this could have been remarkably handy if the labyrinth had been under water, and had they wanted to get fish out of it without the fish seeing them and becoming suspicious. It also points out something special in the total path followed by Theseus through the Labyrinth: it is contractible!
Dungeons we may pass over, as well as the castle and siege warfare. Chain mail is about as old. But Somewhen between the Visigoths and Polyphony was discovered knitting. These, like most mechanical inventions, rely on metric phenomena to make their topology useful, but it is unquestionably their underlying topology that is used. (As a side-note, there's a lovely film-reference back to Ariadne in The Name of the Rose, where Adso returns his knit sweater to a trivial topology in order to escape a non-planar labyrinth! I'm curious how the Vandal painters would fare here!) If you'll forgive a jump-forward, the topologist Poincaré seems to have observed a woman knitting, and independently invented purling on the spot. I don't know what the full topological significance of that is, but the ubiquity of What Is certainly makes itself plain to those who can see it; for which give thanks to God, I think. About such other oddities as the Borromean rings we have remarked elsewhere.
I have no idea what's coming next; but this isn't a bad lot. We've more-or-less covered ±1800. Prof. Cauchemar
If I may proceed,
The Cretan king Minos seems to have hit upon the psychological trick of bewildering his prisoners into imagining a topological obstruction where in fact the obstacle was only metric: that of the Labyrinth (with intimidating monster to keep you distracted). The solution that let Theseus escape the Labyrinth has found renewed popularity of late, and so is worth elaborating. Ariadne's reasoning might conceivably have run thus:
• If one enters the labyrinth and comes to its center, then one has got (and can get) from there to here
• Walking the same way backwards, one can get there from here
• What is wanted is some means to remember the path one took in getting here, and then to reverse it
• Since I don't know how long a path one might need to follow, the keeper of the memory had better be long!
In other words, we keep more information than just the fact that two points are connected by a path: we remember the whole path between them. Some recenter mathematicians more inclined to vandalism have suggested painting the walls of the labyrinth to remind yourself of where you have been --- which is sufficient data to escape, if you paratroop into the maze under cover of darkness; but the solution proposed by Ariadne and adopted by Theseus makes it easier to tidy-up afterwards: unroll a string as you walk along, and then follow it in reverse, winding the string again!A point worth noting, which may have escaped Ariadne and Theseus in their flight, is that the re-winding and the follow-backwards phases of the solution can be performed in either order if only the string itself is slippery enough --- this could have been remarkably handy if the labyrinth had been under water, and had they wanted to get fish out of it without the fish seeing them and becoming suspicious. It also points out something special in the total path followed by Theseus through the Labyrinth: it is contractible!
Dungeons we may pass over, as well as the castle and siege warfare. Chain mail is about as old. But Somewhen between the Visigoths and Polyphony was discovered knitting. These, like most mechanical inventions, rely on metric phenomena to make their topology useful, but it is unquestionably their underlying topology that is used. (As a side-note, there's a lovely film-reference back to Ariadne in The Name of the Rose, where Adso returns his knit sweater to a trivial topology in order to escape a non-planar labyrinth! I'm curious how the Vandal painters would fare here!) If you'll forgive a jump-forward, the topologist Poincaré seems to have observed a woman knitting, and independently invented purling on the spot. I don't know what the full topological significance of that is, but the ubiquity of What Is certainly makes itself plain to those who can see it; for which give thanks to God, I think. About such other oddities as the Borromean rings we have remarked elsewhere.
I have no idea what's coming next; but this isn't a bad lot. We've more-or-less covered ±1800. Prof. Cauchemar
Sunday, July 29, 2012
From a whisk-user
Dear Gelatinifers,
I wish to register a complaint; specifically, without endorsing the heresy sometimes wrongly known as "Luddite", it is nonetheless my firmly-held belief that recipes artificially biased in favour of electromechanical mixing apparatus should only be packaged with electromechanical mixing apparatus (and otherwise appear in books describing what they are), not with foodstuffs. I'm sure I shall, after one or two more trials, adapt the method so as not to melt the cream I've just whipped, but I don't see why I should have to use up the whole box of whatever it is you actually sold me, before it is properly useful, after you pretend to suggest a recipe.
As it is, I'm rather tempted to just get me a bottle of orange-infused curaçao (or maybe kirsch!) and go back to meringue mousse.
a disgruntled cook
I wish to register a complaint; specifically, without endorsing the heresy sometimes wrongly known as "Luddite", it is nonetheless my firmly-held belief that recipes artificially biased in favour of electromechanical mixing apparatus should only be packaged with electromechanical mixing apparatus (and otherwise appear in books describing what they are), not with foodstuffs. I'm sure I shall, after one or two more trials, adapt the method so as not to melt the cream I've just whipped, but I don't see why I should have to use up the whole box of whatever it is you actually sold me, before it is properly useful, after you pretend to suggest a recipe.
As it is, I'm rather tempted to just get me a bottle of orange-infused curaçao (or maybe kirsch!) and go back to meringue mousse.
a disgruntled cook
Tuesday, July 17, 2012
Jigsaw
Dear Crowsfort,
I have a jigsaw puzzle. The pieces look like squares \[ \begin{array}{ccc} A & \overset{f}{\to} & B \\ g\downarrow & \Downarrow & \downarrow h\\ C & \underset{k}{\to} & D \end{array} \] ... actually, $f,g,h,k$ all know what their corners are, so we could leave out the $A,B,C,D$, but this gets distracting. Also, the $\Downarrow$ deserves to have a name, only I can't think of a good way to make it all fit. Which particular $\Downarrow$ a square has in it makes a difference, later!
