Topologies of Thoughts

What Kat Zhang actually built

Before any theorizing, it's worth being precise about the project. the.poet.engineer — Kat Zhang, a Brooklyn-based media artist, engineer, and researcher at Gray Area — extracted embeddings from her personal Obsidian vault, then rendered those embeddings as an interactive 3D network inside TouchDesigner, across three distinct topologies. Her X post describes the three modes exactly:

Kat then followed up with a technical thread that is more revealing than the demo itself. The full quote deserves reading carefully:

"what's not obvious from the hand tracking demo: it's actually a hybrid approach of both using the embeddings of the notes and asking llms to try to make sense of the calculations. for example, the medoid note i got from embeddings and what llm deemed as the central note are actually different, and yet both are about the evolution of interfaces; used clustering to categorize the notes and had llm name the categories articulated what these notes are about; i think the last one is the most interesting, aka using k-nn to build edges among notes and then have llms to make sense of how exactly the notes are connected." — @poetengineer__, 2/18/26, ~600K impressions

This thread reveals three things the demo obscures: (1) the pipeline is a deliberate hybrid — embeddings and LLMs doing different jobs, not one replacing the other; (2) the centrality question has two answers that disagree (the geometric medoid vs. the LLM's semantic judgment), and that disagreement is itself data; (3) Kat considers the k-NN + LLM edge labeling the most interesting part — a ranking that shapes how we should read the whole project.

The Jupyter notebook code is available to her subscribers, meaning the pipeline is reproducible Python running on standard ML libraries — not a TouchDesigner magic trick. TouchDesigner is purely the renderer.

Kat describes her broader practice at Gray Area as investigating "how computation can reveal the unconscious and poetic aspects of the human and machine psyche." This project sits squarely in that lineage. It's not a productivity tool. It's a computational self-portrait — one that can disagree with itself about who you are.

From markdown to geometry

The project follows a pipeline that's now emerging as a small standard in the personal knowledge management (PKM) ecosystem. Understanding each step reveals where the interesting design decisions live.

Step 1: Text → Vector (Embedding)

Each note in the Obsidian vault is fed to an embedding model — likely OpenAI's text-embedding-3-small or similar — which maps it to a point in a high-dimensional space (typically 1,536 dimensions). Notes that discuss related concepts land near each other; unrelated notes are geometrically distant.

Step 2: 1,536D → 3D (Dimensionality Reduction)

To render in TouchDesigner, the embeddings need to collapse to 3 coordinates. UMAP (Uniform Manifold Approximation and Projection) is the standard choice here — it preserves local neighborhood structure better than PCA and is far faster than t-SNE at scale. The key parameters: n_neighbors controls how local vs. global the projection is; min_dist controls cluster tightness.

Step 3: The Medoid — and why it disagrees with the LLM

Before building edges, there's a prior question: what is the center of your thought? The embedding approach gives a precise answer: the medoid — the note whose embedding vector minimizes total distance to all other notes. Crucially, unlike a computed average (which would produce a phantom vector corresponding to no real note), the medoid is always an actual note you wrote. You can read it.

The distinction matters. The "average" of your notes would be a meaningless blur of all your topics at once — a document nobody wrote. The medoid is the note that, on average, is closest to every other note in your vault. It is the note with the most reach — the one that could most plausibly talk to everything else you've written. It is your intellectual center of gravity, revealed by geometry rather than chosen by intention.

This is the uncanny part: you never decide what your medoid is. Kat never chose "the evolution of interfaces" as her central theme. The mathematics found it. The medoid is a mirror that shows you something about your own thinking that you didn't put there deliberately — the unconscious thesis of a body of work.

Kat then discovered that the medoid and the LLM's judgment of "central note" were different notes — and yet both were about the evolution of interfaces. This is not a failure. It is the interesting result. The geometric center and the semantic center of a mind can diverge. The medoid captures where most of your thinking lives; the LLM captures what most of your thinking refers back to. A person could have a medoid in "productivity systems" — because they wrote about it constantly — but an LLM center in "self-worth" — because that concept threads invisibly through everything else.

Step 4: k-NN edge construction + LLM labeling

For the distributed topology, edges are built using k-nearest neighbors in the original high-dimensional embedding space. Each note connects to its k most similar notes by cosine distance. Then an LLM articulates how those neighbors relate — generating a relationship sentence that exists in neither note. Kat explicitly called this the most interesting part of the whole project.

Step 5: Clustering + LLM naming (decentralized topology)

The vault is partitioned into clusters (likely k-means or HDBSCAN on UMAP coordinates), and an LLM names each cluster based on its member notes. The hand-tracking interface in TouchDesigner lets you navigate this physically — pointing at clusters to reveal their LLM-generated names. TouchDesigner is purely the renderer; the pipeline is Python, available as a Jupyter notebook to subscribers.

Each topology makes a different claim about what thought is. Centralized says thinking is convergent — all roads lead to a thesis. Decentralized says thinking is modular — islands of expertise with bridges. Distributed says thinking is relational — the connections themselves are the content, and an LLM needs to name them because you haven't consciously articulated them yet.

The most interesting part nobody is talking about

The distributed topology has a feature that deserves its own section: edges are labeled by an LLM describing how ideas connect. This is not a visualization trick. It is a fundamentally different epistemological move.

In a standard knowledge graph, edges are either explicit (you made a wikilink) or implicit (cosine similarity exceeded a threshold). Both approaches treat connection as a binary: related or not. The LLM edge label introduces a third kind of connection: the articulated relationship.

