Notes on Norman Wildberger's Six: A mathematical exploration — lecture 2 of an elementary course in pure mathematics built around the combinatorics of six objects. This page works through one specific claim from the lecture: that there are exactly 45 meets, and why the 15 × 3 construction generates all of them without overlap.
A meet is a 2-set of non-incident edges — two edges that share no node — on six nodes. The claim is that there are exactly 45 meets. The argument has two parts:
If the sorting puts every meet into exactly one bucket — no meet skipped, no meet in two buckets — then the total is just 15 × 3 = 45.
Given any meet e₁ · e₂, the two edges use 4 different nodes. The remaining 2 nodes form a third edge, which I'll call the complement. That complement is the bucket label.
Example: the meet 1-2 · 3-4 uses nodes {1,2,3,4}. The leftover nodes are {5,6}, so the complement edge is 5-6. This meet goes in bucket 5-6.
The complement is unique. Given a meet, the 4 nodes it uses are fixed, so the 2 leftover nodes are fixed, so the complement edge is fixed. There is no choice involved. One meet → one complement → one bucket.
This means:
The argument above rules out one failure mode (a meet appearing in two buckets). The other failure mode is a meet appearing in zero buckets — being missed entirely. To rule that out, take an arbitrary meet and trace it through the construction.
Pick any meet — say 2-5 · 3-6. Where does it show up?
1-4.1-4. When it does, it generates the 3 pair-ups of the 4 leftover nodes {2, 3, 5, 6}:Our meet 2-5 · 3-6 is the middle one. The construction produces it — we didn't have to know about it in advance, and we didn't have to search for it. It fell out automatically when we processed bucket 1-4.
The general argument. Take any meet M:
So M must be produced. Since M was an arbitrary meet, every meet is produced.
| Concern | Why it can't happen |
|---|---|
| A meet ends up in two buckets (counted twice) | A meet has only one complement, so only one bucket matches. |
| A meet ends up in zero buckets (missed) | A meet always has a complement, and the bucket for that complement always generates it. |
Together: every meet appears in exactly one bucket. The buckets partition the meets.
Fix a bucket — say bucket 5-6. The meets in it use only the 4 leftover nodes {1,2,3,4}. So the question becomes: in how many ways can we split 4 nodes into 2 pairs?
Pick node 1. It must pair with one of {2, 3, 4} — that's 3 choices. The remaining 2 nodes are then forced into the second pair. So 3 splits:
This is the same calculation no matter which edge is the bucket label — it always reduces to "split 4 leftover nodes into 2 pairs," which always has 3 answers.
Each card is a bucket. The bucket label (the dashed gray edge) is the complement. The two solid colored edges are the actual meet.
This is not a "list everything and count" proof. It's a structural proof: it pairs up two ways of counting the same set.
15 × 3.Both ways count the same thing — the set of meets — so they must give the same answer. The structural way is easier because the local question ("how many meets share this complement?") has a clean answer (3) that we can compute without enumerating everything.
The key step is convincing yourself the sorting rule is well-defined: every meet has a complement, and only one. Once you believe that, the partition is automatic, and 15 × 3 = 45 follows.
Affine geometry has only one tool: parallelism. No ruler, no protractor. From this single primitive, an entire grid system can be built — and the grid that results is necessarily made of parallelograms, not squares.
A natural question: if we want a grid, why not draw squares? The answer is that a square requires more structure than affine geometry has — it smuggles in a 90° angle and a length comparison across directions, neither of which exists yet.
Source: WildLinAlg 1 — Introduction to Linear Algebra by Norman Wildberger.
This lecture basically explains this paper visually.
