An Alternative Interpretation of
the Cone
It is possible to view the cone as an
altered plane.
Simply consider a plane with less than a 2pi angle:
An arbitrary angle has been subtracted out of the plane and the
plane
rejoined so that its continuous nature is preserved.
The same process can be done with an angular increase:
This saddle type shape has a definite vertex. Informationally
this
structure contains more than the usual plane.
It does not appear to have an upper limit of angular increase. Here is
a biplane:
The biplane can be folded flat without any need for surgery.
The folding exposes how it was constructed:
There does not appear to be any upper limit on how much angle
may be
added. Here is a triplane:
The triplane can also be folded flat. It is capable of lots of
rotation.
Informationally the triplane contains three times the information of
the standard plane. These instances lay upon a continuum of angle.
According to the definition of curvature these surfaces all have zero
curvature away from the vertex. At the vertex the standard cone has
undefined positive curvature. The inverted forms have undefined
negative curvature at the vertex.
The vertex is a special position on this structure.
Curvature is a concept nearby to the generalized cone yet they seem to
be two different concepts.
The saddle is readily apparent in both yet I have not found any saddle
with a
vertex in existing constructions. If you are aware of such a
construction I would greatly appreciate your feedback.