Four-square gap records as points

Every whole number is a sum of four squares, n = x² + y² + z² + w² with x ≥ y ≥ z ≥ w ≥ 0 (Lagrange). Pick the most balanced one — the one whose largest part x is as small as possible. Strip off the equal-parts diagonal (1,1,1,1) and what is left is the number's shape: a point in a 3-D shape space where the origin is perfect balance and points drift outward as the four parts grow uneven.

The headline number is the gap = x − ⌈√(n/4)⌉: how far the largest part must exceed the smallest value arithmetic allows. A record is a number whose gap is larger than every earlier gap. The records line up on a handful of family rays from the origin: those are the special numbers whose balanced shape simply doubles when you multiply by 4.

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the number
224
balance gap — how far the largest part overshoots
noticeably uneven
Record numbers — where the balanced decomposition sets a new record for the gap x − ⌈√(n/4)⌉. Click a flag to jump. Flags are coloured by family (its ×4 ray).
Gap 0 means the largest part reaches the lower bound ⌈√(n/4)⌉; exact equal parts occur only at n = 4m², where x = y = z = w = m. For the record families 96·4ʲ, 224·4ʲ, 2816·4ʲ the balanced decomposition doubles under ×4, so each is an outward ray; the gap doubles with it. Other numbers do not scale this cleanly.

Seed-shape shortcut: in this 3-D view the equal diagonal component is removed. The 2816 core then collapses to the same direction as the 96 core: (5,3,3,1) − (1,1,1,1) = (4,2,2,0) = 2·(2,1,1,0). That is why the 96·4ʲ and 2816·4ʲ families fall on the same projected line even though they are different record families.  ·  the gap-record-location sequence · exploratory higher-power studies · about