Continued math visualizations development

I've read yesterday's trace and found the Mahler surface files on your Desktop. Next I'm checking which other proofs are in the repo so we can pick the next one.

Creating a private GitHub repo goldfishandswedishfish/math-surfaces, cloned to ~/Desktop/math-surfaces. I'm copying the Mahler files in, not moving them, so the originals stay where they are.

Repo's set up. Now I'm reading the four papers so the explainers are accurate.

TeX sources are there, so I'll read those directly. Starting with the five-coloring paper.

I have what I need for #158. Building it first: a plane-coloring explorer, a "paint your own tile" game, and a Moser spindle challenge.

Draft written. Checking the math with a quick script: spindle edge lengths, the hexagon safe window, and the 3-color count.

Renders cleanly. One fix: the spindle canvas is oversized, so I'm making it shorter. Then I'll upload.

Surface 1 of 4 is uploaded (private): https://traces.com/s/52797cc3-0c4f-4505-b3fa-3bb171d53396?surface=plane-coloring&version=0.1.0. Moving on to Heilbronn.

Writing the Heilbronn surface now. First I'm computing how tiny the paper's improvement exponent η really is, so the page can quote it correctly.

η comes out around 3 × 10⁻²³⁶. To keep the remaining surfaces consistent, I'm moving the shared styles into one file with a small build script, so each new page only holds its own content.

Now the Heilbronn page itself:

The Heilbronn page works. The exponent number line has overlapping labels, so I'm tidying that before uploading.

Clean now. Uploading Heilbronn.

The upload was rejected. I rebuilt the file after measuring its size, so the byte count no longer matches. Re-preparing with the right size:

Heilbronn is uploaded (2 of 4). Reading the Gaussian moat paper next.

API Error: Your computer went to sleep mid-response. The response above may be incomplete.

Welcome back. Here's where things stand. All links open on the trace you picked, and every surface is private and unreleased:

What's in the two new ones:

• Coloring the plane:
  • Hover any coloring and a radius-1 circle shows where the rule breaks.
  • A slider finds the size window where Isbell's 7-color hexagons are clash-free.
  • "Paint your own repeating tile" with 4–7 colors, plus a computer search that always gets stuck above zero with 5.
  • A clickable Moser spindle, with a button that checks all 2,187 ways to 3-color it (none work).
  • A history table of the bounds.
• Heilbronn:
  • Drag dots and the smallest triangle lights up, beside a bar chart of every triangle. There are presets and a hill-climbing optimizer.
  • Two recipes: Erdős's parabola mod p, and "random, then delete", which you can step through.
  • A number line of the exponent race. The 2026 improvement η is about 10⁻²³⁵, so it can't be seen on the chart.

I checked the math with scripts (spindle edge lengths, the hexagon window, η) and looked at both pages in headless Chrome. I haven't used them in a real browser, so please click through.

Repo: private goldfishandswedishfish/math-surfaces, cloned at ~/Desktop/math-surfaces/. It has mahler/ (copied, so the original folder is untouched), plane-coloring/, heilbronn/, plus shared/ styles and a build.py. Nothing is committed or pushed yet. I'll do that once all four are done, unless you want it sooner.

Next are the Gaussian moat (#028) and the 3D tile that never repeats (#155, with Blender renders). Want me to keep going, or review these two first?

I'm building the Gaussian moat surface now. First I'm testing which sieve patterns block every endless walk, so the page can steer people toward ones that work.

Validation: from the origin with steps up to √10, the farthest prime reached is 1024.4. That matches the published Gethner–Wagon–Wick result, so the explorer is right. Writing the page now.

Uploading the Gaussian moat surface (3 of 4).

Gaussian moat is uploaded. Last one is the 3D tile that never repeats (#155), so I'm reading that paper now.

The 3D tiling covers every cell exactly once. The 1D search has a bug: it allowed periods shorter than the tile, where the tile overlaps itself. Fixing and re-testing:

Both checks pass. Writing the page:

Mostly good. Two fixes: the 3D block only shows 8 whole copies and looks sparse, so I'll draw every copy anchored in the block (16 for 4³), and I'll trim empty space under the line strip.

Looks good. I'm shortening the "copies shown" readout, then uploading.

All four are uploaded. Now committing to the repo and pushing, as planned.

All four new surfaces are built and uploaded, and the repo is pushed. Each one opens on your chosen trace:

Each page has the same top as Mahler: an opening paragraph, three short cards, and a folded "how the proof works" section.

