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Negacones.
Let us now build 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 form a triangle with three of them. If you measure the sum of the angles, you will see that it is equal to 180° - q. We will say that it defines a concentration of negative curvature.
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 formed by the three geodesics does not contain the summit S, which contains all the (negative) angular curvature, the sum will be that of Euclidean geometry: 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 neighboring summits. Then you would obtain a surface with constant negative curvature density : a horse saddle. But this surface will never close.
In general, the geometer calls it a surface with constant negative curvature. (32)
"Blunt negacone".
In a previous section, using a surface with constant positive curvature density (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 ensure 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 construction.
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 it is difficult to see how to roll a negacone on a flat plane. It is therefore easier to roll the plane on a negative curvature matrix.
Gutenberg invented the printing technique. A design in relief is carved on a plane. Then ink is applied and it is pressed onto a plane.
Later, the printing matrix was converted into a cylinder, for newspaper printing (rotary press).
But no one, as far as I know, has used the conical press.
In any case, the important point is to bring the two surfaces into contact, regardless of the method. Either you move the matrix, or you roll the paper (the plane surface).
As shown in figure (34), you can use a conical matrix to print something on a plane. Some conical newspapers, laid flat. (34)
It is not definitively certain that no one will ever use this. Suppose you want to produce dresses, with a special design corresponding to conical symmetry. Suppose you have to produce thousands of such dresses. You could engrave the design on a conical matrix, then use it to print on the fabric. The customer could buy it and make the "conical" dress, being sure that the pattern obtained would be correct everywhere.
On figure (35) is what you get when printing with a negative curvature matrix. On the right, a negacone laid flat. (35)
On figure (36) is shown how to join the horse saddle to the portion of negacone.
By the way, you may ask:
- How can I measure the negative angular curvature contained in my horse saddle?
In some places in Texas, near mathematics departments, when you buy a horse saddle, the corresponding angular curvature is indicated on the attached ticket. If not, by comparing the perimeter of the edge, or the area, to the Euclidean value calculated from the radius of this negative curvature disk, you can deduce the corresponding angular curvature. Consider this as a fruitful exercise. (36)
(37)
Now we can use our sticky tape, draw geodesics and project them onto a plane, as shown in figure (38).
(38)
As usual, this plane projection refers to our "mental world", the wall of Plato's cave. The appearance 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 reality, all of this should stem from the underlying geometry.