With proper choice of units this relationship could be made explicit: the area of a triangular region on the sphere is precisely the amount by which its angle sum exceeds degrees. Why didn't Ptolemy realize that this was an example of a non-Euclidean geometry, where the important Euclidean theorem that the angle sum equals degrees simply does not hold? The answer is that he did not think of the relationships among points of a sphere and great circle arcs as a geometry.
To qualify as a geometry, a system would have to have elements corresponding to points and lines, and the first four axioms would have to be satisfied. The system consisting of points on a sphere and lines given by great circle arcs did satisfy the third and fourth axiom, and even the second if we interpret it correctly, but it did not satisfy the first axiom.
Although two nearby points on the sphere determine a unique great circle arc, there are point pairs for which this is not true. More than one great circle arc joins the north and south poles, and in fact there are infinitely many half-circles of longitude joining these two points, all of the same length.
Thus spherical geometry did not qualify as a non-Euclidean geometry, although later on in this chapter we will see that it was closely related to one. Each of them realized that it was possible to construct a two-dimensional geometry with points and shortest distance lines satisfying the first four axioms of Euclidean geometry, but not the fifth.
The parallel postulate required that for any given point not on a given line, there is exactly one line through the point that does not meet the given line. There were two ways for this postulate to fail--if every line through the point meets the given line, or if there were two or more distinct lines through the point not meeting the line.
The inventors of non-Euclidean geometry found systems based on both alternatives to the fifth axiom. The alternative to the fifth axiom in hyperbolic geometry posits that through a point not on a given line, there are many lines not meeting the given line. The alternative axiom stating that there could be more than one line through a given point not meeting a given line led to hyperbolic geometry.
The theorems deduced bv Bolyai and Lobachevski seemed quite strange, but they were as consistent as Euclidean plane geometry. The diagrams that accompanied their demonstrations certainly did not look like those in Euclid's text, and mathematicians searched for some visual representation that could make the new geometry easier to comprehend. One of the most successful expositors of this geometry, in England as well as in his native Germany, was the scientist Hermann von Helmholz.
Appealing to the same thought experiment introduced by Gauss he used the dimensional analogy to explain a way of imagining a non-Euclidean two-dimensional geometry. Helmholz asked his readers to consider a two-dimensional creature constrained to slide along the surface of a piece of marble statuary, measuring lengths of curves and sizes of angles.
For example, a flatworm living on the surface of a cylindrical column would decide that for any region bounded by three shortest distance curves the angle sum would be degrees, just the way it is on a plane, but if the column were in the shape of a long trumpet, the intrinsic geometry would be very different.
The surface he suggested was a pseudosphere , invented by the Italian geometer Eugenio Beltrami. Although this surface had a sharp edge, it still illustrated most of the important properties of hyperbolic geometry, a geometry satisfying the first four axioms, but not the fifth. For any point and any shortest line, there were many lines through the point not meeting the line, and every triangle on the surface had an angle sum strictly less than degrees!
The alternative axiom that every line through a given point would meet any other line led to elliptic geometry. This case was reminiscent of the geometry of the sphere, where every two great circles necessarily meet.
The trouble with spherical geometry is that its straight lines meet twice. The radical solution was to throw away half the points of the sphere and just use the points in the southern hemisphere, below the equator. If the points of the southern hemisphere were the points of the geometry, and great circle arcs were the lines, then any two points did determine a unique line.
For two thousand years, mathematicians accepted this postulate as true, but it had always caused problems. Daina Taimina: Mathematics is supposed to be an axiomatic system, but the fifth postulate does not look like an axiom, rather like a theorem.
And so there were many attempts to try to derive the fifth postulate from the others. Nobody was able to do that. So the question eventually arose: Well, what happens if this postulate is not true? Spherical geometry is pretty easy to understand because we see spheres all around us.
Daina, in you worked out how to make a physical model of the hyperbolic plane using crochet. How did that discovery come about? Some of the models had great aesthetic appeal, especially given the enormous variety of repeating patterns that are possible in the hyperbolic plane.
After the geometer Donald Coxeter explained these conceptual models to Escher, he used patterns based on these models in several of his prints. Many students and mathematicians, including the two of us, wanted to have a more direct experience of hyperbolic geometry—an experience similar to handling a physical sphere.
In , the Italian mathematician Eugenio Beltrami had described a surface called a pseudosphere , which is the hyperbolic equivalent of a cone. Beltrami made a version of his model by taping together long skinny triangles—the same principle behind the flared gored skirts some folk dancers wear. In the s, the American geometer William Thurston had described a model of hyperbolic space that could be made by taping together a series of paper annuli, or thin circular strips.
All these models were time-consuming to make and hard to handle; they are fragile and they tear easily. I grew up in Latvia doing these handicrafts and I decided to try and make one.
At first I tried knitting, but after a while you had so many stitches on the needles it became impossible to handle. I realized that crochet was the best method. I have crocheted a number of these models and what I find so interesting is that when you make them you get a very concrete sense of the space expanding exponentially.
The first rows take no time but the later rows can take literally hours, they have so many stitches. DT: We use these models a lot in our classes at Cornell. One thing is that you can physically experience straight lines. You can fold the crochet models and see how straight lines behave and how they intersect.
It really helps the students to understand very quickly the intrinsic properties of hyperbolic geometry. Moreover, as the vertices of the triangle get further away, the interior angles approach zero. When the points are at infinity, which is the largest triangle you can draw on the hyperbolic plane, the interior angles sum to zero. So on any particular hyperbolic plane all ideal triangles are congruent—they all have the same area—which is pretty amazing I think. What do other mathematicians think of this incursion of feminine handicraft into their domain?
DT: When they see the pictures in our book, lots of people want to have models themselves. These days people have to want one very badly and keep on calling me, then maybe I will make them one.
For a long time people thought that hyperbolic space was just some mathematical abstraction, but at the talk you gave at the Institute For Figuring last spring you brought along a lot of hyperbolic lettuce leaves. Gauss was one of the first to understand that the truth or otherwise of Euclidean geometry was a matter to be determined by experiment, and he even went so far as to measure the angles of the triangle formed by three mountain peaks to see whether they added to Because of experimental error, the result was inconclusive.
Our present-day understanding of models of axioms, relative consistency and so on can all be traced back to this development, as can the separation of mathematics from science.
0コメント