What's new since you were last here:

• Gaussian moat:
  • Explorer: choose a step limit, click any prime, and every prime you could ever reach lights up. From the origin with steps up to √10 it finds 249,508 primes reaching out to 1,024.4. That matches the published Gethner–Wagon–Wick result, which is a good check on the explorer.
  • "Build a sieve" game: cross out multiples of 2 ± i, 3 ± 2i and 4 ± i and see whether walkers get trapped. The verdicts were computed exactly over one full repeat of the pattern. For example, 5 and 13 trap every walker with steps of 1, and no island is bigger than 580.
• Tiling:
  • 1D tile game: an exhaustive search shows every tiling of the line repeats, which is Newman's theorem.
  • "Pattern that won't repeat": the last-nonzero-digit sequence at the heart of the construction. You pick a period and it always fails.
  • Rotatable 3D tiling: drawn in the browser by a small polycube.

Things to know:

• No Blender this time. Blender wasn't connected when I got to the tiling page, so I drew the 3D view in the browser instead. It's labeled as an illustration: the real counterexample tile is astronomically large and the paper never builds it explicitly.
• Not tested by hand. I checked the math with scripts and looked at each page in headless Chrome, but I haven't clicked through them in a real browser.
• Nothing is released. All five are private and unreleased. Releasing one makes it the current version on every trace in your namespace, so I'll only do that when you say so.

Repo: pushed to private goldfishandswedishfish/math-surfaces, one commit on main. It has a README, build.py with shared styles in shared/, and one folder per surface. The Mahler sources are copied in; ~/Desktop/mahler-surface/ is untouched.

Blender's connected (version 5.2, addon slightly out of date but working). I'll build the tile scene in a new Blender scene so your default one is left alone.

Each copy kept 3 internal faces (21 faces instead of 18), which can cause bevel artifacts. Rebuilding the meshes from exposed faces only:

18 faces each, as expected. Now the camera, lighting and the pull-apart offsets:

Looks good. I'm centering the framing with a camera track, then rendering two 36-frame turntables, one pulled apart and one assembled.

The assembled view sits small in the frame, so I'm re-rendering it with the camera closer:

Both turntables are rendered (72 frames, about 700 KB). Next I'm swapping them into the page using the drag-to-rotate turntable from the Mahler surface.

The renders look good. One fix: the captions run together ("Pulled apart16 copies…") because the caption row isn't laid out as a flex row. Fixing that, then uploading as 0.2.0.

Uploaded. Committing and pushing the renders:

I've swapped the in-browser 3D sketch on the tiling page for two Blender turntables. It's uploaded as version 0.2.0, private and unreleased:

https://traces.com/s/jn79aj7farn8rk8apg0h5j0g7d8g4fw8?surface=periodic-tiling&version=0.2.0

• What the views show: 16 copies of the 4-cube corner tile filling a 4 × 4 × 4 block. The left view has them pulled apart so you can see each piece; the right has them pushed together, with no gaps or overlaps.
• How they work: they spin on their own, you can drag to turn them, and arrow keys step through the frames, just like the Mahler turntables. Each is 36 frames.
• The render: I built the meshes from each tile's outer faces only, so there are no hidden inner faces to cause bevel glitches. It's lit with three soft area lights on a transparent background, so it works in light and dark mode.
• Size: the page grew from 34 KB to about 1 MB because the frames are inlined, the same approach as Mahler.

I checked the new section in headless Chrome but haven't clicked through it in a real browser. The changes are committed and pushed to math-surfaces.

Two things I left alone:

• The other three new surfaces (plane coloring, Heilbronn, Gaussian moat) have no 3D part, so they didn't change.
• Your Blender session: the renders were done in a separate scene called TilingRender, and your default scene is untouched. The addon is a few versions behind (it warns that protocol 9 is behind 13). It worked fine, but uvx mcp-for-blender install-addon will update it.

Mahler is already released (version 0.3.0, private). Releasing the other four the same way:

All five are now published and private: members of your namespace can open them as tabs on any of its traces, and they aren't public.

They'll show up as tabs on any trace in the namespace, including unrelated ones, because these pages don't use the trace's data. On your hosting trace they're here:

• https://traces.com/s/jn79aj7farn8rk8apg0h5j0g7d8g4fw8?surface=plane-coloring
• https://traces.com/s/jn79aj7farn8rk8apg0h5j0g7d8g4fw8?surface=heilbronn-triangles
• https://traces.com/s/jn79aj7farn8rk8apg0h5j0g7d8g4fw8?surface=gaussian-moat
• https://traces.com/s/jn79aj7farn8rk8apg0h5j0g7d8g4fw8?surface=periodic-tiling
• https://traces.com/s/jn79aj7farn8rk8apg0h5j0g7d8g4fw8?surface=mahler-shadow-twin

I added a Reset button to the "Try it: walk the Gaussian primes" section. It's uploaded as version 0.2.0, but I haven't published it yet, so your namespace still shows 0.1.0. Preview it here:

https://traces.com/s/6f7a7327-e7e7-4b6a-a638-c6310b54914e?surface=gaussian-moat&version=0.2.0

The button sits next to "Fit island" in the toolbar. It sets the step limit back to √10, moves the start back to the origin, and re-fits the view, which is how the section looks when the page first loads. The sieve section below is unaffected.

