Non-Euclidean Geometry



Euclidean/Non-Euclidean Geometry

About two thousand years ago, Euclid summarized the geometric knowledge of his day. He developed this geometry based upon ten postulates. The wording of one of his postulates, known as the parallel postulate, was very awkward and received much attention from mathematicians. These mathematicians worked diligently to prove that the conclusions in Euclidean geometry were independent of this parallel postulate. A mathematician named Saccheri wrote a book called Euclid Freed of Every Flaw in 1733. His attempt to show that the parallel postulate was not needed actually laid the foundation for the development of the two branches of non-Euclidean geometry. Euclidean geometry assumes that there is exactly one parallel to a given line through a point not on that line. The branch of non-Euclidean geometry called spherical or Riemannian assumes that there are no lines parallel to a given line through a point not on that line. The other branch of non-Euclidean geometry called hyperbolic or Lobachevskian geometry assumes that there is more than one line parallel to a given line through a point not on that line.

Physical models for these geometries allow us to visualize some of their differences. The model for Euclidean geometry is the flat plane. The model for hyperbolic geometry is the outside bell of a trumpet. The model for spherical geometry is the sphere.

I. We have proved that the sum of the angles of a triangle is 180(. On a globe, is it possible to have a triangle with more than one right angle? _________ Is this a Euclidean triangle? _______ Why or why not? _______________________________________

The sides of this triangle (on the globe) curve through a third dimension. The surface upon which the triangle is drawn affects the conclusions about the sum of its angles. Euclidean geometry is true for measurement over relatively short distances (when the surface of the earth approximates a flat plane). Remember the physical experiences possible when this geometry was developed. The geometry of Einstein’s theory of relativity is the geometry of no parallel lines (spherical or Riemannian). Notice that these non-Euclidean geometries are derived from different postulates.

II. A second type of non-Euclidean geometry results when a single definition is changed. Euclidean geometry defines distance “as the crow flies.” In other words, distance is the length of the segment determined by the two points. However, travel on the surface of the earth (the real world) rarely follows this ideal straight path.

On the grid at the right, locate point A with coordinates (-4, -3) and point B with coordinates (2, 1). Use the Pythagorean Theorem to find the Euclidean distance between A and B.

Now consider that the only paths that can be traveled are along grid lines. This distance is called the “taxi-distance.” What is this “taxi-distance” from A to B?

• Points on a taxicab grid can only be located at the intersections of horizontal and vertical lines.

• One unit will be one grid unit.

• Therefore, the numerical coordinates of points in taxicab geometry must always be _____________.

• The taxi-distance between 2 points is the smallest number of grid units that an imaginary taxi must travel to get from one point to another.

1. Two points determine a line segment. (a segment is the shortest distance between two points)

(a) Draw a taxi segment from point A to point B. What is the length of this segment? _________

(b) Is this the only taxi segment between the two points? ______

If not, how many different taxi segments can you draw between points A and B? _______

(c) In taxicab geometry, do two points determine a unique segment? ________

2. A circle is the set of points in a plane that are the same distance from a given point in the plane.

(a) On the grid at the right, draw a taxi-circle with center P and a radius of 6.

(b) Is this the only taxi-circle that can be drawn with this center and this radius? ______ If not, how many different taxi-circles can be drawn? _______

(c) Can you draw a Euclidean circle without lifting your pencil? _______ ; the “taxi-circle”? _____ The “taxi-circle” is an example of discrete mathematics where the sample space is a set of individual points (not a continuous set such as a number line).

3. A midpoint, M, of a segment, [pic], is a point on the segment such that AM = MB.

(a) Find the midpoint of segment PQ.

(b) Is there more than one midpoint? ______

(c) Find the midpoint of segment PT.

(d) What conclusion can you make about the number of midpoints in taxicab geometry?

4. A point is on a segment’s perpendicular bisector if and only if it is the same distance from each of the segment’s endpoints.

|(a) Find all points that satisfy this definition |(b) Find all points that satisfy this definition |(c) Find all points that satisfy this definition |

|in taxicab geometry for segment DE. |in taxicab geometry for segment ST. |in taxicab geometry for segment KL. |

c) What conclusion can you make about perpendicular bisectors in taxicab geometry?

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