Two pieces sharing an edge fit together, so that \[ \begin{array}{ccc} A & \overset{f}{\to} & B \\ g\downarrow & \Downarrow & \downarrow h\\ C & \underset{k}{\to} & D \\ C & \overset{k}{\to} & D \\ g'\downarrow & \Downarrow & \downarrow h'\\ E & \underset{l}{\to} & F \end{array} \] make a rectangle \[ \begin{array}{ccc} A & \overset{f}{\to} & B \\ g\downarrow & \Downarrow & \downarrow h\\ C & \overset{k}{\to} & D \\ g'\downarrow & \Downarrow & \downarrow h'\\ E & \underset{l}{\to} & F \end{array} \] or sometimes \[ \begin{array}{ccc} A & \overset{f}{\to} & B \\ g\downarrow & & \downarrow h\\ C & \Downarrow & D \\ g'\downarrow & & \downarrow h'\\ E & \underset{l}{\to} & F \end{array} \] and can even be squished down to a square \[ \begin{array}{ccc} A & \overset{f}{\to} & B \\ g'g\downarrow & \Downarrow & \downarrow h'h\\ E & \underset{l}{\to} & F \end{array} \] which is handy, though we don't often want to do that.
The good people who cut out my jigsaw puzzle were very nice, and provided an unlimited supply of various standard shapes, guaranteed to fit certain sorts of corners, so that if anywhere in the puzzle you find \[ \begin{array}{ccc} & & A \\ & & \downarrow f\\ B & \underset{g}{\to} & C \end{array} \] you can add in a square \[ \begin{array}{ccc} P_{f,g} & \to & A \\ \downarrow & \lrcorner & \downarrow f\\ B & \underset{g}{\to} & C \end{array} \] and in the same way, if you have \[ \begin{array}{ccc} A & \overset{f}{\to} & B \\ g \downarrow & & \\ C & & \end{array} \] you can fill it in \[ \begin{array}{ccc} A & \overset{f}{\to} & B \\ g \downarrow & \ulcorner & \downarrow \\ C & \to & Q_{f,g} \end{array} \] There is one other sort of handy square, looking like \[\begin{array}{ccc} A & \overset{f}{\to} & B \\ f\downarrow & = & \downarrow g \\ B & \underset{g}{\to} & C \end{array} \] which lets you go around corners, when it looks like a good idea. These have two further special types, \[\begin{array}{ccc} A & \overset{f}{\to} & B \\ f\downarrow & = & \downarrow = \\ B & \underset{=}{\to} & B \end{array} \] ... and there is another of the similar sort that I'm sure you can guess; and there's also vertical and horizontal versions of \[\begin{array}{ccc} A & = & A \\ f\downarrow & = & \downarrow f \\ B & \underset{=}{\to} & B \end{array} \] which also happens to be a $\lrcorner$ and a $\ulcorner$.
Actually, those last two squares are special cases of these two : \[\begin{array}{ccc} A & \overset{f}{\to} & B \\ =\downarrow & = & \downarrow g\\ A & \underset{g f}{\to} & C \end{array}\] and \[\begin{array}{ccc} A & \overset{f}{\to} & B \\ g f \downarrow & = & \downarrow g\\ C & \underset{=}{\to} & C \end{array}\] or reflections of them; but most of these two are neithert $\lrcorner$ nor $\ulcorner$.
They were also kind enough to suggest a few ways to get started, using a special corner called "$*$", or "the point", though it's not really the point of all this. Still, there's always exactly one edge $A\to *$, no matter what $A$ is, and you can also draw it vertically: \[ \begin{array}{c} A\\ \downarrow \\ * \end{array} \] The corner $*$ also has another nifty feature, that the collection of edges $ * \to A$ might as well be called $A$. There's only one corner, $\{\}$ to which you can't draw an arrow from $*$; but on the other hand, there's always exactly one arrow from $\{\}$ to any other corner $A$, including to $*$! So, for instance, there's a nice corner \[ \begin{array}{ccc} \{\} & \to & * \\ \downarrow & & \\ * \end{array} \] and because of the $\ulcorner$ pieces, this gets filled-in as \[ \begin{array}{ccc} \{\} & \to & * \\ \downarrow & \Box & \downarrow\\ * & \to & * + * \end{array} \] although it's more common, among my fellow puzzlers, to call that new thing $\mathbb{S}^0$. It has two points, as you can see. Oh! this one tile happens to be of *both* sorts: it's the standard tile to fill-in those two edges $*\to \mathbb{S}^0$ as well as the standard tile to fill-in the edge $\{\}\to*$ drawn twice from a single copy of $\{\}$. Sometimes it's fun just to look at the special pieces \[ \begin{array}{ccc} A & \to & * \\ \downarrow & \ulcorner & \downarrow \\ * & \to & \Sigma A \end{array} \] which highlight a fascinating sequence of corner labels $A, \Sigma A, \Sigma^2 A, \ldots$ --- the ones you get starting with $\mathbb{S}^0$ are called the spheres (or homotopy spheres) and have the special names $\mathbb{S}^n = \Sigma^n \mathbb{S}^0 $. Going in the other direction --- if you have a favourite arrow $ * \overset{a}{\to} A $, the special square you get is labelled \[ \begin{array}{ccc} \Omega_a A & \to & * \\ \downarrow & \lrcorner & \downarrow a\\ * & \underset{a}{\to} & A \end{array} \] ... to tell you how one is supposed to keep going after that, I have to tell you one last thing about the special squares labelled $\lrcorner$ and $\ulcorner$; given any square at all \[ \begin{array}{ccc} A & \overset{f}{\to} & B \\ g\downarrow & \Downarrow & \downarrow h\\ C & \underset{k}{\to} & D \end{array} \] there are of course the standard two squares \[ \begin{array}{ccc} P_{h,k} & \to & B \\ \downarrow & \lrcorner & \downarrow h\\ C & \underset{k}{\to} & D \end{array} \] and the other one to $Q_{f,g}$; in essence, what it means to be a $\lrcorner$ is that, there's essentially just one edge $ A \overset{w}\to P_{h,k} $ that fits into this puzzle \[ \begin{array}{ccccc} A & \overset{=}{\to} & A & \overset{f}{\to} & B \\ =\downarrow & = & w \downarrow & \Downarrow & \downarrow = \\ A & \underset{w}{\to} & P_{h,k} & \to & B \\ g\downarrow & \Downarrow & \downarrow & \lrcorner & \downarrow h \\ C & \underset{=}{\to} & C & \underset{k}{\to} & D \end{array} \] There's a similar story about unique edges $ Q_{f,g} \to D $ that fit in another puzzle --- try it and see! But particularly, since we always have this square \[\begin{array}{ccc} * & \to & * \\ \downarrow & = & \downarrow a\\ * & \underset{a}{\to} & A \end{array}\] there's exactly one $ * \to \Omega_a A$ that fits in all the necessary puzzles, and this is what lets us keep going to make new spaces $ \Omega^2_a A, \cdots $. Here's a puzzle for you: come up with a good edge $ A \to \Omega_{?} \Sigma A$! This entails finding a way to fill-in that $?$; you should be able to think of perhaps-two.
