cat <<EOF >>/dev/null
Hey! Check it out!!! There's very little difference, he says. Could it be this is where they all come from? And who IS this St. Pod ? Is he or she the long-unheard-of progenitor of POD???
We must look into SB's source for this saint!
Let me know if anything turns up?
EOF
Monday, March 30, 2009
Attn. Cookie-manufacturing magnates
Dear Sir/Madam,
About cookies: they are, generally speaking, too tasty, and too much fun eating. I can't stand it! What are we supposed to do???
An overwhelmed customer
About cookies: they are, generally speaking, too tasty, and too much fun eating. I can't stand it! What are we supposed to do???
An overwhelmed customer
Monday, March 23, 2009
Greetings!
Hello, all new arrivals from good Meredith's For Keats' Sake! I suppose it now behooves me to regale and entertain the two of you? Perhaps I ought to say something witty! OK: um... Three guys walk into a ... oh, you've heard that, have you? Hmm... maybe a rhyme?
Oh dear. Perhaps I should just send you back. Much more edifying over there, anyways. Just ignore any loud echoing bangs as you go!
Cheerio, and thanks for stopping by.
An inconsistent sub-creator creature
There once was a traveler from OrangeActually, I don't think I can finish that one.
Who painted his car bright and ...
Oh dear. Perhaps I should just send you back. Much more edifying over there, anyways. Just ignore any loud echoing bangs as you go!
Cheerio, and thanks for stopping by.
An inconsistent sub-creator creature
Tuesday, March 10, 2009
Not a comedy, not a tragedy
(no substantial spoilers that I can see; not that many people are reading this, anyway...)
Dear Reader,
I've been reading (again!) The Lord of the Rings, which I seem to do periodically. Not quite every year, but most, and sometimes with less than a year between readings. I must be mad, somehow. In any case, I've been thinking back on the first time I read it, and the first time I finished it in particular, back in fifth grade I think.
It made me cry, I don't know why. I certainly didn't understand the ending on that first reading, though since then it's come to make better sense, and the point that constricts my throat and burns my eyes seems to come a bit earlier each time through.
In the present reading, I'm struck by the recurring theme of choice, the necessity of making some choice, the difficulty of making good choices out of imperfect knowledge, the moral imperative to choose the Good. Perhaps the most hopeful thought on this subject is expressed by Aragorn, who answers Éomer's question "How shall a man judge what to do in such times?" — times tumultuous amid unimagined strange happenings:
In other words, the main thing is that we mustn't loose our heads amidst all the possible distractions life in the world may throw at us. If you can recognize good and behave like a normal good person amid abnormal trials, you just might be a hero.
In my present reading, I've just come through one of Tolkien's embedded dissertations, on storytelling and what makes a good story. Here he refers, through his characters, to another of the tales he was continually working on — the tale of Beren and Luthien — and it is interesting that the two speakers in time realize that they belong to, are indeed living out, a long-removed continuation of that very same tale. This inspires them to imagine someone reading their own story out of a book, and to wonder what sort of people would read it, even though they can't see just now how they'll ever get back home or tell anyone of their adventures.
And that set me to thinking about the ending, because I myself do now know how it turns out; yet I want to keep reading anyway, even though I'll probably find myself choking-up for a few minutes towards the end. I'm not sure if it should be called foreshadowing, but having finished before, reading their talk of not foreseeing the end put me in this pensive mood, anyway. It certainly is poignant, in any case.
I begin to wonder if it's a bit like dying, this knowing the end must come, and getting there in time. God willing, though, it'll be years and ages before I find out: too much work to do, first!
An awkward mythophile.
Dear Reader,
I've been reading (again!) The Lord of the Rings, which I seem to do periodically. Not quite every year, but most, and sometimes with less than a year between readings. I must be mad, somehow. In any case, I've been thinking back on the first time I read it, and the first time I finished it in particular, back in fifth grade I think.
