Non- Euclidean Geometry

Non-Euclidean Geometry

By: Ashley Walls

Bolyai- Lobachevsky geometry

In hyperbolic space, lines move away from each other.

There are at least two distinct lines that pass through the point and are parallel to (in the same plane as and do not intersect) the given line.

Plays an important role in Einstein's General theory of Relativity.

hyperbolic geometry is also has many applications within the field of Topology

References

Eves, H. (1990). An introduction to the history of mathematics (6 ed.).

Hoffman , M. (n.d.). Bernhard Riemann. Retrieved from http://www.usna.edu/Users/math/meh/ri emann.html

Latham, L. (2009, September 09). [Web log message]. Retrieved from http://lovecraftismissing.com/?p=2610

Roberts, D. (1998). Euclidean and non- Euclidean geometry. Retrieved from http://regentsprep.org/Regents/math/geometry/G G1/Euclidean.htm

Hyperbolic Geometry

Hyperbolic geometry shares many proofs and theorems with Euclidean geometry, and provides a novel and beautiful prospective from which to view those theorems

Hyperbolic Geometry

Riemann Geometry Continued

Riemannian Geometry

While studying at Göttingen University Gauss was on faculty however because he was unsocial he probably had no impact and may have never met Riemann.

After 1 year Riemann moved to the university of Berlin- here he was able to learn from Jacobi, Steiner, Dirichlet, and Eisenstein

Dirichlet influenced him the most

Bernhard Riemann

Born in 1826 and died in 1866

Father was a Lutheran minister

Showed interest in math and history at an early age and was encouraged by family

At age 16 his math knowledge was noticed while studying at his second prep school

While here he was given a 859 page textbook on number theory 6 days later her returned it saying it was wonderful and he ad mastered it.

Elliptical Geometry

Uses of Elliptic Geometry

Pilots

Ship captains

Something I found interesting…

Did you know that the shortest flying distance from Florida to the Philippine Islands is a path across Alaska? The Philippines are south of Florida - why is flying north to Alaska a short-cut?

Florida, Alaska, and the Philippines are collinear locations in spherical geometry (they lie on a great circle).

A plane geometry that is on the surface of a sphere.

Points are defined in the usual manor.

Lines are defined by the shortest distance between two points lies along them. Thus lines in spherical geometry are great circles.

* Great Circle- largest circle that can be drawn on a sphere, these are lines that divide the sphere in two equal hemispheres.

* Can you think of any examples of a great circle?

Contributors to Non-Euclidean Geometry

Girolamo Saccheri (1667-1733)

John Heinrich Lambert (1728-1777)

Adrien Marie Legendre (1752-1833)

Gauss, Bolyai, and Lobachevsky

Georg Friedrich Bernhard Riemann

Lobachevsky

Spent most of his live at the University of Kasan (student, Professor, rector)

Earliest paper on Non-Euclidean geometry published in 1829

In 1855 (once he was blind) he wrote and published a final treatment

He didn’t live to see his work account to much of anything

Bolyai

1802-1860

Hungairan officer in the Aurstian army

Father was a math teacher and friend of Gauss

Around 1823 he started working towards his Non-Euclidean geometry

“Out of Nothing I have created a strange new universe” ~Bolyai

Review: Euclidean Geometry

In Euclidean geometry, given a point and a line, there is exactly one line through the point that is in the same plane as the given line and never intersects it.

Review: Euclidean Geometry

Greek mathematician Euclid came up with a logical order for axioms, postulates, and propositions.

5th postulate: If a straight line crossing two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if extended indefinitely, meet on that side on which are the angles less than the two right angles.

What is Non- Euclidean Geometry?

Geometry that is different that Euclidean geometry.

Forms of geometry that contain a postulate that is equal to the negation of Euclid’s 5th postulate.

Each Non-Euclidean geometry is a consistent system of definitions, assumptions, and proofs that describe such objects as points, lines and planes.

The two most common non-Euclidean geometries are spherical geometry and hyperbolic geometry.