V08·Act II · Space
From port Gram to Euclidean metric
A 12x12 Gram heatmap dissolves and reforms as the solid it encodes, with the exact icosahedral angles drawn on it.
The relation on screen
Restricting the port inner product to the surviving triplet gives a positive-definite quadratic form, which is a metric.
Symbols
- Gram matrix of port vectors
- the induced spatial metric
computing the Gram
what to look for · Distances are read out of overlaps. No ruler was put in by hand.
Consensus to physics
- 01the limit projector defines a Gram matrix on the 12 ports
- 02the Gram form has unit diagonal and exact icosahedral angles
- 03its range with the induced inner product is a Euclidean 3-space
- 04distance between records becomes meaningful
In standard physics
The birth of a metric tensor
Up to here 'far apart' meant 'many seams away'. Now it means an inner-product distance, and the map from combinatorics to geometry is an isometry fixed up to port relabelling.
boundary · No physical length scale. The metric fixes ratios and angles; metres are nowhere in sight.