# e-book The Non-Euclidean, Hyperbolic Plane: Its Structure and Consistency

Karl Friedrich Gauss, arguably the best mathematician ever, delayed publishing his research on non-Euclidean geometry, fearing that he could compromise his reputation. Throughout the last two centuries several intuitive models of non-Euclidean geometries were proposed. In most of them the definitions of basic geometrical notions challenge our commonly held spatial intuitions.

They are, nonetheless, self-consistent within the model to which they belong. One of the many ways of comparing these geometries and the planar Euclidean geometry is to look at the sum of the interior angles of a triangle in each of them. In the spherical geometry the interior angles always add up to more than two right angles degrees , in the planar geometry they add up to exactly two right angles, while in the hyperbolic geometry they add up to less than two right angles. Here is an example of a triangle on a sphere, with three right angles adding up, therefore, to degrees : .

## The Non-Euclidean, Hyperbolic Plane

It can be shown that in each type of non-Euclidean geometry the sum of the interior angles of a triangle is directly related to the area of the triangle. Area of a circular surface grows differently in each type of geometry. In Euclidean planar geometry it grows proportional with the square of the radius of the circle.

In hyperbolic geometry it grows exponentially with the growth of the radius. In spherical geometry the area grows with the radius but it cannot exceed the area of the whole spherical surface. The three geometries also differ is the system of coordinates best implemented in each. This issue is of great importance for the computational treatment of each type of geometry. We are widely acquainted with the rectangular system of coordinates for the Euclidean plane.

Yet that one is not without ambiguity, as shown in a public radio interview on the subject of Hurricane Katrina. I insert here an instructional module focused on the two most commonly used coordinate systems, planar and spherical:. On a hyperbolic plane the most convenient system of coordinates is also rectangular, as shown in the following picture: . Non-Euclidean geometry can also be introduced and studied in a highly technical manner.

For the reader interested in such an approach we offer a brief bibliography. A few I would recommend are the following in alphabetical order of the authors : Bolyai, Janos. Non-Euclidean Geometry and the Nature of Space. A detailed historical account introduces the reader to the battle of ideas around non-Euclidean geometries.

Henderson, David W. Third edition. This is a textbook used in several undergraduate courses in the U. It provides an inviting, detailed, hands-on, inquiry-based approach to learning non-Euclidean geometry. Especially instructive is the comparative view, property by property. Also good some groundbreaking are the illustrations. Krause, Eugene F. Taxicab geometry: An adventure in non-Euclidean geometry. New York, NY: Dover, This slim booklet is highly entertaining.

It contains many exercises in accessible format. Prekopa, Andras, and Emil Molnar Eds. Non-Euclidean geometries. New York, NY: Springer, This is a collective volume published in the memory of Janos Bolyai. It contains contributions of great variety, both in approach and difficulty. Trudeau, Richerd J. The non-Euclidean revolution.

## The Intellectual History of Non-Euclidean Geometry

Other Popular Editions of the Same Title. Search for all books with this author and title. Customers who bought this item also bought. Stock Image. Kelly, G. New Paperback Quantity Available: Seller Rating:. New Paperback Signed Quantity Available: Book Depository hard to find London, United Kingdom. Published by Springer New Paperback Quantity Available: 1. Kelly G. Published by Springer. New Quantity Available: 1. Published by Springer Verlag Revaluation Books Exeter, United Kingdom.

As the Earth is a sphere, one of the earliest applications of geometry was to the properties of figures drawn on a sphere.

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This is called spherical geometry. For more than two thousand years spherical geometry was studied as a set of results in 3-D Euclidean geometry; but in , Riemann realised that we can also describe it as analogous to the 2-D Euclidean geometry of a plane, but with the points related to each other in a different non-Euclidean way.

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From Riemann's point of view, there is no need for a 3-D space in which the sphere is embedded; attention is confined solely to the points on the surface and the way they are "connected". Let us look at the geometry of the sphere more closely from now we will follow the convention in geometry according to which "sphere" means just the surface, not the solid object. We know that the shortest distance between two points on a sphere is along a great circle.

But a straight line is defined as the shortest distance between two points, so great circles become the "straight lines" of the non-Euclidean geometry of the sphere.

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Since any two great circles will always intersect, in this geometry there are no parallel lines, violating Euclid 's fifth postulate. A circle is defined as the locus of points a given distance from a fixed point. This corresponds to a small circle on a sphere Fig. By replacing Euclid 's postulates with appropriate equivalents, all of spherical geometry can be deduced.

