Gabriel's Horn: A journey In the pursuit of a solid with finite volume and infinite surface area. How can that be? Can we build such a model?
Dr. Andrew Caglieris - known as "Dr. Cags" on campus - is a member of the Math Department at Peddie, where he teaches a number of advanced math courses, including AP Calculus AB.
Following our successful venture two years ago in which students in AP Calculus BC ventured to the Digital Fabrication Lab to build models of solids with different known cross-sections (squares, isosceles right triangles, rectangles, and semicircles) using the laser cutter machine, students in AP Calculus AB recently returned to the newly located Digital Fabrication Lab to build models of Gabriel's Horn, a solid formed by revolving an unbounded region about the x-axis using the 3D printer.
Building on their knowledge relating to volumes of solids of revolution using the method of disks, students first explored analytically the concept of revolving an unbounded region about the x-axis leading to the development of the concept of improper integrals. Through the traditional analytical tools of calculus students were able to establish that the solid generated would have a finite volume, and an infinite surface area. Along the way students developed the notion of calculating both the arc length of a curve as well as the area of the corresponding surface of revolution.
With the theory under their belts, the students proceeded to the Digital Fabrication Lab to seek to produce models of this solid with the fascinating properties described above. The images tell the full story and add a new dimension from the perspective of bringing to life the beauty of mathematics from theory to three dimensional model. Pretty amazing!!
Looking forward to further mathematical explorations in the Digital Fabrication Lab next year.