Gabriel’s horn
Rotate \(y=1/x\), for \(x\geq 1\), around the x-axis. The view shows the horn through \(x=10^6\). Move the cutoff plane to select how much horn and coating to measure, and adjust the coating thickness. All dimensions use the same arbitrary unit.
3D horn · drag to rotate
The slider moves the cutoff plane along the full horn. Its scale is logarithmic.
Thickness is measured radially: the outer radius is 1/x + t.
What changes as the horn extends?
\[V_{\mathrm{inside}}(L)=\pi\int_1^L\frac{dx}{x^2}=\pi\left(1-\frac1L\right).\]
\[V_{\mathrm{coat}}(L,t)=\pi\int_1^L\left[\left(\frac1x+t\right)^2-\frac1{x^2}\right]dx=2\pi t\ln L+\pi t^2(L-1).\]
For every fixed \(t>0\), the coating volume grows without bound as \(L\to\infty\), while the inside volume tends to \(\pi\).
Growth across cutoffs
Horizontal axis: log₁₀ L. Vertical axis: volume in cubic units, logarithmic scale; values below 10⁻² sit on the bottom edge. The dashed horizontal line marks π.