1D Chess: collapse the board to a line

1D·CHESS

collapse the board to a line

LEVEL 1

THE MATH

cov C =

eigenvalues λ₁= λ₂=

θ* = ½·atan2(2S_xy, S_xx−S_yy) =

explained var(PC1) = λ₁/(λ₁+λ₂) =

drag to aim the axis, spread the shadows

PRINCIPAL AXIS LOSS

AIM

0°

drag the board or the dial to aim · scroll / ← → fine.
Find the line the shadows spread along most. That is the principal axis.

the line / axis	PC1, the first principal component (top eigenvector of the covariance): the single direction of greatest spread
CAPTURED / CONTROL%	explained variance, the share of the cloud's total spread your line accounts for
the collapse	projection, flattening the board onto one line (2D → 1D)
PC 1 / 2 bars	the scree, how the variance splits across the components
piece weight	each point's mass in the weighted covariance; heavy pieces dominate the axis