### Tensor algebra love poetry

One of the books Curt gave me for Christmas was Stanislaw Lem’s *The Cyberiad*, which is a wonderfully fantastical collection of related short stories. Here’s a couple of selections:

“Very well. Let’s have a love poem, lyrical, pastoral, and expressed in the language of pure mathematics. Tensor algebra mainly, with a little topology and higher calculus, if need be. But with feeling, you understand, and in the cybernetic spirit.”

“Love and tensor algebra? Have you taken leave of your senses?” Trurl began, but stopped, for his electronic bard was already declaiming:

Come, let us hasten to a higher plane,

Where dyads tread the fairy fields of Venn,

Their indices bedecked from one ton,

Commingled in an endless Markov chain!Come, every frustrum longs to be a cone,

And every vector dreams of matrices.

Hark to the gentle gradient of the breeze:

It whispers of a more ergodic zone.In Riemann, Hilbert or in Banach space

Let superscripts and subscripts go their ways.

Our asymptotes no longer out of phase,

We shall encounter, counting, face to face.I’ll grant thee random access to my heart,

Thou’lt tell me all the constants of thy love;

And so we two shall all love’s lemmas prove,

And in our bound partition never meet.For what did Cauchy know, or Christoffel,

Or Fourier, or any Boole or Euler,

Wielding their compasses, their pens and rulers,

Of thy supernal sinusoidal spell?Cancel me not–for what then shall remain?

Abscissas, some mantissas, modules, modes,

A root or two, a torus and a node:

The inverse of my verse, a null domain.Ellipse of bliss, converge, O lips divine!

The product of our scalars is defined!

Cyberiaddraws nigh, and the skew mind

Cuts capers like a happy haversine.I see the eigenvalue in thine eye,

I hear the tender tensors in thy sigh.

Bernoulli would have been content to die,

Had he but known sucha! → This last line may (or may not) make more sense if you know what the graph of^{2}cos 2φr=alooks like in polar coordinates; for the lazy, here’s a graph with^{2}cos 2φa=2

—pp. 52-3

So they rolled up their sleeves and sat down to experiment–by simulation, that is mathematically and all on paper. And the mathematical models of King Krool and the beast did such fierce battle across the equation-covered table, that the constructor’s pencils kept snapping. Furious, the beast writhed and wriggled its iterated integrals beneath the King’s polynomial blows, collapsed into an infinite series of indeterminate terms, then got back up by raising itself to the

nth power, but the King so belabored it with differentials and partial derivatives that its Fourier coefficients all canceled out (see Riemann’s Lemma), and in the ensuing confusion the constructors completely lost sight of both King and beast. So they took a break, stretched their legs, had a swig from the Leyden jug to bolster their strength, then went back to work and tried it again from the beginninng, this time unleashing their entire arsenal of tensor matrices and grand canonical ensembles, attacking the problem with such fervor that the very paper began to smoke. The King rushed forward with all his cruel coordinates and mean values, stumbled into a dark forest of roots and logarithms, had to backtrack, then encountered the beast on a field of irrational numbers (F_{1}) and smote it so grievously that it fell two decimal places and lost an epsilon, but the beast slid around an asymptote and hid in ann-dimensional orthogonal phase space, underwent expansion and came out, fuming factorially, and fell upon the King and hurt him passing sore. But the King, nothing daunted, put on his Markov chain mail and all his impervious parameters, took his increment Delta k to infinity and dealt the beast a truly Boolean blow, sent it reeling through an x-axis and several brackets–but the beast, prepared for this, lowered its horns and–wham!!–the pencils flew like mad through transcendental functions and double eigentransformations, and when at last the beast closed in and the King was down and out for the count, the constructors jumped up, danced a jig, laughed and sang as they tore all their papers to shreds, much to the amazement of the spies perched in the chandelier–perched in vain, for they were uninitiated into the niceties of higher mathematics and consequently had no idea why Trurl and Klapaucius were now shouting, over and over, “Hurrah! Victory!!”

—pp. 68-70

“Tut-tut, my dears,” he replies. “Better you take a look instead!”

And indeed, they look and gasp–why, there’s nothing there, it’s gone, as if it had never been! And where did it go, vanished into thin air? It beat a cowardly retreat, and grew so small, so very small, you’d need a magnifying glass to see it. They root around, but all they can find is one little spot, slightly damp, something must have dripped there, but what or why they cannot say, and that’s all.

“Just as I thought,” Trurl tells them. “Basically, my dears, the whole thing was quite simple: the moment it accepted the first dispatch and signed for it, it was done for. I employed a special machine, the machine with a big

B; for, as the Cosmos is the Cosmos, no one’s licked it yet!”“All right, but why throw out the documents and pour out the coffee?” they ask.

“So that it wouldn’t devour you in turn!” Trurl replies. And he flies off, nodding to them kindly–and his smile is like the stars.

—pg. 139

“Bestowing happiness by miracle is highly risky,” lectured the machine. “And who is to be the recipient of your miracle? An individual? But too much beauty undermines the marriage vows, too much knowledge leads to isolation, and too much wealth produces madness. No, I say, a thousand times no! Individuals it’s impossible to make happy, and civilizations–civilizations are not to be tampered with, for each must go its own way, progressing naturally from one level of development to the next and having only itself to thank for all the good and evil that accrues thereby. For us, at the Highest Possible Level, there is nothing left to do in this Universe, and to create another Universe, in my opinion, would be in extremely poor taste. Really, what would be the point of it? To exalt ourselves? A monstrous idea! For the sake, then, of those yet to be created? But how are we obligated to beings who don’t even exist? One can accomplish something only so long as one cannot accomplish everything. Otherwise, it’s best to sit back and watch. …And now, if you’ll kindly leave me in peace. …”

—pg. 271