his influential study, âSaggio di interpretazione della geometria non-euclideaâ (A study of the interpretation of non-Euclidean geometry). With the study, a model: the Beltrami pseudosphere. The pseudosphere is odd enough to prompt the suspicion that on its surface, anything might be true.
Illustrations of the Beltrami pseudosphere are vivid and often lovely, but they are incomplete ( Figure IX.2 ). For one thing, the middle of the pseudosphere is girdled by what is plainly a Euclidean annulus or ring. What is that doing there? And for another thing, the embouchure of the pseudosphereâs horns, where a divine trumpeter might have placed his lips, recedes in real life into the infinite distance. This no illustration can show.
F IGURE IX.2. Beltrami pseudosphere
If illustrations of the Beltrami pseudosphere are not complete, they are nevertheless exhilarating. They show a universe coming to quicken. Shapes hidden by Lobachevskyâs proofs come to life in Beltramiâs picture. The triangle is an example. The Euclidean triangle is a familiar shape of art and architecture, its sides soaring or squat. When inscribed on the Beltrami pseudosphere, Euclidean triangles sag inwardly, their sides concave, and their interior angles compromised from the first by the negative curvature of their surface ( Figure IX.3 ). The interior angles of a triangle on a pseudosphere sum to less than two right angles. On a sufficiently curved surface, those interior angles are lucky to sum to anything at all.
F IGURE IX.3. Hyperbolic triangle
I N 1880, THE superb French mathematician Henri Poincaré added another model to the growing gallery of non- Euclidean models, and in the Poincaré disk, another picture. The Beltrami pseudosphere invites the mathematician to seize it by its two horns and, perhaps, give it a toot. No one is about to seize the Poincaré disk. Among mathematicians who understood its nature, no one was tempted to go near it.
The Poincaré disk divides the Euclidean plane into three distinct regions of space. There are those points lying beyond the disk, those on its circumference, and those in its interior. From the outside, the disk is simply a little unit circle, as fixed and finite as a penny. From the inside, it encompasses the whole of the infinite hyperbolic plane. Outside the circle, everything is Euclidean, and inside, everything hyperbolic. Outside and inside are Euclidean from the outside, but hyperbolic from the inside. The inside is accessible from the outsideâ step right in âbut not the outside from the insideâ no exit .
There is no distinction between Euclidean and hyperbolic points. Points are points. Hyperbolic lines are otherwise. They are not like Euclidean lines at all and require careful definition. Those careful mathematical definitions express and make precise something like a dream sequence in which the Poincaré disk floats serenely in the Euclidean plane, a circle among other circles. Now and then, a driftingcircle penetrates the circumference of the Poincaré disk, depositing, before it drifts on, the trace of a circumferential arc on its interior, one meeting the circumference at right angles. These Euclidean arcs are the straight lines of the hyperbolic plane. They are lines because they are lines, and they are straight because, although curved on the outside, they are straight on the inside ( Figure IX.4 ).
F IGURE IX.4. Poincaré disk
I N NON -E UCLIDEAN geometry, ideas that in most circumstances would fly apart stick together. This is especially true of the Poincaré disk. The binding force required to make fly-apart ideas stick together is expressed by the definition of hyperbolic distance. In the Euclidean plane, the distance d ( x, y ) between two points x and y is the length of the straightest line joining them: ds 2 = dx 2 + dy 2 . Square rootsare squared in this formula to preserve the positive character of distance itself. Distances
Illustrations of the Beltrami pseudosphere are vivid and often lovely, but they are incomplete ( Figure IX.2 ). For one thing, the middle of the pseudosphere is girdled by what is plainly a Euclidean annulus or ring. What is that doing there? And for another thing, the embouchure of the pseudosphereâs horns, where a divine trumpeter might have placed his lips, recedes in real life into the infinite distance. This no illustration can show.
F IGURE IX.2. Beltrami pseudosphere
If illustrations of the Beltrami pseudosphere are not complete, they are nevertheless exhilarating. They show a universe coming to quicken. Shapes hidden by Lobachevskyâs proofs come to life in Beltramiâs picture. The triangle is an example. The Euclidean triangle is a familiar shape of art and architecture, its sides soaring or squat. When inscribed on the Beltrami pseudosphere, Euclidean triangles sag inwardly, their sides concave, and their interior angles compromised from the first by the negative curvature of their surface ( Figure IX.3 ). The interior angles of a triangle on a pseudosphere sum to less than two right angles. On a sufficiently curved surface, those interior angles are lucky to sum to anything at all.
F IGURE IX.3. Hyperbolic triangle
I N 1880, THE superb French mathematician Henri Poincaré added another model to the growing gallery of non- Euclidean models, and in the Poincaré disk, another picture. The Beltrami pseudosphere invites the mathematician to seize it by its two horns and, perhaps, give it a toot. No one is about to seize the Poincaré disk. Among mathematicians who understood its nature, no one was tempted to go near it.
The Poincaré disk divides the Euclidean plane into three distinct regions of space. There are those points lying beyond the disk, those on its circumference, and those in its interior. From the outside, the disk is simply a little unit circle, as fixed and finite as a penny. From the inside, it encompasses the whole of the infinite hyperbolic plane. Outside the circle, everything is Euclidean, and inside, everything hyperbolic. Outside and inside are Euclidean from the outside, but hyperbolic from the inside. The inside is accessible from the outsideâ step right in âbut not the outside from the insideâ no exit .
There is no distinction between Euclidean and hyperbolic points. Points are points. Hyperbolic lines are otherwise. They are not like Euclidean lines at all and require careful definition. Those careful mathematical definitions express and make precise something like a dream sequence in which the Poincaré disk floats serenely in the Euclidean plane, a circle among other circles. Now and then, a driftingcircle penetrates the circumference of the Poincaré disk, depositing, before it drifts on, the trace of a circumferential arc on its interior, one meeting the circumference at right angles. These Euclidean arcs are the straight lines of the hyperbolic plane. They are lines because they are lines, and they are straight because, although curved on the outside, they are straight on the inside ( Figure IX.4 ).
F IGURE IX.4. Poincaré disk
I N NON -E UCLIDEAN geometry, ideas that in most circumstances would fly apart stick together. This is especially true of the Poincaré disk. The binding force required to make fly-apart ideas stick together is expressed by the definition of hyperbolic distance. In the Euclidean plane, the distance d ( x, y ) between two points x and y is the length of the straightest line joining them: ds 2 = dx 2 + dy 2 . Square rootsare squared in this formula to preserve the positive character of distance itself. Distances
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