I’m a trainee teacher and one of the pupils asked about this. I thought that there ought to be a perfect/optimum configuration, depending on the size of the sphere and the number of faces. If there is no limit on the smallness of the faces, then I suppose the perfect net would be a tessellating pattern, such as hexagons? But then how would we get it to curve round rather than go on for ever as a flat surface?
The net would not be a tesellation, or as you said, it would just be flat forever.
If you look up, say, the net of a Dodecahedron then you will see ‘cracks’ in the net which allow it to curve back on itself when folded up into the solid shape.
(Dodecahedron net)
http://mathforum.org/alejandre/workshops/dodecahedron.net.gif
To your pupil’s question: For a better approximation of a sphere than a Dodecahedron or Icosahedron, you would have to start using solids with different shaped faces (because there are, of course, only 5 platonic solids).
Using arbitrarily small triangles, you could get an arbitrarily close approximation of a sphere such as in this picture:
http://www.cgal.org/Manual/3.4/doc_html/cgal_manual/Surface_mesher/sphere-surface.png
April 9th, 2010 at 11:11 am
The net would not be a tesellation, or as you said, it would just be flat forever.
If you look up, say, the net of a Dodecahedron then you will see ‘cracks’ in the net which allow it to curve back on itself when folded up into the solid shape.
(Dodecahedron net)
http://mathforum.org/alejandre/workshops/dodecahedron.net.gif
To your pupil’s question: For a better approximation of a sphere than a Dodecahedron or Icosahedron, you would have to start using solids with different shaped faces (because there are, of course, only 5 platonic solids).
Using arbitrarily small triangles, you could get an arbitrarily close approximation of a sphere such as in this picture:
http://www.cgal.org/Manual/3.4/doc_html/cgal_manual/Surface_mesher/sphere-surface.png
References :