| 4 |
Didactic image of a heavenly body ( star,
planet, dense egg)
A star like the Sun is a mass-concentration.
Around : the void, or a portion of space that is "almost empty", for it contains
very rarefied gas and photons. In 2d, the corresponding didactic image is
a blunt (posi) cone :
(18)
You can make it with two components. A portion of a sphere
and a portion of a (posi) cone, glued together. The portion of a sphere is
a constant curvature density surface. The portion of the cone is a flat surface,
a zero local density curvature surface. This last example is an euclidean
surface. The portion of the sphere is a non-euclidean surface (a riemanian
surface).
This is the 2d didactic image of a constant density r
object, surrounded by void.
How to fix the two elements together, in order to ensure
the continuity of the angent plane ? It is simple. Your portion of a cone
comes from a cone whose cut corresponded to an angle q
. Your portion of sphere is supposed to be built with elementary mini-posicones,
so that it contains a certain "amount of angular curvature" q.
If the two angles are equal, the tangent plane will be continuous.
But how to measure the amount of curvature contained in
a given portion of a sphere ?
Total curvature.
We can build a surface, joining elementary posicones.
We can arrange it to get a constant curvature density surface. Then we know
that the surface is a portion of a sphere. If we add more and more elementary
(posi) cones this sphere will be complete. It contains a certain amount of
angular curvature. All the spheres contains the same. The total angular curvature
of a ping-pong ball and the total angular curvature of the earth are equal,
although they have very different weight.
By the way the total curvature of an egg is the same,
for they have the same topology. In principle, hens make eggs with spherical
topology. Personally Ihave never seen an egg with toroidal topology. It would
correspond to some strange snake,with no head, nor tail, or something like
that.
Let us return to ping-pong balls, normal spheres. If this
surface has a constant local angular density it means that the amount of angular
curvature ( the sum of elementary angles Dq )
will be proportional to the area. See figure 19. This area can be limited
by any kind of border. But we can use geodesics of the sphere. Call Sthe
area of the sphere and s the grey area, inside the triangle.
(19)
Above, we saw that the (positive) depart from the euclidean
sum (180°), for a triangle drawn on a surface, depending on the number
of cone's summits that were inside. The sum was 180° plus all the angles
corresponding to those enclosed summits.
Conversely, if Imeasure the depart from the euclidean
sum Ican measure the amount of curvature contained inside the triangle.
A geodesic of a sphere is called a grand circle of the
sphere. See figure (20). Meridians, equator, are grand circles of the sphere.
(20)
We can cut our sphere into eight equal area pieces. See
figure (21). We get eight triangles whose all angles' values are 90°.
Then the depart from euclidean sum is 90° . Each of these triangle contains
an angular curvature equal to 90°. As a conclusion the total curvature,
the total angular curvature of the sphere is 8 x
90° = 720° = 4 p.
(21)
Each grey triangle contains p/2
.
Do you enjoy curved surfaces, riemanian surface geometry
?
If we return back to our blunt cone we see that the angular
curvature is contained inside the circular border, in the constant curvature
density area. The cone's flank, wall, is not a limited surface. You can extend
it to infinite if you want. The amount of angular curvature does not depend
on the perimeter of the border, of the area of the portion of a sphere. This
last can be reduced. See figure (22). Even reduced to a simple point where
it will contain the same amount of angular curvature. That's why we say that
a conical point was a concentrated curvature point. Conversely we can build
smooth surfaces with a set of conical points.
Matter is made of atoms. Atoms can be considered as point-like
objects. They are "concentrated curvature points" in 3d space.
The air you breath is a constant density medium. It is
made of molecules, atoms. It is a set of concentrated curvature points, linked
by euclidean portions of space. You assimilate that to constant curvature
medium.
Next time you breath, think about it. .
(22)
