| 6 |
Negacones.
Let us build now what we will call a "negacone". To build
a posicone, we removed a sector from a plane. Here we add one, corresponding
to an angle q :
(30)
On this surface we can draw geodesic lines, with our sticky
tape and make a triangle with three of them. If you measure the sum of the
angles, you will see that it is 180°-q
. We will say that it defines a negative curvature concentration.
There are objects with negative curvature in your home.
Some seats, for example : (30 bis)
If we take a disk we get the figure (31) :
(31)
Of course, if the triangle composed by the three geodesics
does not contain the summit S, which contains all the (negative) angular curvature,
the sum will be the euclidean sum : 180°.
The horse saddle.
You can build a great number of elementary
negacones
with angle - Dq
and join them. You can do that in order to have an almost constant distance
between two neighbour summits. Then you would build a constant negative
curvature density surface : a horse saddle. But this surface will never
get closed.
In general the geometer call it a constant negative
curvature surface.(32)
"Blunt negacone".
In a precedent section, using a constant positive curvature
density surface (a portion of a sphere) and a portion of posicone we built
a blunt posicone.
Similarly we can build what we will call a blunt negacone.
We have to join a horse saddle to a portion of negacone, along their common
circular border. In order to manage the continuity of the tangent plane the
(negative) curvature contained in the horse saddle must be equal to the negative
curvature involved in the negacone building.
It is relatively easy to build the required portion of
a negacone !
(33)
NB : As the posicone, the negacone can be used as a printing
matrix. But we can hardly see how to roll a negacone on a flat plane. It is
therefore easier to roll the plane on a negative curvature matrix.
Gutemberg invented the printing technique. A design in
relief is carved on a plane. Then one puts ink on it and presses it onto a
plane.
Later the printing matrix was converted into a cylinder,
for newspaper printing (rotative press).
But no one, as far as Iknow, used the conical press.
In all cases, the important point is to put the two surface
into contact, whatever one does. Either you move the matrix, or you roll the
paper (the plane surface).
As shown on figure (34) you can use a conical matrix to
print something on a plane. Some conical newspaper, put flat.
(34)
It is not definitively sure that nobody will use that,
someday. Suppose you want to produce robes, with a special design, corresponding
to conical symmetry. Suppose that you have to produce thousands of robes like
that. You could engrave the native design on a conical matric, then use it
to print on the material,or the fabric. The customer could buy it and make
the "conical" robe, being sure that the obtained patter would be good, all
over.
On figure (35) is what you get if you print with a negative
curvature matrix. On the right a negacone put flat.
(35)
On figure (36) it shows how to join the horse saddle to
the portion of negacone.
By the way, you may ask :
- How can Imeasure the negative angular curvature contained
in my horse saddle ?
In some places in Texas, close to mathematical departments,
when you buy a horse saddle the corresponding angular curvature is indicated
on the joined ticket. If not when you compare the perimeter of the border,
or the area,to the euclidean value, calculated from the radius of this regative
curvature disk, you can deduce the corresponding angular curvature. Consider
this as a fruitfull exercise.
(36)
(37)
Now we can use ou sticky tape, draw geodesics and project
it on a plane, as shown on figure (38).
(38)
As usual this plane projection refers to our "mental world",
the wall of Plato's cavern. The aspect of the projected geodesic would mean
for us that some repulsive force is acting on our reference objects, for example
a repulsive gravitational force . In fact, all that should come from
underlying geometry.