Figure 1. Note that all great circles eventually intersect. All points on a sphere are equivalent, and there is no preferred direction, so the geometry of this 2-D "space" is homogeneous and isotropic. It is common to say that this space is curved, in contrast to "flat" Euclidean 2-D space.

This is a rather misleading description , but we are stuck with it. By convention the curvature is "positive"; we will meet the alternative, negatively curved space, in the next section. Thinking of a sphere as a 2-D space is a useful analogy, but in reality space is three dimensional. What would positive curvature mean for a 3-D space? Simply, that the properties of triangles and circles are exactly the same as in 2-D curved space they are still plane figures, after all.

In 3-D there are infinitely many planes, separated vertically and at different angles Fig. In 3-D the locus of points a given distance say x from a fixed point is a sphere. We can think of this as being built up of circles with every possible orientation, all centred on the fixed point and with radius x. By isotropy, all these circles have the same circumference, so the geometry on the sphere in our curved 3-D space is exactly the same as on a sphere embedded in Euclidean 3-D space, with one exception: the radius of curvature is not x but.

Using the straightest possible lines on the surface shortest distance between two points, as ever , we can draw a triangle, for instance, as shown in the figure. The surface is distorted from a flat plane in the opposite way to the surface of a sphere, so we say it has negative curvature. On the surface shown in the diagram, the curvature changes from point to point. We would have preferred to show the negatively curved equivalent of a sphere, a "pseudosphere" which has constant negative curvature at each point.

Unfortunately it is impossible to construct a 2-D pseudosphere in 3-D flat space, unlike the case for a sphere it is possible in 4-D flat space, but we can't visualise that! But this does not prevent us from working out its geometry; in fact this was the first non-Euclidean geometry to be discovered.

On a pseudosphere, the circumference of a circle with radius x is. We call R 0 the radius of curvature of the pseudosphere. Unlike a sphere, a pseudosphere extends to infinity in all directions. In this case there are an infinite number of "straight lines" passing through a given point that will never intersect a given line, rather than just one parallel. Just as for positive curvature, there is a possibility that our 3-D space is really negatively curved, and the details of the geometry carry over from the 2-D case in the same way. This means that Euclidean geometry is a limiting case of the other two.

By the same token, when we are dealing with lengths x small compared to R 0 , geometry will be effectively Euclidean as we claimed at the beginning. Because a space with Euclidean geometry shows neither positive nor negative curvature is often called flat space. It is often useful to talk about all three geometries simultaneously.

To do this we use the symbol S k to stand for all three cases, according to the value of the curvature constant k , as follows:. It is awkward to use a formally infinite R 0 to describe flat space, and we can avoid doing so if we use the S k notation, because then we can pick any value of R 0 , and it will cancel out:. We have carefully avoided mentioning the most obvious difference between a sphere and a flat plane: on a sphere, if you travel far enough in a "straight line" you will arrive back where you started.

This tells us that, taken as a whole, points on the sphere are linked in a fundamentally different way from points on a plane. These large-scale connections would remain if the sphere was distorted, e. The large-scale connections define the topology of the surface sometimes called "rubber-sheet" geometry because topology is unaffected by stretching or squashing the surface. In contrast, geometry, strictly speaking, is concerned with the actual lengths and angles, and not with large-scale connections. We say that the topology of a sphere is closed , meaning that it has a finite surface area, but no edges.

The topology of a pseudosphere or a flat plane is open , meaning that it extends infinitely in all directions. This suggests that there is a necessary connection between geometry and topology, but this is not correct. For instance, Euclidean geometry also applies on the surface of a cylinder, in the sense that circles have radius exactly 2 r etc.

## Non-Euclidean geometry

This illustrates an important point: if we compare the properties of any two small regions of a cylinder they are the same and independent of direction, so we say that locally the cylinder is homogeneous and isotropic. But topologically, the direction along the cylinder behaves very differently from the direction around it, so we say that globally the cylinder is anisotropic though still homogeneous.

Many other topologies are consistent with Euclidean geometry, including completely closed ones. The simplest closed Euclidean geometry is a torus , generated by connecting together the two ends of a segment of a cylinder. Unlike a cylinder, a 2-D torus embedded in 3-D space e.

In this case it is quite easy to visualise what is going on. Notice that the "edges" only appear because we have to cut the torus to unroll it onto a flat plane; they are not special places as far as inhabitants of the game are concerned.

Try playing this Java version of "Asteroids" written by Mike Hall www.