In October 2019, BAMM hosted an special guest, Maria Garcia Monera, from the University of Valencia. During her visit, she participated in several outreach activities. She kindly donated her surfaces paper models, which are on display at the lobby in the Mathematics Tower. The templates for the models are available on her website.
At the end of the 19th century Felix Klein (1849-1925) and Alexander Von Brill (1842-1935) designed hundreds of surfaces using different materials like paper, wire or plaster. The purpose was to help their students visualize mathematical concepts. In the sliceforms, each slice is the cross section between the surface and a plane.
Most of the models are built by using a technique developed by John Sharp. The pieces for the model are the intersections of the surface with two families of equally spaced parallel planes. The planes in one family are perpendicular to the other.
- Sphere x2 + y2 + z2 = 182
- Cone x2 + y2 = z2
- Paraboloid x2 + y2 = z
- Hyperboloid of one sheet x2 + y2 – z2 = 1
- Hyperbolic paraboloid z = xy
The “IMAGINARY – through the eyes of mathematics” traveling exhibit featured professor Herwig Hauser’s classic algebraic surfaces. These were chosen “in such a way that equations are simple and beautiful; figures are plain and natural and show interesting geometrical facts”. We modeled some of these surfaces with the sliceform method.
- Dullo (apple) (x2 + y2 + z2)2 = x2 + y2
- Heart (x2 + (9/4)y2 + z2)2 – x2z3 – (9/80)y2z3 = 0
- Nepali (xy – z3 – 1)2 = (1 – x2 – y2)3
- Zitrus x2 + z2 = y3(1 – y)3
- Zeck (water drop) x2 + z2 = z3(1 – z)
Some other models are surfaces of revolution, designed by rotating a piece around the surface’s rotation axis. This piece gives some characteristic property to the surface.
- Ruled hyperboloid of one sheet. The rotating piece is a rectangle, which shows that the surface is ruled.
- Torus. The rotating piece is the moon obtained when intersecting the surface with a bitangent plane. ((x2 + y2)1/2 – 2)2 + z2 = 1
It can also be interesting to create a model with non parallel planes. The following model was designed using a tanget plane and rotating it around a line contained in it. It illustrates Meusnier’s Theorem: All curves on a surface passing through a given point p and having the same tangent line at p also have the same normal curvature at p and their osculating circles form a sphere.
The same can be done with a plane that is a certain distance apart from the surface. In the cone model, the plane was rotated in such a way that each piece shows a different conic.
- Cone. In this model, each slice corresponds to a different conic section.
Maria Garcia Monera has a Mathematics degree from the University of Valencia and a PhD on Differential Geometry from the Polytechnic University of Valencia. Her doctoral work was on “r-Critical Points and Taylor Expansion of the Exponential Map for Smooth Immersions in Rk+n“. She is currently an Associate Professor at the University of Valencia and works teaching in high school.
On 2011, María and University of Valencia professor Juan Monterde, started working on sliceforms. Since then, they have designed many models using different techniques.