Hyperbolic Geometry II with NonEuclid Exploration

Objective: Explore properties of hyperbolic polygons using NonEuclid

NonEuclid is a Java program, written by Joel Castellanos, for doing geometry in hyperbolic space. Go to the NonEuclid page and run the applet. You should already be familiar with basic NonEuclid operations.

This exploration reviews theorems and facts of Euclidean geometry. Your job is to determine which of the statements are also theorems and facts in Hyperbolic Geometry.

Angles and Triangles

What is a hyperbolic angle?: Hyperbolic angles are formed by the intersection of hyperbolic rays analogous to the formation of angles in Euclidean geometry. The measure of a Hyperbolic angle, BAC is defined to be the measure of the Euclidean angle, B'AC', formed by the Euclidean tangent lines, AB' and AC'.

1. Construct a triangle. Measure the triangle. Move the vertices around. What are the largest and smallest angle sums you can find? On paper, sketch the triangle that gave the largest angle sum, and sketch the triangle that gave the smallest angle sum.
2. The following is a list of theorems about triangles in Euclidean geometry. Which (if any) are theorems in hyperbolic geometry? Explain your reasoning.
   1. The sum of the angles of a triangle is 180°
   2. The longest side of a triangle is opposite the largest angle.
   3. All three altitudes of a triangle intersect in a single point. (Hint: To construct an altitude of a triangle, use the "Draw Perpendicular" command from the "Constructions" menu. Click the mouse on any two vertices to define the base. Then click on the third vertex to draw the altitude.)
   4. In a triangle, the sum of the lengths of any two sides is always greater than the length of the third side.
3. In a Euclidean triangle, the product "base times height" is the same regardless of which side is chosen as the base. For example, in triangle ABC, (AB) x (the height to C) = (BC) x (the height to A). Describe the construction you did, and the measurements you found. Make a sketch.

Equilateral Triangles

An equilateral triangle is a triangle that has three sides of equal length.

Use the “draw segments of a specific length” command from the constructions menu to create an equilateral triangle.

4. Determine if the following statements from Euclidean geometry are valid in hyperbolic geometry (Indicate True/False). Explain how you checked if these statements were true or not. If the statement is false explain what goes wrong.
   1. It is possible to construct an equilateral triangle.
   2. An equilateral triangle is also equiangular (all three angles have equal measure).
   3. Each angle of an equilateral triangle measures 60°.

Rhombus

A rhombus is a quadrilateral in which all four sides have equal length.

5. It is possible to construct a rhombus. Construct one for yourself. Sketch what this polygon looks like in hyperbolic geometry.
6. The following is a list of theorems about rhombi in Euclidean geometry. Which (if any) are theorems in hyperbolic geometry? Indicate if the statements are true or false. Explain how you checked if these statements were true or not. If the statement is false explain what goes wrong.
   1. The opposite angles of a rhombus are congruent.
   2. The diagonals of a rhombus bisect each other.
   3. The diagonals of a rhombus are perpendicular.
   4. The diagonals of a rhombus bisect the rhombus' angles.

Rectangles and Squares

A rectangle is a quadrilateral with four right (90°) angles. A square is a rectangle with four sides of equal length.

7. The following is a list of theorems about rectangles and squares in Euclidean geometry. Which (if any) are theorems in hyperbolic geometry? Indicate if the statements are true or false. Explain how you checked if these statements were true or not. If the statement is false explain what goes wrong.
   1. It is possible to construct a rectangle.
   2. It is possible to construct a square.
8. In Euclidean geometry a quadrilateral with all sides of equal length and all angles of equal measure must be a square. Is this true in hyperbolic geometry?

Parallelograms

A parallelogram is a quadrilateral with two sets of parallel sides.

9. It is possible to construct a parallelogram. Construct several for yourself. Sketch what this polygon looks like in hyperbolic geometry.
10. In Euclidean geometry, the opposite sides of a parallelogram are of equal length, and the opposite angles of a parallelogram are congruent. Is this also true in hyperbolic geometry? Show why or why not (draw examples).

Handin: A sheet with answers to all questions.