Apocrypha Topologica I

Dear Mathematicelli,

I should like today to introduce an apocryphal history of Topology, both as a phenomenon and as a field of mathematical study. It will necessarily be abbreviated, full of fictions, and other more innocent errors --- hence apocryphal. But it should be ordered to the truth so far as illuminating the modern study of topology itself.

Topology was imposed on the visible Creation by God at least as early as the Second Day, when He said

6 ... "Let there be a firmament made amidst the waters: and let it divide the waters from the waters." 7 And God made a firmament, and divided the waters that were under the firmament, from those that were above the firmament, and it was so.

This highlights the first introduction of a disconnected set, a "you can't get there from here" in the world: to reach the waters above from those below you must cross this firmament, whatever it may be. And this notion of separation, whether in the absolute sense of being mutually inaccessible, or the relative sense of inhabiting disjoint neighborhoods has been a puzzle and inspiration for topologists since before we even had the name topology to describe the field.

Some might argue that the temporal ordering of days already introduced an order topology (or the causal partial-order topology we learn from the Special and General Theories of Relativity). I reply that this is missing something of the point, but if you want to write your own paper on the history of Topology that's quite alright.

Other Biblical features of topological interest in denoting separation: the cherubim posted to keep Adam and Eve out of Eden; the Red Sea; the River Jordan; the Rivers Tigris and Euphrates, that separate the Land Amidst the Rivers from surrounding territories (I mean, you can get in and out of that without crossing either, but then you'd have to cross the ~40km line segment between Palu and Hantepe... a narrow road, as geography goes!)

The next great topological discoveries were knots and chains. Windows are a nifty invention too, I suppose, but knots and chains are much more tangible. The way these work, as I'm sure you've experienced, is that various strands of rope or metal, being extended in one direction, have new ways of becoming separated from conditions they might have liked to achieve: while it would take a wall to stop you, and you can walk around a rope quite happily, a string can get stuck by a rope in one perpetual wandering-around! By the by, I don't mean mechanical knots; these are fascinating, but the mechanical distinctions between various rolling hitches are much fuzzier than the topological fact that there are more ways to get tangled in a net than in a string tied in a single loop. Think of the "Gordian Knot" versus an undoubled slipknot bow as you might use on your shoes.

As another aside, there seems some discrepancy between my predecessors in mathematical apocrypha on the one hand vs. closer accounts of Alexander's encounter with Gordy's knot --- what the knot tied up, whether Alexander really sliced it in two or what, etc., are disputed points. One way or another, there were knots and mechanical facts emphasizing topological separations, and it became a point of proverbs through the march of time.

The Gordian Knot brings us to the relatively recent period and setting known as Greek Mythology, and there, for now we will interrupt our History. Next time: strings graduate from topological features to topological tools!

The Math Prof of Your Nightmares

PSA

Dear Passers-by,

I noticed recently that the blogger profile for this persona didn't offer means of private communication, which was in fact the primary purpose of the /dev/null persona. This must have been the unintended consequence of some policy change within blogger, because other people have been surprised recently to not show contact info; in any case, I've changed that now.

And of course, it's easy to just write here, in Dumbledore Fashion, that mail addressed to qnoodles at gmail will find me. As a purely historical note, the q stands for Quincy, and no, I don't know anyone named Quincy. It just seemed to suit "Mr. Noodles" remarkably well!

your humble host

PS there really is only the one of me.

In deep October

Enter Hamlet, reading (L.C.)
Pol. How does my good lord Hamlet?
Ham. Excellent well.
Pol. Do you know me, my lord?
Ham. Excellent well; you are a fishmonger.
Pol. Not I, my lord.
Ham. Then I would you were so honest a man.
Pol. Honest, my lord!
Ham. Ay, sir; to be honest, as this world goes, is to be one man picked out of ten thousand.
Pol. That's very true, my lord.
Ham. For if the sun breed maggots in a dead dog, being a god, kissing carrion,——Have you a daughter?
Pol. I have, my lord.
Ham. Let her not walk i'the sun: conception is a blessing; but as your daughter may conceive,—friend, look to't, look to't, look to't.
[Goes up stage.]
Pol. (Aside.) Still harping on my daughter:—yet he knew me not at first; he said I was a fishmonger. [Crosses to L.] I'll speak to him again.—What do you read, my lord?

