# Pattern investigations — provenance and scope

These three optional investigations are original implementations of elementary mathematics, inspired by assembl Studio's parametric exploration. They are separate from the [Living Studio creative modes and their source credits](living-provenance.md).

- **Rotation:** repeated geometric petals, with angle `360 / count`.
- **Waves:** two equal-amplitude sine waves, their displacement sum and analytic peak amplitude. This is not a physical vibrating-plate simulation.
- **Regions:** exact two-site nearest-point regions on a flat square, with explicitly represented ties.

The teaching text, JavaScript, styles and canvas diagrams for these investigations were written for assembl Lab. No external renderer, service or third-party Studio algorithm is embedded in these lesson models. Existing Study colours, fonts and Allen artwork are reused. Instrument Sans and IBM Plex Mono use the SIL Open Font License.

## Learning boundaries

These are suggested Years 9–10 explorations; interference and Voronoi regions are extensions. Rotation supports spatial reasoning, with arbitrary-angle symmetry extending beyond the multiples-of-90° example in the Phase 4 teaching sequence. The activities do not cover or assess an entire NZQA achievement standard.

Curriculum references consulted in October 2026:
- [Mathematics and Statistics Phase 4](https://newzealandcurriculum.tahurangi.education.govt.nz/nzc---mathematics-and-statistics-phase-4-years-9-10/5637291579.p)
- [Transforming shapes](https://newzealandcurriculum.tahurangi.education.govt.nz/transforming-shapes/5637187370.p)
- [Tāhūrangi](https://tahurangi.education.govt.nz/our-story)

Geometric patterns are not presented as customary kōwhaiwhai. Cultural contexts require appropriate consultation and attribution.

## Output and privacy

The forty-engine catalogue is a coverage map, not a claim that forty engines are implemented. Three investigations cover related concepts; other families identify proposed paths and existing Study overlap.

These lesson diagrams are event-driven 2D models. Comparisons and optional captions stay in page memory and disappear on reload. PNG creation uses browser Canvas; downloads and optional device sharing are user-triggered. Nothing is uploaded. Printing uses A4 with 12 mm margins; colour depends on printer settings. Mathematical regression tests cover the rotation, wave and region models.