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a103

(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 S was 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 *conical *one 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. I mean : 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.