It made me cry, I don't know why. I certainly didn't understand the ending on that first reading, though since then it's come to make better sense, and the point that constricts my throat and burns my eyes seems to come a bit earlier each time through.
In the present reading, I'm struck by the recurring theme of choice, the necessity of making some choice, the difficulty of making good choices out of imperfect knowledge, the moral imperative to choose the Good. Perhaps the most hopeful thought on this subject is expressed by Aragorn, who answers Éomer's question "How shall a man judge what to do in such times?" — times tumultuous amid unimagined strange happenings:
`As he ever has judged,' said Aragorn. `Good and ill have not chaged since yesteryear; nor are they one thing among Elves and Dwarves and another among Men. It is a man's part to discern them, as much in the Golden Wood as in his own house.'
In my present reading, I've just come through one of Tolkien's embedded dissertations, on storytelling and what makes a good story. Here he refers, through his characters, to another of the tales he was continually working on — the tale of Beren and Luthien — and it is interesting that the two speakers in time realize that they belong to, are indeed living out, a long-removed continuation of that very same tale. This inspires them to imagine someone reading their own story out of a book, and to wonder what sort of people would read it, even though they can't see just now how they'll ever get back home or tell anyone of their adventures.
And that set me to thinking about the ending, because I myself do now know how it turns out; yet I want to keep reading anyway, even though I'll probably find myself choking-up for a few minutes towards the end. I'm not sure if it should be called foreshadowing, but having finished before, reading their talk of not foreseeing the end put me in this pensive mood, anyway. It certainly is poignant, in any case.
I begin to wonder if it's a bit like dying, this knowing the end must come, and getting there in time. God willing, though, it'll be years and ages before I find out: too much work to do, first!
An awkward mythophile.
Friday, February 13, 2009
On the area of a spherical triangle
Dear Self,
For future reference: Of course, this is outlined very nicely in Coxeter's Introduction to Geometry, (second edition), but as always the best way to learn math is to re-work it. Here, a proof-sketch without pictures.
By sphere S signify the level set---in some vector space V---of some positive-definite quadratic form; or equivalently the orbit of a generic point under the orthogonal group of the related inner-product. By a central plane signify any hyperplane including the origin and by hemisphere either of the two separated sets of points of S on one side of a central plane. By a convex lune signify the intersection of (at most) two hemispheres.
We now specialize to the vector space R3 with its usual inner-product, wherein central planes will have dimension two and meet any sphere (as defined!) in a circle, which we will call a great circle. The same great circle may also be refered to as the boundary of a hemisphere on either side of the same central plane. By a spherical triangle we will mean a non-empty intersection of three hemispheres such that the great circles that are their boundaries intersect pairwise, but not all three together. We claim without proof that the notion of angle between vectors corresponding to the inner product on R3 induces a notion of angle between hemispheres as well, and thus also an angle measure for lunes such that if a finite set of great circles and pairwise disjoint lunes has union the whole sphere, then the angles of those lunes have sum equal to 2π. Our final unproved claim is that both the property of being a lune and the angle of a lune are invariant under the action of the orthogonal group.
As a triangle ABC is an intersection of three hemispheres A,B,C, so the pairwise intersections of the same three hemispheres are three lunes AB,AC,BC, and the angles of these three lunes shall be called also the angles of the triangle. Related to the three hemispheres are their oposite hemispheres also A',B',C'; as these have the same respective boundaries, substituting any of A',B',C' for A,B,C, respectively, produces eight disjoint triangles (including ABC) --- we will extend the preceding notation for specific lunes and triangles to name these.
A more economical decomposition, however, is into the two disjoint triangles ABC and A'B'C', and the three lunes AB', BC', A'C. (This is a tedious exercise in propositional logic, or an easy picture to draw). Remark that AB and A'B' have the same angle, as have AB' and A'B.
