Two infinitely tall towers. At ground level their inner walls stand at x = +1 and x = −1; with height y they lean inward along an envelope ±g(y), g(0)=1, g → 0. Each wall is a fractal curve riding on that envelope. The towers meet only at infinity. The object is the spire of space between them: infinite height, width shrinking without ever reaching zero.
Construction. Left wall x = −g(y)·(1 + ρ·T2(y)), right wall x = +g(y)·(1 + ρ·T3(y)), where T2 and T3 are Takagi functions with bases 2 and 3 (period one storey). The walls are mirror-symmetric in envelope but not congruent: different fractal, same ladder. Because the roughness is proportional to g, every storey looks like the last once the x-axis is rescaled by 1/g — which is what “Track the gap” does. Switch it off to see the true spire.
Meeting at infinity. The width 2g(y) is positive at every finite height. With exponential decay the gap's total area is 2/ln(1/q): a finite region of infinite height. With power-law decay and p ≤ 1 the area diverges while the width still goes to zero — the towers converge but the space between them is unbounded.