Opposite of the sphere question, What geometric shape has the smallest volume and largest surface area? ?
The shape that has the largest surface area per unit volume of substance is a flat circular disc that has a thickness that approaches zero. Practically it is a disc only one atom, or one molecule thick.( because it cannot be made thinner than that.). This would be an extreme type of engineering feat. Remember the disc will present a surface area on both sides. Therefore total surface area is 2(pie)r^2, where r = radius of disc and pie = 3,14159. However, A sqaure or rectangle can be shown to have the same total surface area as the disc if it is one atom thick. In fact any flat shape one atom thich would have the same maximum surface area. Therefore there are an infinite number of shapes (mathematically). The only condition being that is it flat and one atom thick. Gold is most likely the best material to make this out of since it can apparently be beaten to a thickness of less than one micron. Thats my thoughts on it. The limit being set by the atomic or molecular dimentions of the sub
