The badge shapes
Claim
Both badges were resolved into exact-arc silhouettes rather than traced outlines, and the 24-point exclusion table was corrected in the process.
Construction
A badge is the centre tile surrounded by a fixed set of circles, with a defined set of arrangements around it. Fill every arrangement at once and the union has an outline: the badge shape.
The 16-point badge has sixteen circles lettered A to P around the centre and sixteen arrangements, one circle-and-diamond fusion each. The 24-point badge was built by expanding the circle grid, keeping 21 circles relettered A to U, discarding the rest, and lettering the twelve diamond gaps a to l; its 24 arrangements are lettered A to X in a sequence of their own, a naming collision the reference flags explicitly.
What it establishes
The silhouettes are exact. Rather than tracing the union, every point on its boundary was matched back to the circle that produced it and the outline reconstructed from real arcs. That is the difference between a drawing that looks right and one that can be scaled, cut or cast without drifting.
The work turned up a structural observation worth keeping: only some circles are ever anchors — the major-arc circle of an arrangement — and the more often a circle is used as an anchor the more completely it ends up interior. Lightly used circles still expose real boundary arcs.
The 24-point exclusion table was worked out by hand and then checked systematically against the rule that two arrangements conflict when they share a circle or a diamond. One missing mutual exclusion was found and corrected.
Note
Which of the two silhouettes do you actually like? They are different objects and the library treats them as one job.
Open questions
- The index names the corrected 24-point exclusion as B–X. The reference records that one gap was found and fixed but does not name the pair, so the specific letters are uncited here.
- The 16-point silhouette currently lives in the interactive model as a traced path mapped in from a drawing at r = 65.3. The exact version and the embedded version are two different objects.
- No file on hand carries the 24-point silhouette's own check, only the statement that it was derived the same way.
Checks
| What | How | Source |
|---|---|---|
| Exact-arc reconstruction | Boundary points matched back to their source circles and rebuilt from real arcs, both badges | nuna_master_reference.md |
| Anchor behaviour | Circles used as anchor more often end up fully interior | nuna_master_reference.md |
| 24-point correction | Exclusion table hand-built, systematically rechecked, one missing mutual exclusion found | nuna_master_reference.md |
| Naming collision | Circles A–U and arrangements A–X are separate sequences | nuna_master_reference.md |
| Traced copy in use | Silhouette path in the model is mapped from r = 65.3 into r = 16 space | nuna-badge-interactive (1).html |