Presets
Melting
Melting moves every edge inward at the same speed (an offset of the outline). Drag the depth slider by hand, or let it run.
Shape
Koch iteration of the slab (0 is the plain triangle).
Camera
Layers
Quality
Model checks
The science
Construction
Replace the middle third of each segment by two sides of an equilateral bump. Nn = 3·4n segments of length 3-n (seed side 1): perimeter P_n = 3 (4/3)^n, area A_n = A0 [1 + (3/5)(1 - (4/9)^n)] → (8/5) A0, box-counting dimension D = ln 4 / ln 3 = 1.2619.
The melting model
The slab is thin, so melting from its sides is treated as a 2-D offset: every point of the outline moves inward by the same distance d (a morphological erosion). The remaining area is A(d) = #{points with distance to the edge ≥ d}, computed from an exact Euclidean distance transform of a 1024×1024 raster of the n-th polygon. For small d the area loss rate is the perimeter: dA/dd = -P. A plain triangle of inradius r gives A(d) = A0 (1 - d/r)^2. Longer perimeter at the same area therefore means a faster initial loss, and the fine bumps vanish first.
The checks on this page
(1) Perimeter summed segment by segment and the shoelace area compared with the formulas at the selected n. (2) Raster area at d = 0 against the shoelace area. (3) The raster erosion of the triangle at d = 0.1 against A0 (1 - d/r)^2. (4) The measured initial slope -dA/dd against the summed perimeter (n up to 4, where the raster resolves the bumps). (5) A box-counting fit on the level-7 curve against ln4/ln3.
Honest simplifications
Real ice melts by heat flow, with curvature effects (Gibbs-Thomson), faceting and a thickness that changes with time; none of that is modelled. The pure offset keeps the geometry honest but not the thermodynamics: it is a geometry of edges, not a thaw simulator. Raster resolution (about 1.5 / 1024 side units per texel) blurs bumps smaller than three texels, which is why the slope check stops at n = 4 and n = 5 is flagged. Real snowflakes are six-fold dendrites grown by diffusion-limited deposition, not Koch curves.
What to try
- Pick n = 0 and n = 5 and melt both by the same depth: the fractal loses its fine bumps first, so its area fraction drops steeply at the start (first graph).
- In the second graph the remaining perimeter starts at Pn and collapses as bumps disappear; the dashed line is the initial value.
- Step n up: the third graph shows P growing by 4/3 per step while the area flattens at 1.6 A0.