Polygons in Spherical and Euclidean Geometry Exploration

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Objective:

Explore the difference between Euclidean and Spherical Geometry regarding the existence of certain polygons. Understand the importance of definitions.


A. 2-Gons

In Spherical geometry we have polygons called 2-gons (also called bi-gons, lunes, bi-angles).

  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.
  • An Equilateral triangle (1): a triangle with all 3 sides congruent
  • An Equilateral triangle (2): a triangle with three 60 degree angles
  • An Equiangular triangle: a triangle with three congruent angles.

In Euclidean geometry all 4 of these definitions really describe one and the same polygon.

  1. Which of these triangles exist in Spherical geometry?
  2. Are the ones that exist describing one and the same shape? Or could they be different? Explain.


C. Squares

In Euclidean geometry we can define a square in at least 2 different ways:

  • Square (1): A 4 gon with congruent sides and congruent angles
  • Square (2): A 4 gon with congruent sides and all angles measuring 90 degrees.

And 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 ones exist in Spherical geometry?
  2. Would you say squares exist on the sphere? Why or why not?

D. A rectangle

In Euclidean Geometry we can define a rectangle in several different ways.

  • R1: A 4-gon with all interior angles 90 degrees
  • R2: A 4-gon with all interior angles congruent.
  • R3: A 4-gon with 2 pairs of parallel sides and all angles congruent.
  1. Which of these definitions cannot possibly work on the sphere?
  2. Is there one that results in possible constructions of 4-gons on the sphere? What will it 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: Opposite sides are parallel and equal in length, and opposite angles are equal
  • P2: A quadrilateral with both pairs of opposite sides parallel and equal in length.
  • P3: A four-sided plane figure with opposite sides parallel.
  • P4: A four-sided plane figure with opposite sides congruent (theorem).
  • P5: A four-sided plane figure with opposite angles congruent (theorem).
  • P6: A four-sided plane figure 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 them?

Handin: A sheet with answers to all questions.