Scale Boundary

The Shape Blowup

Square layers make the growth visible. The book renders a finite grid of exact integer rows, not a smooth curve.

highlighted = computed this step

A finite exact grid

For a square layer with d inputs and d outputs, the count is d times d plus one. The table samples only explicit integer sizes; it is not a smooth curve.

params(d)=d(d+ 1)\text{params}(d)=d(d+\,1)
Shape blowup gridDiscrete exact integer counts.finite square-layer grid, not a smooth curvedd*(d+1)26420872162721001010010001001000discrete exact rows onlyexact mechanisms on toys; scale blocks hand enumeration of real trained values; finite grid only;NOT learning; NOT generalization

The counts jump

At d=2 the count is 6. At d=4 it is 20. At d=8 it is 72. At d=16 it is 272.

2↦6,4↦20,8↦72,16↦2722\mapsto6,\quad 4\mapsto20,\quad 8\mapsto72,\quad 16\mapsto272
Shape blowup gridDiscrete exact integer counts.finite square-layer grid, not a smooth curvedd*(d+1)26420872162721001010010001001000discrete exact rows onlyexact mechanisms on toys; scale blocks hand enumeration of real trained values; finite grid only;NOT learning; NOT generalization

The larger rows

The same exact formula gives 10100 at d=100 and 1001000 at d=1000. These are still exact integer rows, not interpolated values.

100↦10100,1000↦1001000100\mapsto10100,\quad 1000\mapsto1001000
Shape blowup gridDiscrete exact integer counts.finite square-layer grid, not a smooth curvedd*(d+1)26420872162721001010010001001000discrete exact rows onlyexact mechanisms on toys; scale blocks hand enumeration of real trained values; finite grid only;NOT learning; NOT generalization