Temperature Gradients from a Woven Grid

A thermocouple lace isn’t really a thermometer array — it’s a gradient sensor. Each junction reports temperature, but the information is in how temperature changes from junction to junction. That spatial derivative tells you where heat is flowing.

Numerical differentiation from discrete samples is older than digital computers, but the 1970s made it cheap. The central difference formula approximates the derivative at a point by looking at neighbours: f’(x) ≈ (f(x+h) - f(x-h)) / 2h. For a 2D lace, you need both directions.

# Temperature readings from 3×3 lace (°C)
temps <- matrix(c(22.1, 23.4, 24.8,
                  21.8, 22.9, 24.2,
                  21.3, 22.1, 23.3), nrow=3, byrow=TRUE)
spacing <- 0.02  # 2cm between junctions

# Gradient at center junction
dx <- (temps[2,3] - temps[2,1]) / (2 * spacing)
dy <- (temps[3,2] - temps[1,2]) / (2 * spacing)
cat(sprintf("Gradient: %.1f°C/m east, %.1f°C/m south\n", dx, dy))
cat(sprintf("Temperature rises: %.1f° from north\n", atan2(dx, -dy) * 180/pi))

Output: Gradient: 60.0°C/m east, -32.5°C/m south — temperature rises northeast at about 62° from north. The lace tells you which way the heat gradient points, not just how hot it is.

program LaceGradient;
var
  temps: array[1..3, 1..3] of real;
  dx, dy, angle: real;
begin
  temps[2,1] := 21.8; temps[2,2] := 22.9; temps[2,3] := 24.2;
  temps[1,2] := 23.4; temps[3,2] := 22.1;
  
  dx := (temps[2,3] - temps[2,1]) / 0.04;
  dy := (temps[3,2] - temps[1,2]) / 0.04;
  angle := arctan(dx / -dy) * 180 / 3.14159;
  
  writeln('dT/dx: ', dx:6:1, ' C/m');
  writeln('Gradient angle: ', angle:5:1, ' degrees');
end.

The Pascal version does the same math with arrays indexed from 1, the way FORTRAN scientists expected. Both programs turn a grid of numbers into a vector field. Finite differences let you see flow patterns that individual readings hide — which matters when your thermocouples are woven into fabric and you’re trying to understand where the iron’s heat is spreading.