|Scherk surface model with 3D Polyfelt, fixed on a ZOME cube.|
We have "covered" several surfaces with regular polygons: cilinders, cones, toruses and some hyperbolic surface, like the Scherk surface. It has been a very interesting experience, that we would like to share with other Math teachers of primary or secondary schools.
Students can recognise immediately different types of curvatures depending on the configuration of polygons around each point. One can see that around a FLAT point, the sum of the angles is exactly 360º. Around a PARABOLIC point, like in any convex polyhedra, the sum has to be <360º. And around a HYPERBOLIC point the sum must be > 360º.
The next step could be to find all possibe combinations for each of these three types of points. This would lead, for instance, to the classification of semiregular mosaics, or the Archimedian polyhedra. The concepts of Gauss and mean curvatures could be also introduced, if one has soap bubbles, like in our case.
Another interesting concept that could be treated in such sessions is the Euler characteristic (X= vertices-edges+faces) of the constructed tessellations. One can check if the formulas X=2-2g-r (respectively, X=2-g-r) if the surfaces are orientable (respect. non orientable), where g is the genus of the surface and r is the number of boundary components.
This has been a special session organized by Isabel María Romero ( Departamento de Didáctica, de la Universidad de Almería), David Crespo (del colegio Agave, de Huercal de Almería) and myself, for students of the subject "La geometría y la medida en educación primaria", of the Grado en Educación Primarioa, of the University of Almería.