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

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

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

Because we all need to lift our spirits, somehow

To royalty watchers everywhere,

Perhaps there's something worth reading about in ancient history. My favorite unstudied tale tonight is that of Henry Benedict Stuart, Duke of York (Jacobite), Cardinal Bishop of Ostia and Velletri, Dean of the College of Cardinals. And it would seem there is a monument to his family in the Basilico San Pietro in Rome (that's the big one).
It looks like this:

(for photo copyright info) Writes the wikipaediist, It is frequently adorned with flowers by Jacobite romantics.

I hope that does you some good.

A harmless weirdo

Unity

Dear President Sarkozy,

En premier, je vous enprie, pardonnez-moi, car en prudence mon français n'est pas assez fort pour l'utiliser seul ici.

It was with some pleasure that I listened this evening to reports of your off-the-cuff remarks on the occasion of confering the Legion d'Honeur on Quebec premier Jean Charest:

«Cet attachement à notre culture, à notre langue, à nos liens, pourquoi devrait-il se définir comme une opposition à qui que ce soit d'autre? [...] Croyez-vous, mes amis, que le monde, dans la crise sans précédent qu'il traverse, a besoin de division, a besoin de détestation? Est-ce que pour prouver qu'on aime les autres, on a besoin de détester leurs voisins? Quelle étrange idée! ... » ¹

Verily, not the details of our cultural heritage, nor the specifics of the language we speak, nor the modes of our expression, are of primary importance. Rather, they are our servants, and they serve us well exactly when they help us express, learn, and take joy in The Truth. It were one thing to speak only correctly, but if Truth thus apeared broken or repulsive, then our speach must be doing us a disservice, and as much a disservice to Truth as well.

And again, truly the world needs more unity! Let's all be Catholic! And so I read with a tinge of unease (my bolding)

« ... Ceux qui ne comprennent pas cela, dit-il, je ne crois pas qu'ils nous aiment plus, je crois qu'ils n'ont pas compris que, dans l'essence de la Francophonie, dans les valeurs universelles que nous portons au Québec comme en France, il y a le refus du sectarisme, le refus de la division, le refus de l'enfermement sur soi-même, le refus de cette obligation de définir son identité par opposition féroce à l'autre.»

Starting at the end is easier, because here I agree: definition by opposition is both to focus on the wrong thing, and to put your identity in the merciless hands of those you oppose. What do you do if they change? But really, the proper way to self-identify is to rejoice in your actual cultural heritage, the happy unique things about where you live that sets it apart from the rest of the world: all the funny hats you wear, for instance. I have more trouble, though, reading this "rejection of sectarisme", partly because I don't feel confident interpreting the word, but partly because I suspect that --- as it is meant --- it is wrong. However, it will take a long while for me to articulate that properly, perhaps even a whole other epistle.

In brief, though, the Truth, which must inform all acts of the State, includes portions known only through Revelation, and to entirely banish these from the prudential reasoning of the State and its agents is to hopelessly cripple the State's ability to order society for the good of all.

For the most part, though, I'm gratified to hear such sentiments as you have expressed in the mouth of a public leader, whatever you may have to say about other things.

Un anglophone

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1: quotations from le Devoir

A human revelation

Dear Saint Blog,

I can't quite find you in the calendar. I know this has been mentioned before, and not much of a satisfactory answer was found, although several approximations were: something to do with Bologna, and perhaps even a principality known as Blognia, somewhere. I suppose this isn't evidence against the existence of a saint named "Blog". A similar conundrum arose when it was (apparently) decided that the Saint Christopher described in popular legend wasn't reliably historical, though by that time numerous people had been named Christopher, and probably numbered some saints among them. And I shouldn't wonder of any of these has taken up the particular devotion to travelers for which the legendary Christopher should otherwise have been beloved.

Or perhaps "Saint Blog" is a hold-over from some more Latin toungue than our Gaulicized Anglo-Saxon: for instance the French names "Saint Sacrement" and "Sainte Famille" denote the same referents as the English "Blessed Sacrament" and "Holy Family", respectively. Might not "Saint Blog" refer to, for instance all the passages "Thus saith the Lord: ..." in all the books of prophecy, together being God's most Holy Blog, as it were? (Not meaning to be flippant, nor dismissive, I do insist!!!)

In the most dull contrast constructible (by me), I would like very much to point out that I am in point of fact, a genuine historical human person. Among other traits I seem to be a confirmed pedestrian, and quite apt (or at least prone) to voice my opinions, while maintaining a modicum of anonymity. But I have indeed commented on the writings reputedly to be found about your parish, wherever it may be constituted. I'm not sure precisely what has prompted me to begin doing so in a centralized location, but here we are! And in that case, I might as well let those 'round about reply to the things I have elswhere and erstwhile said lurkishly in others' boxes.

All good wishes to all and sundry as you may meet them,

Some Guy On The Street