There are lots of things I haven't mentioned, but of course, that will always be true, even if I say all the things that should come first! You're welcome to play with the jigsaw, too; we'll never run out of pieces!
the joiner
I have a jigsaw puzzle. The pieces look like squares \[ \begin{array}{ccc} A & \overset{f}{\to} & B \\ g\downarrow & \Downarrow & \downarrow h\\ C & \underset{k}{\to} & D \end{array} \] ... actually, $f,g,h,k$ all know what their corners are, so we could leave out the $A,B,C,D$, but this gets distracting. Also, the $\Downarrow$ deserves to have a name, only I can't think of a good way to make it all fit. Which particular $\Downarrow$ a square has in it makes a difference, later!
Two pieces sharing an edge fit together, so that \[ \begin{array}{ccc} A & \overset{f}{\to} & B \\ g\downarrow & \Downarrow & \downarrow h\\ C & \underset{k}{\to} & D \\ C & \overset{k}{\to} & D \\ g'\downarrow & \Downarrow & \downarrow h'\\ E & \underset{l}{\to} & F \end{array} \] make a rectangle \[ \begin{array}{ccc} A & \overset{f}{\to} & B \\ g\downarrow & \Downarrow & \downarrow h\\ C & \overset{k}{\to} & D \\ g'\downarrow & \Downarrow & \downarrow h'\\ E & \underset{l}{\to} & F \end{array} \] or sometimes \[ \begin{array}{ccc} A & \overset{f}{\to} & B \\ g\downarrow & & \downarrow h\\ C & \Downarrow & D \\ g'\downarrow & & \downarrow h'\\ E & \underset{l}{\to} & F \end{array} \] and can even be squished down to a square \[ \begin{array}{ccc} A & \overset{f}{\to} & B \\ g'g\downarrow & \Downarrow & \downarrow h'h\\ E & \underset{l}{\to} & F \end{array} \] which is handy, though we don't often want to do that.
The good people who cut out my jigsaw puzzle were very nice, and provided an unlimited supply of various standard shapes, guaranteed to fit certain sorts of corners, so that if anywhere in the puzzle you find \[ \begin{array}{ccc} & & A \\ & & \downarrow f\\ B & \underset{g}{\to} & C \end{array} \] you can add in a square \[ \begin{array}{ccc} P_{f,g} & \to & A \\ \downarrow & \lrcorner & \downarrow f\\ B & \underset{g}{\to} & C \end{array} \] and in the same way, if you have \[ \begin{array}{ccc} A & \overset{f}{\to} & B \\ g \downarrow & & \\ C & & \end{array} \] you can fill it in \[ \begin{array}{ccc} A & \overset{f}{\to} & B \\ g \downarrow & \ulcorner & \downarrow \\ C & \to & Q_{f,g} \end{array} \] There is one other sort of handy square, looking like \[\begin{array}{ccc} A & \overset{f}{\to} & B \\ f\downarrow & = & \downarrow g \\ B & \underset{g}{\to} & C \end{array} \] which lets you go around corners, when it looks like a good idea. These have two further special types, \[\begin{array}{ccc} A & \overset{f}{\to} & B \\ f\downarrow & = & \downarrow = \\ B & \underset{=}{\to} & B \end{array} \] ... and there is another of the similar sort that I'm sure you can guess; and there's also vertical and horizontal versions of \[\begin{array}{ccc} A & = & A \\ f\downarrow & = & \downarrow f \\ B & \underset{=}{\to} & B \end{array} \] which also happens to be a $\lrcorner$ and a $\ulcorner$.
Actually, those last two squares are special cases of these two : \[\begin{array}{ccc} A & \overset{f}{\to} & B \\ =\downarrow & = & \downarrow g\\ A & \underset{g f}{\to} & C \end{array}\] and \[\begin{array}{ccc} A & \overset{f}{\to} & B \\ g f \downarrow & = & \downarrow g\\ C & \underset{=}{\to} & C \end{array}\] or reflections of them; but most of these two are neithert $\lrcorner$ nor $\ulcorner$.