I am told "Defiant" and asked "Are sparrows blithe?"

I've been having difficulty with this one, because all the words that come into my head are scripture, and it doesn't seem quite right to write a cento on the Bible --- that were only to diminish the worthiest poetry ever set down.

\begin{itemize}
\item "Are not two sparrows sold for a farthing? ... You are worth more than many sparrows."
\item "Consider the birds of the air: they reap not, nor sow, nor gather into barns..."
\item "How lovely are thy tabernacles! ... for the sparrow hath found herself a house, and the dove a nest for to lay her young."
\end{itemize}

What follows instead is an aliterative allegory of wishful thinking. Or something... yeah, I don't know what it is.

Butterflies and sparrows spin blithely by
My window, wonderous marvels in miniature,
Provoking contemplation, to ponder creatures
And Provident God, gracious in peace and plenty.
Against His coming Justice, fearing coldest Gaol,
We rightly cry repentance, and remission crave,
Converting to Love our lagging contrite hearts;
Even as I of Him, each evening,
Beg good grace anew to gain Beatitude
Before dying, as defiant before Death,
To thee would I this warrant of affection
Give; lest lacking love's greeting
We were parted by time's passing wave ---
O! to be so blithe as sparrows!

And, my friends dear, please don't worry. There's nothing more dire looming over me than that recurrent need of the sacraments --- that's sort-of the point, in fact. It's the kind of thing you want to get done *before* the rest looks dire!

Eeek! What a place!

A locus focus, and Oh! what a scary place it is, too. For today we are serving

Detention with Dolores

Harry Potter and the Order of the Phoenix

[Harry Potter] had known this office under three of its previous occupants. [...]

Now, however, it looked totally unrecognisable. The surfaces had all been draped in lacy covers and cloths. There were several vases of dried flowers, each one residing on its own doily, and on one of the walls was a collection of ornamental plates, each decorated with a large technicolor kitten wearing a different bow around its neck. These were so foul that Harry stared at them transfixed, untill Professor Umbridge spoke again.

I'm already shivering!

I also thought about mentioning this locus back in September, for "educational" settings. Indeed, Harry learns much about the nature of evil --- as well as something of the heroic --- in this particular office room between his second and fifth year.

In its fifth-volume avatar, the Defence Against the Dark Arts Proffessor's Office shows us just how ugly good things like kittens can be made, with just the wrong sort of twist; in an interesting parallel, the same room highlights just how ugly such a good thing as devotion to the Truth can become when twisted into "devotion to what I say the Truth is" --- which is an idolatry.

I don't want to say too much more because I believe Enbrethiliel hasn't read this book yet.

That Bat also known as some guy on the street

WHAT ON EARTH?

UPDATED

What has happened to Orbis Catholicus II??? Is the illustrious Mr. Sonnen in some strange unpublished need? Has his server been compromised? Or (weirder) has someone flooded a DNS cache with this sham site?

Bloggians! To the rescue!

UPDATE: it seems to be here now.

Whence 4?

Time for some more math! At my (new! yay!) supervisor's behest I've been trying like a primordial lungfish to breath the fresh air called spectral sequences --- how they ever managed to acclimate to such thin and rarefied reference to anything tangible is beyond me; but then, the calculations themselves are thick as water to my more-terrestrial brain.

Today is not about spectral sequences, but back to geometric measure theory. Some time ago I got as far as to outline the proportion of surface areas vs. angles

$$ S(\triangle qrs):S(\mathbb{S})::\angle qrs + \angle rsq + \angle sqr - \pi:4\pi $$

for a spherical triangle $qrs$ --- although for reasons of presentation, that tale refered to the hemispheres $A,B,C$ described by the arcs $qr,$ $rs,$ $sq$ and containing the triangle $\triangle qrs$. (Do you see some ambiguity creeping into the tale? Don't worry: make your choices and then show that they aren't important!)

So, our task today is to improve the proportionality by establishing, for a unit sphere $\mathbb{S}$, an equality $S(\triangle qrs) = \angle qrs + \angle rsq + \angle sqr - \pi$, which by the complementary proportion will give $S(\mathbb{S})=4\pi$.

