Non-Euclidean Geometry Part I

What is Euclidean Geometry? It is an axiom system about points lines consisting of five axioms two points determine a unique line any terminated line may be extended indefinitely a circle may be drawn with any given point as center and any given radius all right angles are equal

and 5. If two lines lying in a plane are met by another line, and if the sum of the internal angles on one side is less than two right angles, then the lines will meet if extended sufficiently on the side on which the sum of the angles is less than two right angles.

Parallel Postulate Equivalent formulations of “Euclid’s Fifth” If a line intersects one of two parallels, it will intersect the other. Lines parallel to the same line are parallel to each other. Two lines which intersect cannot be parallel to the same line. and

Playfair formulation of the parallel postulate Given a line L and a point P not on L, there exists a unique line parallel to L through the point P. P L

If not the parallel postulate, then what? Girolamo Saccheri (1667-1733) For quadrilateral ABCD, with right angles A,B and AD=BC, one of these holds: C and D are both right angles; C and D are both obtuse angles; C and D are both acute angles. Johann Lambert (1728-1777) With Playfair, proved existence of an absolute unit of length, and considered this a contradiction.

Other Non Euclidean Geometers Gauss (1777-1855) Bolyai (1802-1860) Lobachevsky (1792-1856) Riemann (1826-1866)

Lobachevskian or Hyperbolic Geometry Given a line L and a point P not on L, there exists a unique line parallel to L through the point P. Riemannian or Elliptic Geometry NO

Model for Elliptic Geometry On the surface of a sphere, points are antipodal point pairs lines are great circles (the shortest distance between two points) Any pair of lines must intersect. Saccheri Hypothesis is angles C and D are obtuse. Angle sum of a triangle is greater than π. Triangle area is proportional to excess of angle sum.

Any pair of lines intersect

Saccheri Hypothesis is angles C and D are obtuse.

Angle sum of a triangle is greater than π.

Triangle area is proportional to excess of angle sum. Surface area of unit sphere = 4π Surface area of hemisphere = 2π Surface area of lune of angle A = 2A When a third line is drawn, a triangle ABC is formed. The angles at B and C mark two other lunes of area 2B and 2C, respectively. The triangle is part of each lune. Call the triangle’s area K

Lune A is divided into the triangle of area K and an area I. Lune B is divided into the triangle of area K and an area II. Lune C is divided into the triangle of area K and an area III. The areas I+II+III+K = 2π The areas of Lunes A,B and C sum to 3K+I+II+III = 2 (A+B+C). Subtracting the first equation from the second yields 2K=2(A+B+C)-2π, or K = (A+B+C)-π.

Model for Hyperbolic Geometry On the surface of a pseudosphere, points a line is the shortest path between two points on the line There are multiple lines through a point P not on a line L that do not intersect L Saccheri Hypothesis is angles C and D are acute Angle sum of a triangle is less than π Triangle area is proportional to defect of angle sum.

Curvature The surface of a sphere has constant positive curvature The surface of a pseudosphere has constant negative curvature