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ORIGIN / 04

The Allegory of the Manifold

Pollen grains, electron spheres, and the dodecahedron Plato saved for last: what embedding spaces, unlearning, and enterprise data chaos have in common — the dents are the meaning.

In this entry

On Pollen, Electrons, and the Shape of Everything We Know

Josh Morgan
AI Research Lab, Actian, Inc.


1. Two Problems, One Shape

Most of us look at a flower and see a thing of beauty.

Very few see what Jantina Tammes saw. The first professor of genetics in the Netherlands — appointed in 1919, when the university had barely conceded that women could take exams — she spent the early twentieth century resolving the visible traits of plants into statistics and probability. Flax stems, pollen morphology, the inheritance of continuous characters. Beauty, decomposed into distribution.

Her nephew Pieter was looking at flowers too, but closer. Staring at the pores scattered across a grain of pollen, he posed a question in his 1930 doctoral dissertation at the very same University of Groningen that remains unsolved to this day. Pointing at the distribution of exit places on the surface of pollen grains, he simply asked:

What is the optimal way to pack circles on the surface of a sphere — for every possible number of circles?

At nearly the same moment, a British physicist named J.J. Thomson was circling the same shape from the other side. His question: if you confine a set of electrons to the surface of a sphere, and each one wants nothing more than to flee every other, where do they come to rest?

Pieter asked how to place points. Thomson asked where points would fall on their own. A constraint and a force. Packing and settling.

Neither knew that a century later, both of their problems would be staring out from just beneath the surface of the AI revolution.


2. The Simplicial Complex, Simplified

Imagine a language of only four words, each meaning something wholly its own. You want to train a model to hold these words as vectors in three dimensions, and the single thing the model must accomplish is to keep them apart — four meanings can only carry signal if they never collapse into one another.

That force — repulsion under a roof of finite space — settles them into the most separated arrangement three dimensions can offer: the four vertices of a tetrahedron.

Add words and the shape answers. Twenty of them, still each a perfect island, and the points come to rest as the twenty vertices of a dodecahedron — the solid Plato held back from the four classical elements and assigned to the cosmos itself. Fire got the tetrahedron. Earth the cube. Air the octahedron. Water the icosahedron. But the dodecahedron, the fifth solid, the one with no element, no substance, no matter — Plato gave that one to the heavens. The shape of everything.

Now let the dimensions run to 4,096 and let the words be the whole of human language. I’ll leave the precise topology to the reader’s imagination, but this is the country a word model lives in. Vectors must stay distinct or there is no signal to train on — and yet king minus man plus woman lands near queen only works if the shape can be rotated so that one relation echoes another, so that the same pattern recurs in different corners of the space. The next unremarkable CRM, the next summarization API, the next code assistant — each one is a vast, virtual, hyperdimensional weather system of Tammes and Thomson forces, packing and settling at once, chaos and symmetry locked in a permanent negotiation.

The Tammes force says: keep apart. The Thomson force says: find equilibrium. Training is both at once.


3. The Dent

Here’s the part nobody talks about.

The dodecahedron — the beautiful Platonic answer to twenty unrelated words — is the degenerate case. It’s what happens when every word is an island, when nothing is related to anything, when the only pressure is repulsion. Pure Tammes spacing. Perfect symmetry. Zero information.

Real language isn’t that.

The moment two words are related — “king” and “queen,” “hot” and “cold,” “run” and “running” — they resist the repulsion that wants to push them apart. They’re closer than Tammes would allow, because meaning is pulling them together. The solid dents.

And here’s the thing: the dent is the meaning.

The ideal solid is the prior — the maximum-entropy arrangement, the configuration that says nothing about anything. Every deviation from that ideal, every place where the actual geometry refuses to match the perfect crystal, is a piece of information the model learned. The dents are not noise. The dents are not imperfection. The dents are the entire point.

A trained embedding space doesn’t resemble a Platonic solid. It resembles a Platonic solid that has been deformed by reality, and the deformation is the knowledge.

This reframing matters because it changes what convergence means. The Platonic Representation Hypothesis — the mounting evidence that different models, trained on different data with different architectures, converge toward the same internal geometry — doesn’t mean they’re converging on the same solid. They’re converging on the same dents. The shared structure across independently trained models is agreement about how reality breaks the symmetry. Not agreement on the crystal. Agreement on the deformation.

Different cartographers. Same terrain. They disagree on coordinates but agree on topology — and the topology is the dents.


4. The Fingerprint

Now push this one step further.

If a trained model is a deformed solid — an ideal shape that’s been dented by every sample it ever saw — then each sample has a fingerprint: the specific dent it contributed to the overall shape.

This isn’t metaphor. It’s measurable.