They were also kind enough to suggest a few ways to get started, using a special corner called "$*$", or "the point", though it's not really the point of all this. Still, there's always exactly one edge $A\to *$, no matter what $A$ is, and you can also draw it vertically: \[ \begin{array}{c} A\\ \downarrow \\ * \end{array} \] The corner $*$ also has another nifty feature, that the collection of edges $ * \to A$ might as well be called $A$. There's only one corner, $\{\}$ to which you can't draw an arrow from $*$; but on the other hand, there's always exactly one arrow from $\{\}$ to any other corner $A$, including to $*$! So, for instance, there's a nice corner \[ \begin{array}{ccc} \{\} & \to & * \\ \downarrow & & \\ * \end{array} \] and because of the $\ulcorner$ pieces, this gets filled-in as \[ \begin{array}{ccc} \{\} & \to & * \\ \downarrow & \Box & \downarrow\\ * & \to & * + * \end{array} \] although it's more common, among my fellow puzzlers, to call that new thing $\mathbb{S}^0$. It has two points, as you can see. Oh! this one tile happens to be of *both* sorts: it's the standard tile to fill-in those two edges $*\to \mathbb{S}^0$ as well as the standard tile to fill-in the edge $\{\}\to*$ drawn twice from a single copy of $\{\}$. Sometimes it's fun just to look at the special pieces \[ \begin{array}{ccc} A & \to & * \\ \downarrow & \ulcorner & \downarrow \\ * & \to & \Sigma A \end{array} \] which highlight a fascinating sequence of corner labels $A, \Sigma A, \Sigma^2 A, \ldots$ --- the ones you get starting with $\mathbb{S}^0$ are called the spheres (or homotopy spheres) and have the special names $\mathbb{S}^n = \Sigma^n \mathbb{S}^0 $. Going in the other direction --- if you have a favourite arrow $ * \overset{a}{\to} A $, the special square you get is labelled \[ \begin{array}{ccc} \Omega_a A & \to & * \\ \downarrow & \lrcorner & \downarrow a\\ * & \underset{a}{\to} & A \end{array} \] ... to tell you how one is supposed to keep going after that, I have to tell you one last thing about the special squares labelled $\lrcorner$ and $\ulcorner$; given any square at all \[ \begin{array}{ccc} A & \overset{f}{\to} & B \\ g\downarrow & \Downarrow & \downarrow h\\ C & \underset{k}{\to} & D \end{array} \] there are of course the standard two squares \[ \begin{array}{ccc} P_{h,k} & \to & B \\ \downarrow & \lrcorner & \downarrow h\\ C & \underset{k}{\to} & D \end{array} \] and the other one to $Q_{f,g}$; in essence, what it means to be a $\lrcorner$ is that, there's essentially just one edge $ A \overset{w}\to P_{h,k} $ that fits into this puzzle \[ \begin{array}{ccccc} A & \overset{=}{\to} & A & \overset{f}{\to} & B \\ =\downarrow & = & w \downarrow & \Downarrow & \downarrow = \\ A & \underset{w}{\to} & P_{h,k} & \to & B \\ g\downarrow & \Downarrow & \downarrow & \lrcorner & \downarrow h \\ C & \underset{=}{\to} & C & \underset{k}{\to} & D \end{array} \] There's a similar story about unique edges $ Q_{f,g} \to D $ that fit in another puzzle --- try it and see! But particularly, since we always have this square \[\begin{array}{ccc} * & \to & * \\ \downarrow & = & \downarrow a\\ * & \underset{a}{\to} & A \end{array}\] there's exactly one $ * \to \Omega_a A$ that fits in all the necessary puzzles, and this is what lets us keep going to make new spaces $ \Omega^2_a A, \cdots $. Here's a puzzle for you: come up with a good edge $ A \to \Omega_{?} \Sigma A$! This entails finding a way to fill-in that $?$; you should be able to think of perhaps-two.
There are lots of things I haven't mentioned, but of course, that will always be true, even if I say all the things that should come first! You're welcome to play with the jigsaw, too; we'll never run out of pieces!
the joiner
Tuesday, July 3, 2012
To the credit of Mr. Bumble
or, a foreigner's comments.
Of course, the question of whether a given law is logically consistent with a narrow scope of (written, "foundational" or, paradoxically "superior") law is an important one, in that the police should be able to tell whether they should be arresting citizens or legislators; not that the latter is envisioned or practised much, according to a particular and narrow scope of law.
It is a sad thing, however, that this seems to be the only competence of various superior courts; elsewhere several people are remarking that more important than whether a document like Veritatis Splendor is infalible is whether it's true. Similarly, more important than whether a given law is legal ("constitutional") is whether it is just; or, under the maxim that an unjust law is no-law-at-all, whether a given piece of legislation is indeed a true law.
Alas...
no beadle me
Of course, the question of whether a given law is logically consistent with a narrow scope of (written, "foundational" or, paradoxically "superior") law is an important one, in that the police should be able to tell whether they should be arresting citizens or legislators; not that the latter is envisioned or practised much, according to a particular and narrow scope of law.
It is a sad thing, however, that this seems to be the only competence of various superior courts; elsewhere several people are remarking that more important than whether a document like Veritatis Splendor is infalible is whether it's true. Similarly, more important than whether a given law is legal ("constitutional") is whether it is just; or, under the maxim that an unjust law is no-law-at-all, whether a given piece of legislation is indeed a true law.
Alas...
no beadle me
Thursday, May 3, 2012
Drip drip drip... (a lackadaisical rambling)
"Oh, Hi", as they say one says. I suppose I ought to write something.
As I begin to write in earnest, it is indeed starting to rain. The day has been brooding on this evening moment from early hours; but only now are lightnings and dropplets together falling. The heavens roar and pour forth to wash my dusty city: Vidi aquam egredientem...
