Imagine what happens if we bend the triangle along the
edges between the colored regions, so that it's not flat any
more. It bends away from the plane that it was in originally.
The 3 corner angles in a "bent" triangle like this don't add up to
180. They add up to something more than 180. Informally speaking,
that's because the triangle doesn't lie in a plane any more; it's
bent, so there's room for wider angles. The actual number that
the angles add up to depends on the amount of bending; the greater
the bending, the greater the sum of the angles.
Since the intersection point between the three colored regions
is where the three bending lines meet, we call that point
the "bending point". The location of the bending lines
is completely determined by the position of the bending point.
As long as the corner angles in our bent triangle are still
nice, an even number of them will fit around
each corner and so we can still make a "tiling" with such a triangle.
What happens in this case is that the triangles curve around and fill
out a solid object in space.
Since there are lots of triples of nice
angles which add up to more than 180 degrees, there are lots of bent
triangles which can be used to make solid shapes like these. The
choices are infinite, but we'll focus on the cases in which the
angles are (90,60,36), (90,60,45), and (90,60,60), because these
cases correspond to "widening" the top angle in the (90,60,30)
triangle we had above.
As we move the bending point in the original triangle,
the lines of bending move, giving rise to different shaped
polyhedra in the final tiling. With certain special positions
of the bending point we get nice symmetric polyhedra
in the final tiling. In fact, this gives rise to all 5 of
the Platonic Solids and 11 of the 13 Archimedean Solids.
You can make your own virtual polyhedra like this one by moving
the bending point around with
You can also make real polyhedra like this with your own