Train a model on a dataset. Train the same architecture on the same dataset plus one additional sample. The difference between those two models — in parameter space, in activation space, in the geometry of the embedding manifold — is the fingerprint of that sample. Its specific contribution to the deformation.

Now, is that fingerprint exact? No. Training isn’t commutative. The order matters, the learning rate matters, the batch composition matters. If you’re thinking “but the path through parameter space is chaotic and the same sample has different effects depending on when it’s seen” — you’re right. In the same way that the actual vector for “queen” doesn’t land at exactly king − man + woman. The arithmetic is approximate. The geometry is real.

The fingerprint of a sample is its relational effect — how it bends the model away from what the rest of the distribution alone would have produced. And that relational quantity can be far more stable than the absolute trajectory that created it. Two different training runs reach different parameter values, but the marginal pull of a given sample against its background distribution points the same way in both. The dent it makes is recognizable.

This leads somewhere interesting.

If a sample’s contribution is a dent, and that dent is measurable, then in principle you can ask: what happens if I un-dent it?

Not by literally reversing the training step — that path is gone, the trajectory is nonreversible. But by identifying the geometric signature the sample left behind and applying the inverse deformation to the manifold. Not “subtract the parameter delta.” Instead: “restore the distributional geometry to what it would have been without this sample’s pull.”

That’s a different operation. And it’s much more plausible than the naive version.

There’s a beautiful tension hiding here, and it’s worth naming because it’s the kind of thing that generates research programs. The samples whose fingerprints are most stable and most composable are the ones whose information is most redundantly represented elsewhere in the training set — many other samples make similar dents, reinforcing the same geometric feature from multiple directions. King − man + woman works because the gendered-royalty parallelogram is supported by thousands of co-occurrences, not one. The dent is robust because it’s redundant.

But those are precisely the samples you can’t cleanly un-dent. Remove one instance of a redundantly-encoded concept and the rest of the distribution holds the dent in place. The solid springs back. You’ve pulled one finger off the clay and nine others are still pressing.

Conversely, the samples you’d most want to un-dent — a memorized outlier, a single piece of private data, a rare capability you wish the model hadn’t learned — are the ones with the least distributional support. Lone spikes sticking off one vertex, held by almost nothing. Easy to snap off, but with the most chaotic, least stable fingerprints, because nothing else in the distribution reinforces them.

Fingerprint stability and un-denting difficulty are the same quantity measured from opposite ends.

That’s not a refutation of the theory. That’s the theory’s central prediction, and it’s testable. Plot the curve. If it shows up — if the correlation between distributional redundancy and fingerprint stability holds, and if un-denting efficacy anti-correlates with both — you’ve identified a conserved quantity governing when unlearning is even possible.

The dent defines its own limit.


5. The Grid and the Graph

So we have a shape. A deformed solid in impossibly many dimensions, its geometry carrying the compressed residue of everything the model was trained on. Every dent a piece of meaning. Every stable sub-structure a concept. Every local neighborhood a small crystal of related ideas, and those crystals interpenetrating, sharing axes, occupying the same space without colliding because high-dimensional near-orthogonality gives you exponentially more room than you’d expect.

Now zoom out. Look at a company.

Every enterprise has the same problem, and almost none of them know it. Their data lives in dozens of systems — catalogs, lakehouses, governance platforms, architecture management tools, knowledge bases — each one a monolith with its own model of reality, its own schema, its own vocabulary. Each system insists on its own view because collapsing two into one destroys the local information that makes each one useful. But they all describe the same underlying enterprise, which means there are forces pulling them toward coherence, toward shared structure, toward a shape that none of them can see alone.

Repulsion between local views. Attraction toward a common reality. Points on a sphere, refusing to collapse but unable to ignore each other.

Sound familiar?

My colleague Ole Olesen-Bagneux has been working this exact problem — he calls the architecture that emerges the Meta Grid, the third wave of data decentralization after microservices and Data Mesh. Where those are big, complex, and fast, the Meta Grid is “small, simple, and slow”: a coordination layer that lets the grid emerge rather than imposing it. Jean-Georges Perrin, working out of Actian’s CTO Office, has been attacking the same terrain from the standards side — his work chairing the Linux Foundation’s Bitol project produced the Open Data Contract Standard, which is essentially a way to formalize the dents: to make the relationships between data domains explicit, contractual, and governed before they calcify into chaos. (Between Ole’s Fundamentals of Metadata Management, JGP’s Implementing Data Mesh, and a few more O’Reilly volumes currently in progress — stay tuned — the roadmap is being written in real time.)

The embedding space is a knowledge graph that no one designed. The knowledge graph is an embedding space that humans can read. Same object, different resolutions.

But here’s where it gets interesting — and where the dent theory stops being a metaphor about models and starts being a blueprint for data systems.