Here comes a thick bit, right now! Oh, what a noise! A hundred, a thousand snare drums without a Drum Major to coordinate them. They march off gradually, leaving a pancake-sizzle sound behind.
Another period of dripping, of quiet, though it's all one rain. The same flood that purged the Earth also floated Noah along; the difference was a matter of disposition, of being inside or outside the Ark, the Barque... being inside, and it not being a flood, this quiet rainy bit is making me sleepy, so I'll turn off the drippy tap, now, and I'm sure you all won't mind. Maybe a good evening for a warm bath...
Good night.
God willing, some of us will see eachother in the morning.
As I begin to write in earnest, it is indeed starting to rain. The day has been brooding on this evening moment from early hours; but only now are lightnings and dropplets together falling. The heavens roar and pour forth to wash my dusty city: Vidi aquam egredientem...
* * *
The storm passes over with varying intensity, like lumpy oatmeal, or a battle waxing and waning as new troops weary and new forces drawn up. Sometimes the thunder recalls a lion purring --- not immediately violent, not quite safe. Sometimes the air in the house veritably tingles with a hunter's anticipation!Here comes a thick bit, right now! Oh, what a noise! A hundred, a thousand snare drums without a Drum Major to coordinate them. They march off gradually, leaving a pancake-sizzle sound behind.
* * *
After prayer, what is one supposed to do, if one doesn't know what to do? Obviously, beyond thinking, too --- it's the thinking that seems to get me into trouble, you see. I think and think and usually come up with no decision. Most unhobbitlike am I, in that way. Do you remember, perhaps, when Bilbo said it?
"Go back? No good. Go sideways? Impossible. Go on? Only thing to do!Somehow (perhaps I forget the eliminateds and return to them?), I neither reach a "yes, that sounds good" nor an "only thing to do!". Wandering about, getting dizzy... and forgetting why.
* * *
But that's OK. Gradually, we are walling-up the side-routes to traps and blinds; gradually, we learn to keep direction (that is, "of straightness") in the darkness we once took for seeing. True sight, in the true light, takes some getting-used-to.Another period of dripping, of quiet, though it's all one rain. The same flood that purged the Earth also floated Noah along; the difference was a matter of disposition, of being inside or outside the Ark, the Barque... being inside, and it not being a flood, this quiet rainy bit is making me sleepy, so I'll turn off the drippy tap, now, and I'm sure you all won't mind. Maybe a good evening for a warm bath...
Good night.
God willing, some of us will see eachother in the morning.
Monday, April 23, 2012
On Being Wrong
It's often been brought to my attention, recently, that a graduate student --- indeed, any honest research professional --- spends most of his time being confused. Sometimes this confusion is consciously felt; sometimes it isn't. When it isn't consciously felt, it can lead to people declaring things like "Of Course the Riemann Hypothesis Ought to be True" (this is something most number theorists expect, and can't prove yet) or "Of Course You Can Square the Circle" (this is something actually true, in the right context, but not the context most people claiming it intend) or "Of Course All Widgets are Thingumy" (may or may not be true, depending on W and T, but usually not).
This happens to (honest) research professionals all the time, as well as to the more-normal core of Humanity; but there is a key difference between (honest) r.p.s and a particular subgrouping, not really quite fitting in the latter; which is that an h.r.p. is likely not to mind having been wrong, and will happily acknowledge it and receive correction. The weirdos are convinced that they Are Not Wrong, and no ammount of argument will convince them --- because the wrong conclusion derives from honest unknown confusion. Exposed to truth, honest unknown confusion learns only deceptive felt confusion, and prefers what it doesn't find confusing (even unknown confusion). I trust I make myself clear?
Anyways. I get confused a lot, and sometimes it shows. It's actually kind-of fun! Have a good day, everyone.
This happens to (honest) research professionals all the time, as well as to the more-normal core of Humanity; but there is a key difference between (honest) r.p.s and a particular subgrouping, not really quite fitting in the latter; which is that an h.r.p. is likely not to mind having been wrong, and will happily acknowledge it and receive correction. The weirdos are convinced that they Are Not Wrong, and no ammount of argument will convince them --- because the wrong conclusion derives from honest unknown confusion. Exposed to truth, honest unknown confusion learns only deceptive felt confusion, and prefers what it doesn't find confusing (even unknown confusion). I trust I make myself clear?
Anyways. I get confused a lot, and sometimes it shows. It's actually kind-of fun! Have a good day, everyone.
Monday, April 9, 2012
Monday Musings
Dear Herecomeseverybody,
If I may be allowed to imagine, I should imagine that today would be the liturgical anniversary of "Twin" Thomas' assertion "nisi videro ... non credam". Certainly, it wasn't yesterday, and it will certainly come before the Octave, when we shall hear "beati qui non viderunt et crediderunt".
At other times I've wondered why might Thomas not have been there that Easter Sunday; today, being in a mood for punning, it occurs to me that, in Vulgate and Douay, the Low Sunday Beatitude is, as all beatitudes, written in the perfect tense: "blessed are they who have not seen, and have believed". But it raises the curious question of whether this particular beatitude applies in that moment to anyone at all. The only candidates that spring to mind are Mary the Mother of Our Lord, Peter, and the disciple whom Jesus loved --- but that disciple ("he that saw these things... his testimony is true, and he knows his testimony is true...") was careful to write, earlier
Nor is Our Lord's visible revelation of his resurrection made to make up for what is wanting in his disciples' faith, but to make up for what is wanting in ours. It's one thing for the Eleven survivors among the Twelve to learn of the empty tomb and believe, but another for them to say so to those same priests and scribes and lawyers who, mere pages earlier were crying "surely thou hast a devil". That is, not only has He given them faith in His rising, he has also given them the power to say and we have seen Him. I have not seen, but I believe because Peter saw.
