Three-square spread records as points

Write a whole number as three squares, n = x² + y² + z², and read (x,y,z) as a point in space — not a brick. Every representable number's most balanced triple is a dot in the x ≥ y ≥ z ≥ 0 corner. The cube direction (1,1,1) is "perfectly balanced"; a number's imbalance is the spread x − z between its largest and smallest part. Which way a dot leans — toward an axis (flat) or a corner — is its shape.

A dot's direction is its shape (x,y,z)/√n; its distance from the origin grows with size (log n). So each "× 4" chain k, 4k, 16k, … marches outward along one ray — that fixed direction is the proved doubling law, made visible. A dot with z = 0 rides out along the floor (x–y plane); y = z = 0 rides the x axis. No flat-brick paradox — a zero is just a point on a wall.

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the number
445
imbalance — spread x − z
shape — how far it leans from the cube line
noticeably leaning
Plot your own point: , ,
Record numbers — where the best triple's spread x − z beats every smaller number, setting a new record for the absolute spread. Click a flag to jump. Flags are coloured by family (its doubling ray).
The cube line is reachable (imbalance 0) only when n = 3·m² (then x = y = z = m). Multiplying n by 4 doubles the imbalance (the spread x − z) but leaves the direction (shape) unchanged — the dot steps outward along the same ray; press ×4 to watch. Numbers of the form 4a(8b+7) (7, 15, 28, …) are not a sum of three squares at all.  ·  the record-locations sequence · about