VI. CASE STUDIES
Consequently we will analyze each study case with their respec-
tive topologic approach. As it was stated the architectural roofs to
study will be: Dome, toroidal vault and free form surface.
6.1. ARCHITECTURAL DOME
Exist many ways and technics to design a Dome including its
structural efficiency, such as the funicular shapes. Because of the
second framework research we will present mathematical-geo-
metrical models which are homeomorphic to a sphere.
Figure C1: according with the Hairy Ball theorem states that there is no
nonvanishing continuos tangent vector field on dimensional spheres. “Every
smooth vector field on a sphere has a singular point”. [Petit, 1999] It was first
proved in 1912 by Brouwer. That means that in a sphere always there are some
singularities.
6.1.1. POLYHEDRAL SURFACES AND TOPOLOGICAL
ASPECTS
Polyhedron is a solid with three dimensions with flat polygonal
faces, straight edges and sharp vertices. A polyhedron is homeo-
morphic to sphere. This geometrical models include the platonic
and archimedian (Figure B6). It is worthy to mention that it
would be possible to create a structural membrane with an hybrid
tensile system properties, but for this document we will choose
the basics and also the solids which are related to the Euler-Poin-
care formula.
Figure C2: Left: shape with genus 2. Rigth: shape with genus 0. The euler char-
acteristic is a topological invariant that describes a topological space´s shape
or structure regardless of the way is bent. For any polyhedron that doesn´t
intersect itself is always 2. We can consider any polyhedron homeomrofic to a
shpere.
Figure C3: polihedron examples. Above: Platonic solids. Below: Arhimedean solids.
- Petit, J.P. 1999. Le topologicon.
17