Polygons in Spherical and Euclidean Geometry Exploration

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Objective:

Explore the existence of certain types of polygon in Euclidean geometry and spherical geometry. Understand the importance of definitions.


A. Biangles

In Spherical geometry, two sided polygons (2-gons) exist. They are also called biangles, bi-gons, and lunes.

  1. Draw an example of a 2-gon on a sphere.
  2. Why are there no 2-gons in Eucidean geometry? Which axiom is “responsible” for the difference between the two geometries?

B. Triangles

In Euclidean geometry we can define

  • A regular triangle: any 3-gon with congruent sides and angles.

At least two definitions of equilateral triangle are possible:

  • ET1: a triangle with all 3 sides congruent.
  • ET2: a triangle with three 60 degree angles.

Finally, define

  • An equiangular triangle: a triangle with three congruent angles.

In Euclidean geometry all four of these definitions describe the same polygons.

  1. Which of these triangles exist in spherical geometry?
  2. Of the ones that exist, do they define the same shapes? Or could they be different? Explain.

C. Squares

In Euclidean geometry we can define a square in at least two different ways:

  • S1: A 4-gon with congruent sides and congruent angles.
  • S2: A 4-gon with congruent sides and all angles measuring 90°.

Compare this to:

  • A regular 4-gon: A 4-gon with congruent sides and congruent angles.

In Euclidean geometry these are one and the same thing.

  1. Which of these shapes exist in spherical geometry?
  2. Would you say squares exist on the sphere? Why or why not?

D. Rectangles

In Euclidean geometry we can define a rectangle in several different ways:

  • R1: A 4-gon with all interior angles 90°.
  • R2: A 4-gon with all interior angles congruent.
  • R3: A 4-gon with two pairs of parallel sides and all angles congruent.
  1. Which of these definitions cannot possibly work on the sphere?
  2. Is there one that defines 4-gons which are possible to construct on the sphere? What will such a 4-gon look like? What would you call it?

E. Parallelograms

In Euclidean geometry the following statements can all be used to define a parallelogram. They all describe the exact same family of polygons.

  • P1: A quadrilateral with opposite sides parallel and equal in length, and opposite angles equal.
  • P2: A quadrilateral with both pairs of opposite sides parallel and equal in length.
  • P3: A quadrilateral with opposite sides parallel.
  • P4: A quadrilateral with opposite sides congruent (theorem).
  • P5: A quadrilateral with opposite angles congruent (theorem).
  • P6: A quadrilateral whose diagonals bisect one another (theorem).
  1. Which definition cannot possibly work on the sphere? Why?
  2. Of the other definitions, which ones correspond to polygons we can construct on the sphere? Can they be used interchangeably? What would you call these polygons?

Handin: A sheet with answers to all questions.