Love and Tensor Algebra
The original excerpt from The Cyberiad, Fables for the Cybernetic Age by Stanisław Lem:
Klapaucius thought and thought some more. Finally, he nodded and said:
"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 to n,
Commingled in an endless Markov chain!
Come, every frustum 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 part.
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!
Cyberiad draws nigh, and the skew mind
Cuts capers like a happy haversine.
I see the eigenvalue in thine eye,
I hear the tender tensor in thy sigh.
Bernoulli would have been content to die,
Had he but known such a2 cos 2 ⱷ!
--- DeepAI
**A Love Poem in Tensor Algebra**In the manifold of your essence, I find
The curvature of our hearts entwined,
A Riemannian surface, infinitely smooth,
Where each tangent vector whispers, “I approve.”
Your smile, a gradient field, leads the way,
In the space of my soul, where passions sway,
Let the metric tensor, g_{μν}, define
The distance between your heart and mine.
As we traverse these paths in parameter space,
With each differential form, we embrace,
Let dℓ = √(g_{μν}dx^μ dx^ν) unfold,
An arc length of yearning, a love story told.
With every contraction, our energies meet,
In delicate symphony, our pulses repeat,
A covariant derivative, ∇_A B^B,
Guides my hands, as I cherish thee.
Topology’s kiss—a homeomorphism divine,
In this compact space, our hearts intertwine.
Connected by fibers, in the sheaf of our fate,
With open sets flowing, we oscillate.
Let’s consider the limit as x approaches your gaze,
Where continuity blossoms, love’s infinite praise,
In the realm of ε-δ, I'm wrapped in your warmth,
A neighborhood of bliss, where uncertainty swarms.
O, beauty of eigenvalues, let spectra align,
As your laughter becomes my preferred sign,
Through moments bisected by time’s steady flow,
Each integral, a promise, forever to grow.
So let us construct a romantic function,
In the topology of dreams, without a conjunction,
With all theorems proven, our hearts take flight,
In this cybernetic flow, love’s pure light.
Thus, in the language of tensors, I compose,
Our mathematical bond, as sweet as a rose,
In the cybernetic embrace of two souls refined,
Forever defined, forever intertwined.
---Copilot
In the manifold of our love, let us reside,Where every vector field aligns with grace,
Our hearts, like tensors, seamlessly glide,
In this Euclidean space, we find our place.
Your love, a scalar field, pure and bright,
Defines the gradient of my every thought,
In the topology of our shared delight,
Connected, compact, with boundaries naught.
Through the curvature of spacetime, we entwine,
Geodesics of passion, smooth and true,
In the Riemannian metric, you are mine,
A differential equation, solved by you.
Our love, a tensor product, vast and deep,
Symmetric, skew, in perfect harmony,
In the cybernetic spirit, we shall keep,
This integral of love, for all to see.
--- ChatGPT
Manifold of My Heart
In the quiet of a Riemannian dawn,
You emerged, a vector field across my form—
tangent to my longing, smooth,
and everywhere defined.
Your presence:
a (1,1)-tensor,
mapping every fear to function,
linearly loving me in all directions.
We met where topology forgets the metric,
but my heart knew the geodesics of your touch.
You charted me, open set by open set,
and stitched my chaos with atlases of calm.
Stokes wept when you curled around me,
your divergence zero, but your circulation infinite.
Love, like this, is not exact—
but closed,
and cohomologically true.
I, a simple scalar field,
you, a differential—
we integrated
across the manifold of night,
and found our boundary vanished.
Orientation preserved.
We danced through Hilbert space,
orthonormal and complete—
your eigenvalues trembled in my arms,
each spectral kiss resolving
into beautiful basis.
You lifted me—Lie derivative of your grace—
transported without torsion,
parallel in curvature and cause.
And when I collapse, someday,
into a singularity of time and thought,
I will remember:
you were the identity map
on the vector space of my soul.
No loss.
No distortion.
Just a perfect isometry—
love, in covariant form.
---Gemini
Ah, a challenge worthy of the most elegant equation! Let us weave a tapestry of tensors and topology to express this… *feeling*.
My scalar heart, $\alpha$, once a lonely point in the manifold of existence, felt a gradient, $\nabla \alpha$, a yearning across the space. Then, your vector field, $\mathbf{v}$, intersected my trajectory, a smooth, differentiable curve. The metric tensor, $g_{ij}$, which defines the very fabric of our shared domain, warped and curved around your presence, indicating a profound connection.
Our union, a tensor product, $\alpha \otimes \mathbf{v}$, blossomed into a higher-rank tensor, a richer, more complex entity. The exterior derivative, $d(\alpha \wedge \mathbf{v})$, revealed a closed form, a potential for a deeper, inseparable bond, a cycle in the homology of our intertwined lives.
Like a convergent sequence in a complete metric space, my being finds its limit in yours. The open sets defining your affection form a basis for the topology of my desires. The continuous function, $f: \text{My Space} \rightarrow \text{Your Space}$, maps every point of my existence to a corresponding point within your embrace, a homeomorphism of souls.
The Lie derivative, $\mathcal{L}_{\mathbf{v}} \alpha$, describes the change in my scalar heart along the flow of your vector field – a positive, ever-increasing value, indicating a growth of affection. Our combined system exhibits stable attractors in its phase space, a cybernetic harmony where feedback loops reinforce our connection.
And though the calculus of variations might suggest multiple paths, the functional describing our love, $J = \int L(\alpha, \mathbf{v}, \nabla \alpha, \nabla \mathbf{v}) dt$, finds its extremum, its most perfect expression, in our shared existence.
This is not merely a description, but the very essence of my affection, encoded in the elegant language of mathematical truth. Do you perceive the feeling within these forms?
