The Strange New Worlds: The Non-Euclidean Geometries Presented by: Melinda DeWald Kerry Barrett

Euclid’s Postulates 1. To draw a straight line from any point to any point. 2. To produce a finite straight line continuously in a straight line. 3. To describe a circle with any center and distance. 4. That all right angles are equal to one another. And the fifth one is …

Euclid’s fifth postulate: If a straight line falling on two straight lines makes the sum of the interior angles on the same side less than two right angles, then the two straight lines, if extended indefinitely, meet on that side on which the angle sum is less than the two right angles.

For 2000 years people were uncertain of what to make of Euclid’s fifth postulate! Euclid’s Work

About the Parallel Postulate It was very hard to understand. It was not as simplistic as the first four postulates. The parallel postulate does not say parallel lines exist; it shows the properties of lines that are not parallel. Euclid proved 28 propositions before he utilized the 5 th postulate. Once he started utilizing this proposition, he did so with power. Euclid used the 5 th postulate to prove well-known results such as the Pythagorean theorem and that the sum of the angles of a triangle equals 180.

The Parallel Postulate or Theorem? Is this postulate really a theorem? If so, was Euclid simply not clever enough to find a proof? Mathematicians worked on proving this possible “theorem” but all came up short. 2 nd century, Ptolemy, and 5 th century Greek philosopher, Proclus tried and failed. The 5 th postulate was translated into Arabic and worked on through the 8 th and 9 th centuries and again all proofs were flawed. In the 19 th century an accurate understanding of this postulate occurred.

Playfair’s Postulate Instead of trying to prove the 5 th postulate mathematicians played with logically equivalent statements. The most famous of which was Playfair’s Postulate. This postulate was named after Scottish scientist John Playfair, who made it popular in the 18 th century. Palyfair’s Postulate: Through a point not on a line, there is exactly one line parallel to the given line. Playfair’s Postulate is now often presented in text books as Euclid’s 5 th Postulate.

Girolamo Saccheri Saccheri was an 18 th century Italian teacher and scholar. He attempted to prove the 5 th postulate using the previous 4 postulates. He tried to find a contradiction by placing the negation of the 5 th postulate into the list of postulates. He used Playfair’s Postulate to form his negation. Through a point on a line, either: 1.There are no lines parallel to the given line, or 2.There is more than that one parallel line to the given line. The first part of this statement was easy to prove. The second part was far more difficult using the first four postulates, he found some very interesting results but never found a clear contradiction. He published a book with his findings: Euclides Vindicatus

Revelations of Euclid Vindicated In the 19 th century Euclid Vindicated was dusted off and revisited by four mathematicians. Three of whom started by considering: Can there be a system of plane geometry in which, through a point not on a line, there is more than one line parallel to the given line? Carl Friedrich Gauss (German) looked at the previous question, but did not publish his investigation. Nicolai Lobachevsky(Russian) produced the first published investigation. He devoted the rest of his life to study this different type of geometry. Janos Bolyai (officer in Hungarian army) looked at the same question and published in 1832.

Revelations Continued These three came up with the same result: If the parallel postulate is replaced by part 2 of its negation in Euclid’s postulates, the resulting system contains no contradictions. They settled once and for all that the parallel postulate CAN NOT be proved by the first four postulates. This discovery lead to a new type of plane geometry, with an entire new theory of shapes on surfaces….

Non-Euclidean Geometry

Riemann Geometry The fourth person to take a look at Euclid Vindicated was Bernhard Riemann. He was looking at part one of the negation of the parallel postulate he wondered if there was a system when you are given a point not on a line and there are NO parallel lines. He found a contradiction but it depended on the same assumptions as Euclid, that lines extend indefinitely. Riemann observed that “extended continuously” did not necessarily mean they were infinite. –Ex: Consider an arc on a circle: it is extended continuously but its length is finite. This contradicted Euclid’s idea of using the postulate. Amounted to alternate version of the postulate. THERE IS A NEW SYSTEM!

Differences from the Euclidean System After Non-Euclidean geometry was discovered other types of geometry were distinguished by how they used parallel lines. –In Reimann’s geometry for instance parallel lines did not exist. New systems of Lobachevsky and Riemann were formally called Non-Euclidean geometry. The differences about parallelism produces vastly different properties in the new geometric systems. –Only in Euclidean geometry is it possible to have two triangles that are similar but not congruent. –In Non-Euclidean geometries if corresponding angles of two triangles are equal then the triangles must be congruent.

More Differences The sum of the angles of a triangle: Euclid: exactly 180 degrees Lobachevsky: less than 180 degrees Riemann: greater than 180 degrees Euclid Lobachesky Riemann

Another Difference The ratio of the circumference “C” of a circle to its diameter “D” depends on the type of geometry being used. –Euclid: exactly Pi –Lobachevsky: Greater than Pi –Riemann: Less than Pi

Balloon Activity Get in pairs. Draw two dots (at least an inch apart) on a balloon. Connect them with a straight-line. Blow up the balloon to a relatively large size. Have string go from one point to the other over the originally drawn line. Try to find a line from one point to the other that is shorter than the string.

Geometry as a Tool Geometry should be used as a tool to help deal with our world. -The type of geometry a person uses depends on the situation that person is faced with. Euclidean geometry “makes sense” in most people’s minds because that is what we are taught as children. Euclidean works well for the construction world. Riemann’s geometry is good for astronomers because of the curves in the atmosphere. Lobachevsky’s geometry: Theoretical physicists use this system.

Timeline 2 nd century: Ptolemy tried and failed to proved the Parallel postulate 5 th century: Greek philosopher, Proclus tried and failed to prove the Parallel Postulate. 8 th and 9 th centuries: The 5 th postulate was translated into Arabic and worked on, but again all proofs were flawed 18 th century: Playfair’s Postulate was made popular by the Scottish scientist John Playfair. 18 th century: Girolamo Saccheri published his work on “proving” Euclid’s Fifth Postulate in Euclides Vindicatus 19 th century Euclid Vindicated was dusted off and revisited by four mathematicians (Bernhard Riemann, Carl Friedrich Gauss, Nicolai Lobachevsky, and Janos Bolyai) 19 th century mathematicians played with logically equivalent statements to Euclid’s Fifth Postulate (thus coming to the accurate conclusion that this statement was indeed a postulate). 19 th century: Non-Euclidean Geometry was created.

References Heath, Thomas L., Sir. The Thirteen Books of Euclid’s Elements. New York: Dover Publications, Inc., 1956 Non-Euclidean Geometry. 13 November Rozenfeld, B. A. Istoriia Neevklidovoi Geometrii (English). New York: Springer-Verlag, 1988