Lesson 7: Non-Euclidean Geometries Quiz

Who are credited with the independent discovery of Hyperbolic Geometry?

• Euclid, Bolyai, and Lobachevsky

• Euler, Fermat, and Poincaré

• Pythagoras, Newton, and Gauss

• Archimedes, Descartes, and Riemann

What is the characteristic postulate of Hyperbolic Geometry?

• Through a point P not on a line l, there exists exactly one line parallel to l.

• Through a point P not on a line l, there exist at least two lines parallel to l.

• Through a point P not on a line l, there is no line parallel to l.

• Through a point P not on a line l, there exist infinitely many lines parallel to l.

What is Poincaré's model of Hyperbolic Geometry?

• Points are ordinary points inside the unit circle, lines are diameters of the circle or arcs of circles perpendicular to the boundary.

• Points are points on the surface of a 2-dim sphere with antipodal points identified, lines are straight lines extending to infinity.

• Points are points on the surface of a 2-dim sphere with antipodal points identified, lines are great circles (longitudes or meridians).

• Points are ordinary points inside the unit circle, lines are straight lines extending to infinity.

In Hyperbolic Geometry, what is the sum of the angles of a triangle?

• Equal to 180 degrees

• Depends on the type of triangle

• Greater than 180 degrees

• Less than 180 degrees

What did Enrico Beltrami prove in 1868?

• He proved the existence of similar triangles in Hyperbolic Geometry

• He proved the equivalence of Euclidean and non-Euclidean (hyperbolic) geometries

• He proved the existence of parallel lines in Hyperbolic Geometry

• He proved the existence of infinitely many lines parallel to a given line through a given point in Hyperbolic Geometry

What is the characteristic postulate of Elliptic Geometry?

• Through a point P not on a line l, there exist infinitely many lines parallel to l.

• Through a point P not on a line l, there is no line parallel to l.

• Through a point P not on a line l, there exist at least two lines parallel to l.

• Through a point P not on a line l, there exists exactly one line parallel to l.

What is Riemann's spherical model of Elliptic Geometry?

• Points are points on the surface of a 2-dim sphere with antipodal points identified, lines are straight lines extending to infinity.

• Points are ordinary points inside the unit circle, lines are diameters of the circle or arcs of circles perpendicular to the boundary.

• Points are points on the surface of a 2-dim sphere with antipodal points identified, lines are great circles (longitudes or meridians).

• Points are ordinary points inside the unit circle, lines are straight lines extending to infinity.

In Elliptic Geometry, what is the sum of the angles of a triangle?

• Depends on the type of triangle

• Less than 180 degrees

• Greater than 180 degrees

• Equal to 180 degrees

What did Einstein use as the underlying mathematical basis of his theory of general relativity?

• Elliptic Geometry

• Hyperbolic Geometry

• Euclidean Geometry

• Riemannian Geometry

According to Poincaré, what is the relationship between different geometries?

• One geometry is more true than another

• One geometry is more convenient than another

• One geometry is more complex than another

• One geometry is more popular than another