Let us think on these things with humble joy, and take every help to live as people new-raised from the death that is sin.
Happy Easter,
a simple one among everybody
If I may be allowed to imagine, I should imagine that today would be the liturgical anniversary of "Twin" Thomas' assertion "nisi videro ... non credam". Certainly, it wasn't yesterday, and it will certainly come before the Octave, when we shall hear "beati qui non viderunt et crediderunt".
At other times I've wondered why might Thomas not have been there that Easter Sunday; today, being in a mood for punning, it occurs to me that, in Vulgate and Douay, the Low Sunday Beatitude is, as all beatitudes, written in the perfect tense: "blessed are they who have not seen, and have believed". But it raises the curious question of whether this particular beatitude applies in that moment to anyone at all. The only candidates that spring to mind are Mary the Mother of Our Lord, Peter, and the disciple whom Jesus loved --- but that disciple ("he that saw these things... his testimony is true, and he knows his testimony is true...") was careful to write, earlier
4 ... they both ran together, and that other disciple did outrun Peter, and came first to the sepulchre.Nonetheless, let us not think on these things into thinking of ourselves as being better than Peter or John were in those moments of great amaze. They had to live through the Gospel before telling it to anyone, whereas we grow up hearing about Easter as a thing accomplished, every year about this time --- we know the ending of this story long before we know what the story is even about, or where it really started.
5 And when he stooped down, he saw the linen cloths lying; but yet he went not in.
6 Then cometh Simon Peter, following him, and went into the sepulchre, and saw the linen cloths lying,
7 And the napkin that had been about his head, not lying with the linen cloths, but apart, wrapped up into one place.
8 Then that other disciple also went in, who came first to the sepulchre: and he saw, and believed.
Nor is Our Lord's visible revelation of his resurrection made to make up for what is wanting in his disciples' faith, but to make up for what is wanting in ours. It's one thing for the Eleven survivors among the Twelve to learn of the empty tomb and believe, but another for them to say so to those same priests and scribes and lawyers who, mere pages earlier were crying "surely thou hast a devil". That is, not only has He given them faith in His rising, he has also given them the power to say and we have seen Him. I have not seen, but I believe because Peter saw.
Let us think on these things with humble joy, and take every help to live as people new-raised from the death that is sin.
Happy Easter,
a simple one among everybody
Saturday, April 7, 2012
Tomorrow, we'll sing...
echo << eof >> /dev/null
eof
II 1 So the heavens and the earth were finished, and all the furniture of them. 2 And on the seventh day God ended his work which he had made: and he rested on the seventh day from all his work which he had done.
eof
Wednesday, March 28, 2012
Thursday, March 22, 2012
More Pythagoras than Pythagoras
So, a nifty theorem: that in a tetrahedron of which the three angles at one vertex are right angles, the square of the opposite face is the sum of the squares on the three adjacent faces. By homogeneity, it follows that the square of any plane area in 3-space is the sum of the squared areas of its three orthogonal shadows on any three orthogonal planes. Here, by homogeneity I mean that orthogonal shadows of parallel plane figures are proportional to eachother as those plane figures are.
A sketch of the hypothesis, as stated:
The "opposite" face is left implicit; the adjacent faces coloured blue, red, and grey.
There are, of course, several approaches to this theorem.
One might calculate relations among the sides and areas, and proceed algebraically to conclude that the two quantities described have equivalent expressions (e.g., in terms of the three adjecent edge lengths).
Alternatively, one might take as prior what we have stated as corollary, and argue (by suitable means) that the sum-of-squared-shadows is independent of which three orthogonal planes are chosen for catching the shadows; then, since the equation obviously holds when our plane figure lies in one of the three planes --- for then it is its own shadow on its own plane, and its other two shadows have no area, and the full proposition follows.
I rather like circles, lately, so I'll do something between the two: a special plane shape (indeed a circular disc) has three orthogonal shadows on three orthogonal planes, and their squares sum to the square on the circle; one returns to general plane figures by homogeneity again.
The niftyness of this arrangement begins with the observation that the orthogonal shadow of a circular disc is an ellipse; and moreover that the major axis of this ellipse equals the diameter of the circle. Indeed, the shadow doesn't depend on where the disc lies, only on its relative attitude. Supposing the disc's center were on the shadow plane, then the shadow and disc meet in a line, precisely this major axis. Since the area of an ellipse is universally proportional to the product of its principal axes, this reduces the problem to arguing that the squares on the minor axes of the three elliptical shadows sum to the square of the circle's diameter.
To get there, we must be sneaky: consider a line segment perpendicular to our circular disc, and equal to a diameter. With some thought, one can see (this means "prove", but it's not too hard) that the minor axis and the shadow of this perpendicular lie on the same line; but they are shadows of equal and perpendicular line segments, so their squares must sum to the square on the diameter --- once in each ellipse.
At the same time, the three orthogonal shadows of the perpendicular segment squared sum to twice the square on the diameter (this is a fun exercise) adding up six squared line shaddows in three ellipses should give three times the squared diameter (since each ellipse gives one), so the three squared minor axes sum to a single squared diameter, as promised.
A sketch of the hypothesis, as stated:
The "opposite" face is left implicit; the adjacent faces coloured blue, red, and grey.
There are, of course, several approaches to this theorem.
One might calculate relations among the sides and areas, and proceed algebraically to conclude that the two quantities described have equivalent expressions (e.g., in terms of the three adjecent edge lengths).
Alternatively, one might take as prior what we have stated as corollary, and argue (by suitable means) that the sum-of-squared-shadows is independent of which three orthogonal planes are chosen for catching the shadows; then, since the equation obviously holds when our plane figure lies in one of the three planes --- for then it is its own shadow on its own plane, and its other two shadows have no area, and the full proposition follows.
