Gabriel’s Horn and the Painter's Paradox in Perspective

Abstract

Gabriel’s Horn is usually discussed as the painter’s paradox. The horn can hold a finite volume of paint, but its inner surface area is infinite and, therefore, cannot be painted. This may seem counterintuitive at first. In this paper, we provide the following perspective: Any finite volume consists of an infinite number of area layers, which amounts to an infinite surface area. This is shown using an example of a “mathematical” ice cube which melts into an infinitely thin film of infinite surface area. Students can appreciate this before they encounter calculus, which is normally used to establish the painter’s paradox. So, we show a perspective that is accessible to a wider range of students, and which is also applicable to all volumes besides just Gabriel’s Horn.