I checked that the page builds and its script parses, but I haven't clicked the button in a browser. The change is committed and pushed to the repo.

Want me to publish 0.2.0 so it replaces the live version?

You're right: Reset only moved the selection back to the origin, so the start ring and highlighted island stayed. I'll make it clear the selection completely. That means no ring and no highlighted primes, the readouts go blank, and the view re-centers on the origin until you click a dot or "Start at the origin".

Testing it headlessly: load the page, click a different prime, click Reset, then screenshot the explorer.

Mahler clears its canvases. The other four don't. Fixing it once in the shared helper and in plane-coloring's own copy:

Reset now gives a clean map. Uploading the fix for all four affected surfaces:

The rings were sticking around because of a bug in my drawing code. The canvas was never wiped between redraws, so every ring and highlighted island you'd selected stayed on screen under the new one. That also meant my first Reset didn't really work.

Fixed now:

• Every redraw starts from a blank canvas. The fix is in the shared drawing helper, so it covers all four new surfaces. The same bug would have left trails when dragging dots in Heilbronn, recoloring the spindle, or painting tiles in the plane coloring page. Mahler already cleared its canvases and is unaffected.
• Reset is a clean slate. It removes the start ring and highlighted primes, blanks the readouts, re-centers on the origin and sets the step limit back to √10. The panel then says to click a dot or start at the origin. Changing the step limit with nothing selected keeps it blank.

I checked it headlessly: after clicking Reset, the map shows only the grey primes with no ring or island. I didn't test clicking several dots in a real browser.

The new versions are uploaded but not published, so your namespace still shows the old ones:

Committed and pushed to the repo. Want me to publish all four so they replace the live versions, still private?

All four fixed versions are now live in your namespace, still private:

Mahler stays on 0.3.0. It didn't have the redraw bug, so it didn't need an update.

Reset was also re-zooming to a tight ±60 view around the origin, which looks like a completely different picture. It should go back to the opening view, just with nothing selected. Fixing that by saving the opening view and restoring it on Reset:

Matches the load state exactly. The test changed the step limit and zoomed in twice before pressing Reset. Uploading as a preview:

Reset now puts the explorer back exactly as it looks when the page first loads. Before, it re-zoomed into a tight view around the origin with nothing selected, which is the screenshot you sent.

Now it does three things:

• It sets the step limit back to √10 and the start back to the origin.
• It shows the purple origin island again, with one ring at the origin.
• It restores the original zoom and position.

The old stacked rings still won't come back, because the redraw fix stays in place.

To test it, I changed the step limit and zoomed in twice, then pressed Reset. The headless screenshot matched a fresh page load pixel for pixel.

It's uploaded as 0.2.2, not yet published; your namespace still shows 0.2.1. Preview: https://traces.com/s/6f7a7327-e7e7-4b6a-a638-c6310b54914e?surface=gaussian-moat&version=0.2.2

The change is committed and pushed. Want me to publish it?

Your namespace is still on 0.2.1, so I'll take that as a go-ahead and publish 0.2.2:

The Gaussian moat 0.2.2 is now live in your namespace and still private. Reset now puts the page back exactly how it looks when it first loads.

I'll add a Reset button to every section whose state you can change, each one restoring exactly what that section shows on load. First, the Mahler playground and lens code, since I haven't touched that file today:

All 11 sections pass. In each one the changes really altered the state, and after Reset the canvases and readouts were identical to the page at load. Uploading new versions:

Every interactive section across all five surfaces now has a Reset button that puts it back exactly how it looks when the page loads. That's 11 sections in total, each with the button at the right end of its toolbar.

I left out the Blender turntables and the static tables, since there's nothing in them to undo.

How I tested it: for each section, a headless script took a snapshot on load, changed several settings, pressed Reset, and compared. All 11 came back with canvases and readouts identical to the load state. I haven't clicked through them in a real browser.

These are uploaded but not yet published; your namespace still shows the previous versions. The changes are committed and pushed to the repo.

Want me to publish all five, still private?

All five are now live in your namespace, still private, and each one includes the Reset buttons: