Egg Carton
z = sin(x) · cos(y + t)
Open in the app The dials and keys named below are the app's.
What it draws
The expression is a product of two one-variable pieces. sin(x) depends on x alone and never moves. Its zeros, x = nπ, are straight nodal lines, and the sheet is pinned flat along every one of them. cos(y + t) depends on y and the clock. It slides along −y at 1 unit per second at Speed 1, so the whole pattern repeats every 2π ≈ 6.28 seconds.
Extremes sit where both factors are extreme, at x = π/2 + mπ and y + t = nπ. That is a grid of hills and hollows spaced π ≈ 3.14 apart in each direction. Their signs alternate like a chessboard, which is the egg carton, and each one reaches ±1 before Height scales it.
Why the pattern
Split the product into a sum and the moving parts appear: sin(x) · cos(y + t) = (sin(x + y + t) + sin(x − y − t))/2. Those are two plane waves of equal strength running along the two diagonals. Each has a wavevector of length √2, so each travels at 1/√2 ≈ 0.71 units per second, and their sum solves the two-dimensional wave equation with that speed. Where they meet in step a hill rises, where they meet out of step a hollow drops, and along x = nπ they cancel exactly for all time. A product of a fixed profile and a travelling one is the same thing as two waves crossing. The picture only decides which description is easier to see.
Try
- Top in the deck: the chessboard, with the still nodal lines running the length of the sheet.
- Stop the clock inside the formula, sin(x) · cos(y): a fixed carton, the standing pattern the two crossing waves make.
- Halve the cells in one direction, sin(2x) · cos(y + t): twice as many nodal lines, spaced π/2 ≈ 1.57.
- Pull Mesh to its top stop: the hills and hollows read as one lit solid, and the flat lines between them stand out.