In this video, Christine Breiner from MIT, provides an explanation of the mathematics for a solid with finite volume and infinite area
Infinity can be a hard concept to understand. In this pod cast Steve Patterson and Dr. Danial Isaacson
"cover two definitions for infinity, discuss the metaphysics of mathematics, and cover some basic "intuitionist" concepts"
To help understand and visulize shapes with infinite surface area
and finite volume I have created 2 applets. You can move the 3D shapes
around to get a 360 view of the objects. Torricelli's Trumpet has
sliders and check boxes for a more indepth understanding and the Menger Sponge
provides an example of a fractal solid