I rather like circles, lately, so I'll do something between the two: a special plane shape (indeed a circular disc) has three orthogonal shadows on three orthogonal planes, and their squares sum to the square on the circle; one returns to general plane figures by homogeneity again.
The niftyness of this arrangement begins with the observation that the orthogonal shadow of a circular disc is an ellipse; and moreover that the major axis of this ellipse equals the diameter of the circle. Indeed, the shadow doesn't depend on where the disc lies, only on its relative attitude. Supposing the disc's center were on the shadow plane, then the shadow and disc meet in a line, precisely this major axis. Since the area of an ellipse is universally proportional to the product of its principal axes, this reduces the problem to arguing that the squares on the minor axes of the three elliptical shadows sum to the square of the circle's diameter.
To get there, we must be sneaky: consider a line segment perpendicular to our circular disc, and equal to a diameter. With some thought, one can see (this means "prove", but it's not too hard) that the minor axis and the shadow of this perpendicular lie on the same line; but they are shadows of equal and perpendicular line segments, so their squares must sum to the square on the diameter --- once in each ellipse.
At the same time, the three orthogonal shadows of the perpendicular segment squared sum to twice the square on the diameter (this is a fun exercise) adding up six squared line shaddows in three ellipses should give three times the squared diameter (since each ellipse gives one), so the three squared minor axes sum to a single squared diameter, as promised.
Monday, March 19, 2012
Since we're re-writing actual real positable Law like nobody's business...
Dear Sir Isaac,
I just had a bit of brie fall, if you will believe it, to the flooring of my computer's tower case (er... I keep it open because, being an eleven-year-old obsolescence it overheats otherwise... Joseph Fourier and William Thomson, you're next on my list!)
And so I'm writing to protest, sir, this so-called "law" of so-called "universal" gravitation. Obviously we don't want cheese in our computers (any more than we can avoid it) so obviously, it's time we extended to dairy curd the same exceptions that Helium and small birds seem to have been enjoying for centuries!
Yours, most gravely,
a physical progressive
I just had a bit of brie fall, if you will believe it, to the flooring of my computer's tower case (er... I keep it open because, being an eleven-year-old obsolescence it overheats otherwise... Joseph Fourier and William Thomson, you're next on my list!)
And so I'm writing to protest, sir, this so-called "law" of so-called "universal" gravitation. Obviously we don't want cheese in our computers (any more than we can avoid it) so obviously, it's time we extended to dairy curd the same exceptions that Helium and small birds seem to have been enjoying for centuries!
Yours, most gravely,
a physical progressive
Sunday, March 18, 2012
A little more, about quadrilaterals
Before getting quite lost in why the isoperimetric problem should have exactly one solution (the answer, in brief, is relative convexity, but that's a mouthful, before we know how to spell it), a little bit about Steiner's proof, adapted slightly.
Now, last time it was pointed out that rotating a figure preserves both its area and its perimeter; and this in fact remains true of separate pieces of the figure; so, for instance, pertaining to the area within the black curve below:
one may rotate the mauve or blueish pieces, preserving their area as well as the lengths of both black and red curve or line segments. This is clearly a nifty thing to consider, as it means also that if there's a quadrilateral with the same sides as the red quadrilateral, but with greater area, then aranging the mauve and green bits around that gives a new shape with the same black perimeter, and greater area (the mauve and bluey bits will be the same, and the transparent bit in the middle can be made greater).
It just so happens that the best arrangement of vertices for the red quadrilateral is (surprise!) as a cyclic quadrilateral, aka "chord" quadrilateral, which is to say, such that the vertices are all on a circle. Perhaps the other surprise is that this is always doable. There is a bit of annoyance about proving the optimality of a cyclic arrangement, in that it works out to be more natural to maximize the square of the area --- in a calculation attributed to one Bretschneider, this is a sum of two terms, one of which is determined entirely by the four side lengths, the other proportional to their product and otherwise depending only the sum of opposite vertex angles... you can see how this is getting messy, yes? The calculus is quite straight-forward, but the geometry gets to be rather icky. I think there ought to be a 4-dimensional scissors congruence proof of this fact, considering how similar it is to Heron's formula. But we mustn't jump to conclusions! (Also, such things are difficult to read).
Anyways. Suppose, then, that the curve we had wasn't a circle. Then there must be some four points of our curve that weren't cyclic. (Three points determine a circle, a fourth is either on that circle or off it). And this in turn means there was then another curve with the same perimeter, and containing greater area.
Steiner's original proof relied on a simpler case of this quadrilateral argument, specifically that of parallelograms with fixed sides. It's easier to see that the greatest such area is a rectangle; but reducing to this case uses slightly more surgical trickery beforehand. I'm undecided, just now, which approach is tidier.
Now, last time it was pointed out that rotating a figure preserves both its area and its perimeter; and this in fact remains true of separate pieces of the figure; so, for instance, pertaining to the area within the black curve below:
It just so happens that the best arrangement of vertices for the red quadrilateral is (surprise!) as a cyclic quadrilateral, aka "chord" quadrilateral, which is to say, such that the vertices are all on a circle. Perhaps the other surprise is that this is always doable. There is a bit of annoyance about proving the optimality of a cyclic arrangement, in that it works out to be more natural to maximize the square of the area --- in a calculation attributed to one Bretschneider, this is a sum of two terms, one of which is determined entirely by the four side lengths, the other proportional to their product and otherwise depending only the sum of opposite vertex angles... you can see how this is getting messy, yes? The calculus is quite straight-forward, but the geometry gets to be rather icky. I think there ought to be a 4-dimensional scissors congruence proof of this fact, considering how similar it is to Heron's formula. But we mustn't jump to conclusions! (Also, such things are difficult to read).
