# Cookie Tools construction models

Original mathematical models, schema 1. Not a reproduction of Minecraft world generation.

Cone/pyramid: centered coordinates use c=(D−1)/2. Layer radius interpolates from D/2 to 0.5, or sqrt(0.5) for an even circular base, over H layers. Include a cell when its center lies inside the circle or square. Hollow models keep the bottom, horizontal boundary and cells exposed above.

Spiral: Y advances the angle by 2π/S, where S is the rise per full turn. Each radial band is sampled at at least four points per block-length of arc. A Manhattan supercover joins rounded samples, so adjacent layers share their endpoint and horizontal paths stay connected. Width counts radial bands. Check headroom separately.

Roof: symmetric height f(x)=floor(min(x,span−1−x)×rise/run). A shell contains one block per (x,z) position. Filled construction includes all Y from 0 through f(x). Count for a shell is span×length; count for a solid roof is length×Σ(f(x)+1).

Bounds prevent excessive allocations. All counts are exact for these discrete definitions; a blueprint is not a promise about collision clearance or stair/slab orientations.