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Concept
Gabriel's horn
Summary
A Gabriel's horn (also called Torricelli's trumpet) is a type of geometric figure that has infinite surface area but finite volume. The name refers to the Christian tradition where the archangel Gabriel blows the horn to announce Judgment Day. The properties of this figure were first studied by Italian physicist and mathematician Evangelista Torricelli in the 17th century.
These colourful informal names and the allusion to religion came along later.
Torricelli's own name for it is to be found in the Latin title of his paper De solido hyperbolico acuto, written in 1643, a truncated acute hyperbolic solid, cut by a plane.
Volume 1, part 1 of his Opera geometrica published the following year included that paper and a second more orthodox (for the time) Archimedean proof of its theorem about the volume of a truncated acute hyperbolic solid.
This name was used in mathematical dictionaries of the 18th century, including "Hyperbolicum Acutum" in Harris' 1704 dictionary and in Stone's 1726 one, and the French translation Solide Hyperbolique Aigu in d'Alembert's 1751 one.
Although credited with primacy by his contemporaries, Torricelli was not the first to describe an infinitely long shape with a finite volume or area.
The work of Nicole Oresme in the 14th century had either been forgotten by, or was unknown to them.
Oresme had posited such things as an infinitely long shape constructed by subdividing two squares of finite total area 2 using a geometric series and rearranging the parts into a figure, infinitely long in one dimension, comprising a series of rectangles.
Gabriel's horn is formed by taking the graph of
with the domain and rotating it in three dimensions about the x axis. The discovery was made using Cavalieri's principle before the invention of calculus, but today, calculus can be used to calculate the volume and surface area of the horn between x = 1 and x = a, where a > 1.
Official source
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