Hyperbolic Geometry

Author:Jacinta Johnny, Marcinek, Tibor

Topic:Geometry

Hyperbolic Shape

Investigate Triangles

List down the properties of a triangle that you have learned before.

Check my answer

Now construct a triangle using the hyperbolic applet below. What do you notice?

Investigate Rectangles

List down the properties of a rectangle that you have learned before.

Check my answer

Construct a rectangle using the hyperbolic applet below. What do you notice?

Lambert & Saccheri Quadrilaterals

In this world (hyperbolic space), rectangles cannot exist because

the surface "curves away" from the corners.
a four-right-angled shape is a logical impossibility.

So, we call these shapes the Saccheri and Lambert quadrilaterals. They are special shapes that try to be rectangles but end up with acute (sharp) angles.

Exploration

Try out using the Poincaré Disk Model applet below to 1. Form a quadrilateral with two equal sides perpendicular to a base. Observe all their angles. • Draw points A and B to create a hyperbolic line on the Poincaré disk. • Using the 'Hyperbolic Perpendicular Bisector' tool, construct the perpendicular bisector of segment AB at E. • Using the 'Hyperbolic Perpendicular at Point' tool create two perpendicular lines to AB at point A and B respectively. • Set another point F on the perpendicular bisector of AB. • Using the 'Hyperbolic Perpendicular at Point' tool, create a line perpendicular to EF at F

Now try using the Poincaré Disk Model applet above to 1. Form a quadrilateral with three right angles. Observe the fourth angle.

Note:

A Saccheri quadrilateral has two equal sides perpendicular to a base. The summit angles of a Saccheri quadrilateral are acute. A Lambert quadrilateral has three right angles. The fourth angle of a Lambert quadrilateral is acute.