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(positive) Curvature.
When we draw a triangle, using our sticky tape, on a plane,
the sum of the angles' value is 180°. This is an euclidean surface. We
will say that it contains no curvature. It is really a flat surface. The sum
of the angles of our triangle is the euclidean sum. When we drew the triangle
on our cone, our "posicone", and when the summit Swas outside it, the sum
was still 180°. Oppositely, when the summit was inside, the sum was 180°
plus the angle q (the
cut we managed to build our posicone, see figure (8)).
This summit is a peculiar point of the surface, a conicalone and we will say that it contains some (positive) concentrated curvature.
Its a concentrated (positive) curvature point.
Now we can manage two cuts, corresponding to angles q1
and q2 . See
figure (13). Then we get some strange surface with two conical points S1
and S2. See figure (14).
(13)
(14)
Now you can draw as many geodesic triangles you want,
correspnding to different cases.
-
If they contain no conical summit, the sum of the angles
is 180°. -
If they contain the summit S1 , the sum is
180° plus q1. -
If they contain the two summits, q1
and q2, the
sum is 180° + q1+
q2
(15)
Imagine now that you can make a large number of tiny posicones
and glue them together, as shown on figure (16). Each tiny posicone corresponds
to an elementary angle Dq
. You can arrange these mini-cones in a regular way. Imean : the distance
between a summit and summits of the neighbours' mini cone would be almost
constant everywhere.
(16)
If your mini-cone get smaller and smaller, as well as
their associated elementary angle Dq
, you will build a portion of regular surface with constant curvature density.
A sphere is a surface with constant local curvature density.
In a simpler way, on says that the sphere is a constant curvature surface.
If you arrange your mini-cones differently, you can build
a variable local curvature density surface . For an example an egg.
The egg of a hen is a variable local curvature density surface. But a ping-pong
ball is a constant curvature density surface. That's so that the hen recognizes
its egg and makes the difference with the ping-pong ball. It draws geodesics
with sticky tape, and so on...
In fact, the hen does not physically figure geodesics
on the object. It does it mentally .
(17)
In general relativity one identifies mess density r
to local curvature.
Of course the general relativity playground is not a 2d
surface. You can imagine a 3d hypersurface. You can imagine a 3d hypersphere.
But who can imagine a 4d hypersurface ?
By the way, the 4d curvature of the 4d hypersurface called
"universe" has special features we are not going to explore here. This shows
that didactical models are limited. But they are good to stimulate the imagination
and to open the mind towards somewhat different worlds.