The connection between two notes is not just a similarity score. It is a sentence. And that sentence can surprise you with what it says. — synthesis of distributed topology logic

Concretely: if note A is about "creative block" and note B is about "ADHD hyperfocus", a cosine similarity edge just says "0.74 related." An LLM edge label might say "Both explore the relationship between attention dysregulation and creative output — but from opposite poles of productivity." That sentence is itself a new idea, not present in either note.

What the open-source community built around this problem

Kat's project sits in a rapidly growing ecosystem of tools solving adjacent parts of the same problem. Understanding the landscape reveals what is solved, what is unsolved, and where the research frontier actually is.

What brain network science says about
the right topology

The choice to render thoughts as three topologies isn't just aesthetic. There is a 30-year literature in cognitive neuroscience doing exactly this — mapping brain regions as nodes, functional connections as edges, and analyzing the resulting topology for what it predicts about cognition. That literature has something specific to say about each of Kat's three modes.

The centralized topology is neurologically naive

Kat's centralized view — one core idea connecting all — creates a star graph. In brain network terms, this is the worst possible topology for cognition. It forces every signal through a single hub, creating long path lengths everywhere except through the center. Real brains don't work this way. They exhibit small-world topology: dense local clusters connected by a handful of long-range hub nodes. This structure minimizes average path length while maintaining local specialization — and shorter path length correlates with higher performance on working memory tasks. The centralized topology is useful as a diagnostic (what is the center?) but as a model of how thought actually works, the neuroscience says it's wrong.

The distributed topology is closest to how brains are wired

Kat's distributed view — k-NN edges with LLM labels, no hierarchy — most closely approximates the small-world structure that real cognitive networks exhibit. But it's missing something the neuroscience considers essential: betweenness centrality. In brain networks, nodes with high betweenness sit on the shortest path between many other node pairs — they are structural bridges. Remove them and the network fragments. These hub nodes correspond to the default mode network and prefrontal cortex — regions associated with creativity, working memory, and insight. Applied to a personal note graph: the notes with highest betweenness are not your most-written-about topics. They are the concepts that connect your intellectual domains. A 400-word note on "systems thinking" might be the most structurally important note in a vault because it bridges engineering and philosophy. Kat's pipeline computes the medoid but not betweenness centrality — and the two would likely disagree, producing a third kind of "center" that neither the medoid nor the LLM captures.

The decentralized topology predicts where new ideas live

Kat's decentralized view — clustered hubs — maps directly onto what neuroscience calls modular organization. Brain networks are modular: densely connected communities with sparse bridges between them. The participation coefficient of a node measures how many of its connections cross community boundaries. Nodes with high participation are integrators — and in the neuroscience literature, these cross-boundary nodes are where the most generative cognitive work happens. In a personal vault, these are the notes that connect your clusters to each other. The decentralized topology makes them visible. But it also reveals the inverse: notes with zero participation — the orphans that belong to no cluster. The neuroscience treats these as noise. Section 5 argues they might be the most interesting notes in the vault.

Better intellectual performance was associated with shorter characteristic path length and higher nodal centrality of hub regions in the salience network. — Farahani et al., Frontiers in Neuroscience (2019), reviewing van den Heuvel et al.

The problems this project opens

1. The temporal connectome

Kat's project treats the vault as a static snapshot. But thought evolves. If you embed notes with timestamps and visualize the trajectory of cluster centroids over time, you get a temporal connectome of the self — a record of how your obsessions form, peak, drift, and dissolve. The note on "grief" that was a cluster center in 2021 may have merged into a broader "impermanence" cluster by 2024. This is autobiography as graph dynamics.

2. Prompted image generation isn't creating in the medium of AI

Kat said this explicitly on Threads: "prompted image generation, which represents most of the AI art we see, isn't really creating in the medium of AI." Her note visualization project is the enacted counterargument. She is using AI (the embedding model, the LLM edge labeler) not as an image generator, but as a cognitive prosthetic that reveals structure you couldn't see. The AI is doing something it is uniquely positioned to do — collapsing high-dimensional semantic space — not just producing pretty images.

3. The notation connection — Ian Arawjo's reply

In the same thread, Ian Arawjo — HCI researcher at Université de Montréal and Mila, author of "How Notations Evolve: A Historical Analysis with Implications for the Incremental Formalization of User-Defined Abstractions" (arXiv 2602.01525, submitted to CHI 2026) — responded: "I see 'evolution of notation in the arts and sciences'! Check out our paper (maybe something to add to the cool Obsidian graph!)"

The connection is non-trivial. Arawjo's paper finds that notation evolution proceeds from informal ideas that borrow from prior culture, especially the remixing of prior notations and the application of conceptual metaphors of linking and grounding — and that a notation is revised and extended as it comes into contact with diverse situations and people. The relevance to Kat's project: the note graph itself is a notation. The act of choosing topology — centralized, decentralized, distributed — is a notational choice that encodes values and makes certain thoughts easier to have than others.

The distributed topology with LLM edge labels is arguably a new kind of notation: one where the connections between concepts are auto-generated text rather than lines, arrows, or formal symbols. Arawjo's historical analysis of how notations become institutionalized maps directly onto the question of whether the LLM edge label is a notation that could stabilize and circulate — or whether it is too personal, too generative, too context-dependent to ever formalize.

4. The orphan inversion

Clustering algorithms like HDBSCAN mark low-density notes as outliers — notes that don't belong to any cluster. The standard treatment is to filter them as noise. But in a personal knowledge base, an orphan note may be something entirely different: an idea that has no predecessors yet, a thought that hasn't found its cluster because the cluster doesn't exist yet. The orphan note is not a failure of organization — it may be the most genuinely novel thing in the vault. Inverting the assumption about outliers — treating them as signals rather than noise, candidates for future cluster seeds rather than discards — is a research direction nobody has taken seriously.