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The horse saddle.
...One can (at least in thought) manufacture a large number of mini negacones and join them to one another. If the vertices are distributed in a roughly regular manner over this surface, one will manufacture a surface element with constant negative curvature density. But it won't be like with the sphere. That surface will not close back up.
...Let us imagine that we have manufactured a piece of surface with constant negative curvature density (we shall say with constant negative curvature). One can take any point, a nail, a string, and trace a circular contour, obtaining what is called a horse saddle (surface with constant negative curvature).
The blunted negacone.
...Earlier, we had manufactured a blunted cone (a "blunted posicone") starting from a spherical cap and bordering it with a truncated cone. We had even found how to go about it so that there would be no discontinuity of the tangent plane. The angle that had served to manufacture the cone from which this truncated posicone was extracted had to be equal to the quantity of angular curvature contained in the spherical cap.
...There is a way to measure the quantity of (negative) angular curvature contained in a horse saddle, by comparing the perimeter and the radius, an operation that is performed rather rarely in saddleries. Let us suppose that we know it. We can manufacture the truncated negacone accordingly.
NB: The negacone is, like the cone, a developable surface. That means that one can "apply" it onto a plane. But the operation seems more delicate. It is hard to see how one could easily roll a negacone on a rigid plane.
...Rather than a rolling operation, it is preferable to envisage an operation of application, or more tellingly, of printing. What does printing mean? It is applying a surface bearing inked reliefs onto another surface.
...In Gutenberg's time, one pressed one plane onto another plane. When one prints a pattern on a fabric, one rolls a cylinder over it. One runs a ribbon of paper between two cylinders, on a rotary press, and one prints on the support the relief pattern borne by one of them.
...Finally, it hardly matters how one goes about pressing the sheet onto the matrix, as long as the latter is a developable surface. Instead of rolling a cone on a rigid plane, one could just as well press the paper onto the cone, by hand, little by little, avoiding slipping. If one bore relief signs on the cone, and these, inked, printed themselves on the sheet, one would obtain at the end of the operation this
...One could proceed in the same way by pressing a sheet of paper, a flexible object, onto a negacone bearing inked reliefs. One would then obtain this (having reproduced all the patterns on the sheet).
Simple game to illustrate the fact that the negacone is indeed a developable surface.
On such an object, as on the blunted posicone, we can trace geodesics, then project them onto a plane.
...The plane projection indicates to us how we would perceive the trajectories according to our Euclidean vision of the world. The object equivalent to our horse saddle would produce trajectories whose shape would evoke a repulsive gravitational force.
...It is proposed to call this bizarre surface a "blunted negacone". It is only a word, but one has to find one. We had seen earlier that one could pass continuously from the blunted cone (posicone) to the pointed cone, the inverse operation consisting in blunting the object.
...Same thing for the continuous passage from the blunted negacone" to the negacone bearing a concentrated (negative) curvature point. .
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