Fun with TeX and paper

echo <<eof >>/dev/null

We have calling cards!

That is, these are the sort of card you leave on a silver tray under the eyes of a lady's Mother or their doorman, when said lady isn't home or disposed to receive you, to say that you did, in fact, come to "call" --- without the bother or nuissance to your neighbors of actually calling out with voice to see if she answers. Some folk call them business cards these days, what with the advent of the inescapable mobile phone, in principle allowing men to pester decent ladies with person-to-person "calls" at any hour of night or day. Since precious little business gets done here (thank goodness) to call these cards "business cards" would be farcical beyond even our usual fare.

The obverse face is a tweaked version of this TeX file --- it took some doing, too, may I say. I was quite worn-out by the end of it. The reverse are filled with a rhombic variation of those famous Penrose tilings, one of several impressively economical files here exhibiting the illegible efficiency of Adobe's postscript virtual machine.

Another picture:

eof

Is it scary here? Or is it just me?

I seem to linger on one or two authors; this time I feel like another Tolkien spot, not far from last time's locus.

After stumbling along for some way along the stream, they came quite suddenly out of the gloom. As if through a gate, they saw the sunlight before them. Coming to the opening they found that they had made their way down through a cleft in a high steep bank, almost a cliff. At its feet was a wide space of grass and reeds; and in the distance could be glimpsed another bank almost as steep. A golden afternoon of late sunshine lay warm and drowsy upon the hidden land between. In the midst of it there wound lazily a dark river of brown water, bordered with ancient willows, blocked with fallen willows, and flecked with thousands of faded willow-leaves. The air was thick with them, fluttering yellow from the branches; for there was a warm and gentle breeze blowing softly in the valley, and the reeds were rustling, and the willow-boughs were creaking.

`Well, now I have at least some notion of where we are!' said Merry. `... This is the River Withywindle!...'

I suppose there's not much frightening about that, as far as it goes --- but you have to read the book! I love the way that "willow" note recurs, like an ostinato counterpoint. Or perhaps it's a flatted dominant? For myself, "willow" is one of my "cellar door"s, quite apart from how I admire willow-trees in person. But here...

... There were armies of flies of all kinds buzzing around their ears, and the afternoon sun was burning on their backs. At last they came suddenly into a thin shade; great grey branches reached across the path. Each step forward became more reluctant than the last. Sleepiness seemed to be creeping out of the ground and up their legs, and falling softly out of the air upon their heads and eyes.

Are you thinking what I'm thinking?

Go read the book, now, if you don't remember it. But it's worth noting that willows, while lovely and graceful, are not the trustiest of trees; their wood is rather too soft for structural uses, and they tend to fall apart as they get older. A venerable old specimen in one of my favourite boyhood parkly haunts was dismantled recently, to forestall it's falling down on unsuspecting visitors. Tolkien's rather more willful Old Man Willow character, who very much defines this place, is an alarming extrapolation of my poignant beloved trees.

--Some Guy on the Street, for

The River Withywindle
The Lord of the Rings: The Fellowship of the Ring
J.R.R. Tolkien

Something a little more Aedifying?

Dear Exegete,

I gave myself to LifeChain for an hour today; and whilst standing there the hymn occurred to me that, for no good reason, I know best in the Russian chant setting. Not that I could tell you which words mean what, nor how they fit together...

Anyways, I thought it made a nice contrast to the more commonly-referenced De Profundis; which is apposite, of course, but has so many other uses, as well. The latter is good for commemorating those lives already cut short; but Save, O Lord, Thy people is looks from the present to the future, and recalls souls that thrive still on Earth and for whom there is hope.

awaiting your reply,
he would be more sad, if not for life

Parce Domine populo tuo
et ne des hereditatem tuam in obprobrium
ut dominentur eis nationes
quare dicunt in populis "ubi est Deus eorum"?
zelatus est Dominus terram suam et pepercit populo suo

I need to fill my horrid writing fix somehow!

Dear Strongbad,

Its seems to like be a long tim you hasnt write email ansers. why is that? do you not get emailz anymor? Are you cant think to smart way of write back them? when do you write agane?

yores truly
guido quiscumque
INTERNETS.