These four lunes are disjoint and, together with the boundaries of A and B, they have union the whole sphere, so the angles of AB and AB' are suplementary. Similarly are the angles BC and BC', AC and A'C. The three lunes AB', BC', A'C, thus have angles summing to 3π less the sum of the angles of ABC. There is no finite set of great circles whose union together with that of the three triangles is the whole sphere; if three lunes orthogonally equivalent to AB', BC', A'C together with one more did give the whole sphere, then the fourth lune must have angle equal to the sum of the angles of ABC less π.
Sometimes a classical geometer
For future reference: Of course, this is outlined very nicely in Coxeter's Introduction to Geometry, (second edition), but as always the best way to learn math is to re-work it. Here, a proof-sketch without pictures.
By sphere S signify the level set---in some vector space V---of some positive-definite quadratic form; or equivalently the orbit of a generic point under the orthogonal group of the related inner-product. By a central plane signify any hyperplane including the origin and by hemisphere either of the two separated sets of points of S on one side of a central plane. By a convex lune signify the intersection of (at most) two hemispheres.
We now specialize to the vector space R3 with its usual inner-product, wherein central planes will have dimension two and meet any sphere (as defined!) in a circle, which we will call a great circle. The same great circle may also be refered to as the boundary of a hemisphere on either side of the same central plane. By a spherical triangle we will mean a non-empty intersection of three hemispheres such that the great circles that are their boundaries intersect pairwise, but not all three together. We claim without proof that the notion of angle between vectors corresponding to the inner product on R3 induces a notion of angle between hemispheres as well, and thus also an angle measure for lunes such that if a finite set of great circles and pairwise disjoint lunes has union the whole sphere, then the angles of those lunes have sum equal to 2π. Our final unproved claim is that both the property of being a lune and the angle of a lune are invariant under the action of the orthogonal group.
As a triangle ABC is an intersection of three hemispheres A,B,C, so the pairwise intersections of the same three hemispheres are three lunes AB,AC,BC, and the angles of these three lunes shall be called also the angles of the triangle. Related to the three hemispheres are their oposite hemispheres also A',B',C'; as these have the same respective boundaries, substituting any of A',B',C' for A,B,C, respectively, produces eight disjoint triangles (including ABC) --- we will extend the preceding notation for specific lunes and triangles to name these.
A more economical decomposition, however, is into the two disjoint triangles ABC and A'B'C', and the three lunes AB', BC', A'C. (This is a tedious exercise in propositional logic, or an easy picture to draw). Remark that AB and A'B' have the same angle, as have AB' and A'B.
These four lunes are disjoint and, together with the boundaries of A and B, they have union the whole sphere, so the angles of AB and AB' are suplementary. Similarly are the angles BC and BC', AC and A'C. The three lunes AB', BC', A'C, thus have angles summing to 3π less the sum of the angles of ABC. There is no finite set of great circles whose union together with that of the three triangles is the whole sphere; if three lunes orthogonally equivalent to AB', BC', A'C together with one more did give the whole sphere, then the fourth lune must have angle equal to the sum of the angles of ABC less π.
Sometimes a classical geometer
Wednesday, February 11, 2009
Warum nicht?
Dear Deutsche Welle,
I am greatly enjoying your series of lessons "Deutsch -- warum nicht" ; although that kobold Ex can be a bit trying to listen to. In any case, many thanks!
Ein Student
I am greatly enjoying your series of lessons "Deutsch -- warum nicht" ; although that kobold Ex can be a bit trying to listen to. In any case, many thanks!
Ein Student
Monday, February 9, 2009
Reading is a joy
Dear Libraries and Librarians of the World,
Do please take very good care of your books! A good book is indeed a joy to read, but a book falling apart can be quite frustrating.
That's all I've got to say, today. Goodnight!
An avid reader
Do please take very good care of your books! A good book is indeed a joy to read, but a book falling apart can be quite frustrating.
That's all I've got to say, today. Goodnight!
An avid reader
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