Multiplying Gaussian humps — interactive experiment

Numerical experimentation · an exact product identity

Two landscapes, one product

Move the Gaussian bumps and watch their overlap. Multiplication sharpens the peaks and suppresses bumps whose centers are far apart.

Bumps in each mixture 5 n bumps in each factor give n² terms in the product. Move the bumps 0.18 A smooth loop; widths and coefficients stay fixed.

Scale each colorbar to its own sampled maximum Paused

First mixture f

5 humps

Sampled max —

Second mixture g

5 humps

Sampled max —

Their product fg

25 humps

Sampled max —

Each colorbar has its own scale; compare its numbers to see the product’s smaller amplitude.

Terms in the exact product: 25
Sum of product coefficients: —
Identity residual (sampled): —

The humps are exp(−|x − a|²/(2s²)), with height one before multiplication by their coefficient. Positive coefficients sum to one in each factor; these are not probability-density normalizations. Every panel uses the same square [−2.5, 2.5]² with equal axis scales. Smooth colors sample a 181 × 181 grid; contours occur at 20%, 40%, 60%, and 80% of the displayed colorbar. The residual compares the n²-term Gaussian expansion with pointwise multiplication on a separate 17 × 17 grid.