\(\mathbb{F}^2 \quad v=(x, y) \quad A=\left(\begin{array}{ll}a & b \\ b & c\end{array}\right)\) fixed then define
\[\begin{aligned} & v_1 \cdot v_2=v_1 A v_2^{\top}=\left(x_1 y_1\right)\left(\begin{array}{ll} a & b \\ b & c \end{array}\right)\binom{x_2}{y_2} \\ & =a x_1 x_2+b x_1 y_2+b x_2 y_1+c y_1 y_2 \end{aligned}\]
| Item | Date | Status |
|---|---|---|
| Stacker DSL (reader + macro expander) in Racket | Jan 20–22 | Completed |
| Matrix-vector multiplication module in Racket | Jan 26 | Completed |
PowerShell git-summary alias |
Jan 30 | Completed |
| Single-neuron malt training demo | Jan 16 | Completed |
| UHG visual explorer (Klein disk, drag-point) | Feb 13 | Completed + discarded |
| Quadrance computation table | Feb 17 | Completed + discarded |
cross_ratio_proj() Python function |
Feb 18 | Completed |
quadrance() + loss() PyTorch embedding functions |
Feb 23 | Written, framing paused |
| instacomments README fix | Feb 19 | Shipped |
| Bilinear form Scheme DSL stubs | Mar 2 | Exploratory |
“Seeing Tools” as a design philosophy — DSLs, visualizers, tables as instruments for making invisible structure visible. Coined and elaborated across Feb 13–25. “Inventing on Principle at its finest! Bret Victor would be proud of me.”
“Probing” as a research epistemology — when facing an opaque system, don’t demand a theory first; send cheap experiments and read the echoes. Crystallized Feb 26. “This is how working mathematicians actually experience the subject.”
UHG as an “API to the hyperbolic plane” — Universal Hyperbolic Geometry as a kernel/seeing tool for hyperbolic ML, replacing classical logmap/expmap. Feb 28. “Universal Hyperbolic Geometry is an interface to the API of the hyperbolic plane, one whose implementation I could not take a peek at.”
DSL-as-cheaper-neural-net argument — structured domains don’t need neural nets if you design the right language. Jan 18. “Often, creating a domain specific language is a much cheaper solution!”
“Math as DSL / proofs as programs” — arrived at the Curry-Howard correspondence organically through the stacker work. Jan 21. “Now, I see math as choosing the right language. Proofs are just programs that type-check.”
Conceptual Design as constraint analysis — a design without forcing functions cannot evolve. Applied to UHG Feb 15. “Limitations are forcing functions. But right now UHG has no forcing functions.”
REPL + Claude Code learning loop (DrRacket + timed work sessions) — Jan 15 onward, sustained. “REPL-based interactive development might solve my problem with slow feedback loops.”
uv Python project workflow — adopted Jan 23, ongoing. Figured out through trial and error: uv init → uv add → uv sync → uv run.
Intermediate artifacts for learning hard math — build a series of visualizations, each making one property salient while hiding others. Feb 18. “The quadrance computation table (makes the 1/(1-r²) blowup salient, hides the projective formalism).”
Live-blog while working (Roam capture → Twitter) — Feb 18, active. “This workflow just grew organically during this work session. Everything that turned out well in my life followed the same design process.”
Reply-to-build-audience Twitter strategy — high-quality replies to niche-popular users rather than broadcasting to a general audience. Working: DefenderOfBasic turned on notifications, +6 followers from one reply.
Somatic processing before cognitive work — exercise/emotional release as prerequisite to focus. Feb 12–21. “I felt depressed until 6 pm. Then, I decided to go on a walk and exercise. Now, I’m working again like nothing happened.”
1. Understanding the context—what problems users face, what they're trying to accomplish
2. Proposing forms—software concepts that resolve those problems
3. Testing the fit—whether the concept's operational principle aligns with how users would naturally think and work
So in my framing: the concept (not just “the tool”) would be the form, and the user’s actual practices, goals, and environment would be the context. A well-designed concept achieves “goodness of fit” between these.
]]>Gifford looked at the user data and found that Instagram’s most devoted users all did similar actions. In fact, he found that these devoted users did the same things in the first 10 minutes of using Instagram. In the data, Gifford saw a clear pattern for successful onboarding. […] the Instagram team surfaced what worked best, and they re-designed the system to make those “first 10 minutes” behaviors the default, not just a happy accident.
The next step is showing others how to follow along, which reminds me of Terence Tao’s crowdsourcing project. My prediction: as government funding for universities continues to shrink, more academic work will migrate online, leveraging what Yochai Benkler calls “The Wealth of Networks”. Top researchers like Tao increasingly need contributions from amateur researchers doing independent work in their free time. Political uncertainty is essentially forcing this shift.
How do we create a context where more is being added to the commons? How do we convert rival to non-rival goods?
]]>If we want abundance, we need to expand the commons, especially at the frontiers of human knowledge. All of my work in tech has been because I read Benkler, and one way or another its been political. It’s just a different kind of political. ~ Conor White-Sullivan