The standard framing treats organizational data chaos as a problem to be solved — hire consultants, impose governance, build a catalog, hope people use it. That framing assumes chaos is the enemy.

It isn’t. Chaos is signal.

An enterprise generates usage patterns. Queries, joins, access logs, pipeline dependencies, search behavior — the entire behavioral exhaust of people interacting with data. Those patterns are stress signals. When the same two tables get joined in every report, that’s a Thomson force — the data wants to be closer together, and the system is making humans do the work of bridging the gap manually. When a particular dashboard gets rebuilt from scratch every quarter because nobody can find the last one, that’s a Tammes failure — the organizational packing put two related things too far apart. When three teams independently build pipelines to the same source because none of them know the others exist, that’s a missing edge in the graph — a dent the organization is trying to make that the current structure won’t allow.

These aren’t symptoms of dysfunction. They’re the forces telling you where the structure wants to go.

What if you trained a network on those stress signals?

Not a recommendation engine. Not a chatbot that answers “where’s the Q3 revenue data?” Something deeper. A system that reads the usage patterns the way a geologist reads stress fractures — as evidence of underlying forces that the current arrangement isn’t resolving. A network that learns the dents the organization is trying to make and builds the knowledge graph accordingly. The good dents — the ones supported by consistent, redundant usage across teams — become stable nodes in the federated graph. The bad dents — one-off queries born from confusion, searches that dead-end, pipelines that duplicate effort — get identified and gently un-dented by restructuring the metadata layer to resolve the force that created them.

The institutional knowledge — the existing catalogs, the governance rules, the data contracts, the domain ontologies — is the pretraining. It’s the prior shape of the solid. And the usage patterns are the ongoing training signal, the pressure of reality continuing to deform the structure toward something more efficient.

Today, you build the knowledge graph by hand and pray people follow the governance model. Tomorrow, the data system reads its own stress signals and the graph builds itself — settling into equilibrium the way electrons settle on a sphere, the way embeddings settle in a trained model. Not because someone designed the optimal arrangement, but because the forces were allowed to resolve. The same chaos that drives data dysfunction becomes the training signal that resolves it, provided someone with the right kind of hammer knows where to put the dents.

The metagrid becomes self-organizing. The solid finds its own shape.


6. The Manifold

There’s a quiet consensus forming in representation learning research. Different neural networks — trained on different data, with different architectures, different initializations, different objectives — are converging on the same internal geometry. Not the same numbers. The same structure.

Huh et al. formalized this as the Platonic Representation Hypothesis. Moschella et al. demonstrated that independently trained models can be stitched together zero-shot by projecting into a shared relational space. Jha et al. pushed it further with vec2vec, translating embeddings from unseen models while preserving semantic geometry. And Goldstein et al. showed — in Nature Communications — that neural activity patterns in the human brain align geometrically with deep language model embeddings.

The brain and the transformer are drawing the same map.

If hundreds of independently trained models are converging on the same relational structure, that structure isn’t an artifact of any particular architecture or dataset. It’s something they’re all approximating.

There is a manifold. The models are learning to chart it.

And now, with the language we’ve built: the manifold is the deformed solid. The ideal Platonic shape — the undented, maximum-entropy, zero-information crystal — is the prior. Reality is the force that dents it. Every model that trains on data drawn from reality converges toward the same deformation, because reality is the same for all of them. The convergence isn’t mysterious. It’s inevitable. Different sculptors working the same clay arrive at the same shape because the clay is reality and the hands are mathematics and there’s only one way the clay wants to sit.

The allegory of the manifold is Plato’s allegory, updated. The prisoners in the cave see shadows on the wall — low-dimensional projections of higher-dimensional objects. A trained embedding model sees the same shadows, cast by the same fire, projected onto the same wall. The manifold is the thing casting the shadows. We can’t see it directly. We can only see its projections — in a language model, in a vision model, in a brain. But the projections agree, which means the thing is real.

Plato assigned the dodecahedron to the heavens because it was the shape with no element, no substance — the form of the whole, not of any part. He was right about more than he knew. The manifold our models are converging on is exactly that: the shape of the whole. Deformed by everything. Defined by nothing in particular. Bigger on the inside than any single model can chart, because high-dimensional space holds exponentially more structure than low-dimensional intuition predicts.

And everything under the sun is in tune. But the sun is eclipsed by the moon.


7. What Comes Next

This paper is a provocation, not a proof. But provocations have a way of becoming research programs when the right people read them.

The Tammes-Thomson framing gives us a vocabulary for what embedding spaces are doing — packing and settling, repulsion and equilibrium, a constant negotiation between separation and coherence. The dent gives us a vocabulary for what training means — not memorization, not function approximation, but geometric deformation of an ideal shape by the pressure of reality. The fingerprint gives us a vocabulary for what individual samples contribute — and, more provocatively, for what it might mean to take that contribution back.

