Significance and Application

With what may seem like limited mathematical tools Torricelli was able to prove the function f(x) = 1/x 1 ≤ x ≤ ∞ and rotated about the x axis creates a solid with infinite surface area and finite volume. This acute hyperbolic solid, known as Torricelli's Trumpet/Gabriel's Horn created a frenzy not only in mathematical world but in the philosophical world as well. Scholars from both areas debated, denied and defended the idea with some believing it was genius others thinking it was nonsense. Regardless of the opposition this discovery pushed mathematics and the idea of infinity in a new direction. Philosophers began debating what infinity was, redefining and building new ideas. Mathematicians began pushing the limits of their current understanding testing, proving and developing new math. Whether you agree or disagree with the existence of acute hyperbolic solids you can't deny the very concept help push math into uncharted territories and led the way in new discoveries.

Today acute hyperbolic solids have a class of their own classified as super solids by William P. Love. He classifies these into five different areas, Super cone, Super cube, Super pyramid I, Super pyramid II, and Super sine solid. Using basic infinite series and standard geometry the first four shapes can be easily created.

So what does all this mean today?

The acute hyperbolic solid and super solids are fascinating object to study. They go beyond our natural understanding of geometric shapes by introducing the paradoxical idea of infinite surface area with finite volume. Introducing students to these shapes and having them compute the surface area will introduce an element of creativity. It will allow the students to think outside the box using calculus techniques to create their own acute hyperbolic solid/super solid. According to Love "most solids can be transformed into super solids if you wrinkle their surface enough. Fractal curves can be revolved to form fractal solids."

These solids provide a good opportunity for math teachers as well. Because these shapes are unique and unordinary. Students may find it fun and exciting to investigate the different properties associated with Torricelli's Trumpet, The Super Cone, or any super solid they create. It may even challenge some to push the limits of their knowledge, challenge current mathematics and become the next great mathematician.