Anyways. Suppose, then, that the curve we had wasn't a circle. Then there must be some four points of our curve that weren't cyclic. (Three points determine a circle, a fourth is either on that circle or off it). And this in turn means there was then another curve with the same perimeter, and containing greater area.
Steiner's original proof relied on a simpler case of this quadrilateral argument, specifically that of parallelograms with fixed sides. It's easier to see that the greatest such area is a rectangle; but reducing to this case uses slightly more surgical trickery beforehand. I'm undecided, just now, which approach is tidier.
Tuesday, March 13, 2012
Dear Orville,
Oatmeal.
I didn't like it much growing up. I still don't like it now. But I'm beginning to appreciate that it has, at least, an inoffensive nutritiousness. Then as now, it's more about the fruity things that can be included to break up its blandness. Also, I see why those very practical Scots formed a notable habit of salting it --- it's perfectly sensible!
Anyway, hope that dried maize thing works out for you,
a cook-to-eat-er
Oatmeal.
I didn't like it much growing up. I still don't like it now. But I'm beginning to appreciate that it has, at least, an inoffensive nutritiousness. Then as now, it's more about the fruity things that can be included to break up its blandness. Also, I see why those very practical Scots formed a notable habit of salting it --- it's perfectly sensible!
Anyway, hope that dried maize thing works out for you,
a cook-to-eat-er
Monday, March 5, 2012
A Little Bit about Isoperimetric Inequalities
So, in Crelle XVIII, Jakob Steiner published the earliest known solution of the ordinary isoperimetric problem --- that is, among measurable plane figures with fixed perimeter, find which has the greatest area. I'm sure something like this result had been known considerably earlier; Euler and later Lagrange had formulated and solved rather general extremum problems for curves, and the isoperimetric example isn't difficult to solve. Variational problems were posed by one of the Bernoulis --- most notably the brachistochrone, of which several solutions were published simultaneously (Oh! For the good old days!). My source for Steiner's priority also claims he's the first person to have formulated the problem (or at least to have published such a formulation). So I'm inclined to suspect that Newton and Leibniz and them all thought the thing so painfully obvious that they never mentioned it.
The isoperimetric problem is considerably simpler than the brachistochrone. In fact, there are a number of similar problems which have strikingly similar solutions; for instance, among rectangles having fixed perimeter, that with the greatest area is a square, which is practically a proposition in Euclid, and quickly generalizes to the elsewhere-famous Cauchy-Shwartz-Буняковский inequality. If we said "triangles" instead of "rectangles", we'd get an equliateral triangle as the solution, and if you want to keep guessing at the best answers, you'll see this post is actually a love-song to symmetry.
The isoperimetric problem itself has rotational symmetry. Since rotating a plane figure changes neither its perimeter nor its area, it should be clear that if we had any extremal isoperimetic figure, then we could get more extremal figures by rotating the first one. That is, we could get more if there were more --- if our extremal isoperimetric figure didn't have the same symmetry as the isoperimetric problem itself. But there aren't more, there's only one. And there's a good reason there's only one, but we'll get to that another time.
For now, pleasant dreams!
The isoperimetric problem is considerably simpler than the brachistochrone. In fact, there are a number of similar problems which have strikingly similar solutions; for instance, among rectangles having fixed perimeter, that with the greatest area is a square, which is practically a proposition in Euclid, and quickly generalizes to the elsewhere-famous Cauchy-Shwartz-Буняковский inequality. If we said "triangles" instead of "rectangles", we'd get an equliateral triangle as the solution, and if you want to keep guessing at the best answers, you'll see this post is actually a love-song to symmetry.
The isoperimetric problem itself has rotational symmetry. Since rotating a plane figure changes neither its perimeter nor its area, it should be clear that if we had any extremal isoperimetic figure, then we could get more extremal figures by rotating the first one. That is, we could get more if there were more --- if our extremal isoperimetric figure didn't have the same symmetry as the isoperimetric problem itself. But there aren't more, there's only one. And there's a good reason there's only one, but we'll get to that another time.
For now, pleasant dreams!
Wednesday, February 29, 2012
How Long, O Lord, How Long?
Dear Lott of Oxford,
We humbly exhort you to gather up your family, the Godzdogz, and the last ten righteous folk. You're standing in the way of the brimstone.
a shepherd in exile
We humbly exhort you to gather up your family, the Godzdogz, and the last ten righteous folk. You're standing in the way of the brimstone.
a shepherd in exile
Friday, February 17, 2012
If you didn't hear me come in, it's because my steps were muffled by the dust.
Dear Belfry,
Where are you? It's getting so that you might think this place were haunted by real bats, not just the pseudonymous kind. Really, if you can't think of some nifty math or other stuff to blog about, we might have to fall-back on that original conceit behind this here --- er ---- whattkhaview. And that means I have to come up with people to write to, and people to pretend to be and all that; of which you will remember what a pain it got to be the first time around, yes? Well, perhaps it'll be fun.
Anyways, get over that cold/depression/choleric and come back, soon, you hear?
Sincerely,
you-know-who
Where are you? It's getting so that you might think this place were haunted by real bats, not just the pseudonymous kind. Really, if you can't think of some nifty math or other stuff to blog about, we might have to fall-back on that original conceit behind this here --- er ---- whattkhaview. And that means I have to come up with people to write to, and people to pretend to be and all that; of which you will remember what a pain it got to be the first time around, yes? Well, perhaps it'll be fun.
Anyways, get over that cold/depression/choleric and come back, soon, you hear?
Sincerely,
you-know-who
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