None of this is proven in the mathematical sense. But neither was the claim that neural networks are polytope systems when we first wrote it down, and that turned out to be experimentally validatable. Neither was the claim that predominantly negative training could match positive supervision, and that turned out to hit 98.37% on MNIST in a single epoch. And neither was the observation that YouTube’s audio fingerprinting — frequency-domain compression via DCT — could be applied to AI embeddings, treating vectors like songs and keeping only the low frequencies that carry semantic weight. That started as “what if AI works the same way as a video codec?” and ended up 30x faster with hardware already deployed on every phone on the planet. It also became the eureka moment behind query-adaptive dimension selection — the realization that not all dimensions matter equally, and the ones that matter change depending on what you’re looking for.

The track record of “this sounds like a metaphor but it’s actually geometry” is pretty good around here.

The research program writes itself:

  1. Measure the dent. Train paired models (with and without specific samples) on small architectures. Characterize the geometric delta in embedding space. Is it stable across initializations? Across architectures?

  2. Test composability. Do fingerprints add? Under what conditions? The prediction: approximate additivity in the linear regime, failure under superposition. The failure is the interesting part.

  3. Plot the curve. Fingerprint stability vs. distributional redundancy. If the anti-correlation with un-denting efficacy holds, you’ve identified the conserved quantity.

  4. Bridge to the graph. Take an enterprise knowledge graph (Ole’s got a few lying around). Take an embedding model trained on the same enterprise’s data. Measure the structural alignment. The prediction: the knowledge graph’s topology will correlate with the embedding space’s local geometry at the concept level, because they’re both projections of the same underlying manifold.

  5. Ask the hard question. If the manifold is real — if it’s the thing all models are approximating — can we learn it directly? Not by training another model. By extracting the invariants across many models, the way you’d extract the terrain from many maps. The shared dents. The universal deformation.

That last one is the program we’ve been building toward across all of these papers. The Emperor’s Gradient argued that optimization is geometrically underconstrained. This Theory Is Not Incorrect showed that networks learn by avoidance, by constraint, by discovering what things aren’t. The Allegory of the Manifold says: the thing they’re all approximating has a shape, that shape is deformed by reality, and the deformation is the knowledge.

We’ve been asking models to teach us facts.

Perhaps we should be asking them to show us the shape of meaning itself.


Acknowledgments

Thanks to:

  • Jantina Tammes, for seeing statistics in flowers before it was fashionable — or permitted
  • Pieter Tammes, for looking at the holes instead of the surface
  • J.J. Thomson, for asking where things fall when they want to be anywhere else
  • Plato, for saving the best solid for last
  • Ole Olesen-Bagneux, for showing that the grid was already there, waiting to be noticed
  • Jean-Georges Perrin, for insisting that the dents deserve contracts
  • Roger Waters, for knowing what happens when everything is in tune
  • The manifold, for being real whether we look at it or not

References

[1] Tammes, P.M.L. (1930). “On the origin of number and arrangement of the places of exit on the surface of pollen-grains.” Doctoral dissertation, University of Groningen.

[2] Thomson, J.J. (1904). “On the Structure of the Atom.” Philosophical Magazine, Series 6, 7(39), 237–265.

[3] Huh, M., Cheung, B., Wang, T., & Isola, P. (2024). “The Platonic Representation Hypothesis.” ICML. https://arxiv.org/abs/2405.07987

[4] Moschella, L., et al. (2023). “Relative representations enable zero-shot latent space communication.” ICLR. https://arxiv.org/abs/2209.15430

[5] Jha, R., Zhang, C., Shmatikov, V., & Morris, J.X. (2025). “Harnessing the Universal Geometry of Embeddings.” https://arxiv.org/abs/2505.12540

[6] Goldstein, A., et al. (2024). “Alignment of brain embeddings and artificial contextual embeddings in natural language points to common geometric patterns.” Nature Communications.

[7] Olesen-Bagneux, O. (2023). The Enterprise Data Catalog. O’Reilly Media.

[8] Olesen-Bagneux, O. (2025). Fundamentals of Metadata Management. O’Reilly Media.

[9] Olesen-Bagneux, O. (2024). “The Meta Grid Manifesto.” https://themetagridnotes.substack.com

[10] Olesen-Bagneux, O. (2025). “Federated Knowledge Graph.” Actian Data Intelligence Platform. https://dataintelligenceplatform.substack.com

[11] Perrin, J-G. (2024). Implementing Data Mesh. O’Reilly Media.

[12] Perrin, J-G. et al. “Open Data Contract Standard (ODCS).” Bitol, Linux Foundation. https://bitol.io


Internal Distribution: AI Research Lab
Status: Thought leadership — handle with imagination
Next Steps: Read it, argue with it, then measure the dents

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