# Introduction be pointless as the holes are infinitely small,

Introduction Gabriel’s horn, the horn which the Archangel Gabriel blows in to announce the day of judgement.

The horn is a shape which is said to have an infinite Surface area but a finite volume. In the Bible, the horn serves as a bridge between divinity and mortality, in mathematics it serves as a bridge between infiniteness and finiteness. The shape was first discovered in 17th century by Evangelista Torricelli who was influenced by Galileo Galilei who was working as Benedetto Castelli’s apprentice. The shape is also known as Torricelli’s trumpet.  The paradox arises when the question of how much paint is needed to paint the horn. Looking at the surface area the answer would be infinite but looking at the volume of the shape there is a finite answer. To answer this question Calculus must be used to find the volume and the surface area.

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http://mathworld.wolfram.com/images/eps-gif/GabrielsHorn_800.gif Rationale  As a music enthusiast, I find this paradox particularly interesting as it allows me to explore a shape of a very famous instrument, the trumpet. The paradox is also very interesting because there are several challenges which arise if someone would like to play the instrument.

The shape is infinitely long so there is no end to blow in, and even if there was you wouldn’t be able to reach it as it is infinitely long, even if you could reach the hole it would be pointless as the holes are infinitely small, and even if you could blow in the whole it would take infinite time for the sound to come out.  Aim To explore Gabriel’s horn and the paradox of a shape with an infinite surface area and a finite volume.