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The saddle shape.
...One can (at least in thought) construct a large number of small negative cones and join them together. If the vertices are distributed roughly regularly over this surface, one will create a surface element with constant negative curvature density. But it won't behave like a sphere. This surface will not close up.
...Imagine we have constructed a portion of surface with constant negative curvature (we'll call it constant negative curvature). We can take any point, drive in a nail, tie a string to it, and draw a circular contour, obtaining what is called a saddle shape (a surface with constant negative curvature).
The blunted negative cone.
...Earlier, we constructed a blunted cone (a "blunted posicone") by starting with a spherical cap and joining it with a truncated cone. We even found how to ensure there was no discontinuity in the tangent plane. The angle used to create the cone from which this truncated posicone was derived had to equal the amount of angular curvature contained within the spherical cap.
...There is a way to measure the amount of negative angular curvature contained within a saddle shape, by comparing the circumference and the radius—an operation rarely performed in saddlery. Suppose we know this value. We can then construct the truncated negative cone accordingly.
Note: The negative cone, like the cone, is a developable surface. This means it can be "applied" onto a plane. However, the operation seems more delicate. It's hard to imagine how one could easily roll a negative cone over a rigid plane.
...Rather than a rolling operation, it's preferable to consider an application or, more vividly, an impression operation. What does it mean to print? It means applying a surface bearing inked relief patterns onto another surface.
...In Gutenberg's time, one pressed one plane onto another. When printing a pattern onto fabric, one rolls a cylinder over it. On a rotary press, a ribbon of paper is passed between two cylinders, and the relief pattern carried by one of them is transferred onto the support.
...Ultimately, it doesn't matter how one applies the sheet to the matrix, as long as the matrix is a developable surface. Instead of rolling a cone over a rigid plane, one could just as well press the paper onto the cone by hand, gradually and carefully, avoiding slippage. If we had raised signs on the cone, and those signs were inked, they would transfer onto the sheet, resulting in the following at the end of the process:
...One could proceed similarly by pressing a flexible sheet of paper onto a negative cone bearing inked reliefs. The result would then be this (having reproduced all the patterns onto the sheet).
Simple exercise to illustrate that the negative cone is indeed a developable surface.
On such an object, as on the blunted posicone, we can draw geodesics and then project them onto a plane.
...The planar projection shows us how we would perceive the trajectories according to our Euclidean vision of the world. The equivalent object to our saddle would produce trajectories whose shape evokes a repulsive gravitational force.
...We propose to call this strange surface a "blunted negative cone." It's just a word, but we need one. Earlier, we saw that one could smoothly transition from the blunted cone (posicone) to the sharp cone, and the reverse operation consists of blunting the object.
...The same applies to the smooth transition from the "blunted negative cone" to the negative cone bearing a concentrated point of negative curvature.
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