# Difference between revisions of "Polygons in Spherical and Euclidean Geometry Exploration"

Line 32: | Line 32: | ||

# Would you say squares exist on the sphere? Why or why not? | # Would you say squares exist on the sphere? Why or why not? | ||

− | + | ===D. A rectangle=== | |

− | R1: A 4-gon with all interior angles 90 degrees | + | In Euclidean Geometry we can define a rectangle in several different ways. |

− | R2: A 4-gon with all interior angles congruent. | + | * R1: A 4-gon with all interior angles 90 degrees |

− | R3: A 4-gon with 2 pairs of parallel sides and all angles congruent. | + | * R2: A 4-gon with all interior angles congruent. |

− | Which of these definitions cannot possibly work on the sphere? Is there one that results in possible constructions of 4-gons on the sphere? What will it look like? What would you call it? | + | * R3: A 4-gon with 2 pairs of parallel sides and all angles congruent. |

+ | # Which of these definitions cannot possibly work on the sphere? | ||

+ | # 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=== | |

− | P1: Opposite sides are parallel and equal in length, and opposite angles are equal | + | In Euclidean Geometry the following statements can all be used to define a parallelogram. They all describe the exact same family of polygons. |

− | P2: A quadrilateral with both pairs of opposite sides parallel and equal in length. | + | * P1: Opposite sides are parallel and equal in length, and opposite angles are equal |

− | P3: A four-sided plane figure with opposite sides parallel. | + | * P2: A quadrilateral with both pairs of opposite sides parallel and equal in length. |

− | P4: A four-sided plane figure with opposite sides congruent (theorem). | + | * P3: A four-sided plane figure with opposite sides parallel. |

− | P5: A four-sided plane figure with opposite angles congruent (theorem). | + | * P4: A four-sided plane figure with opposite sides congruent (theorem). |

− | P6: A four-sided plane figure whose diagonals bisect one another (theorem). | + | * P5: A four-sided plane figure with opposite angles congruent (theorem). |

+ | * P6: A four-sided plane figure whose diagonals bisect one another (theorem). | ||

− | Which definition cannot possibly work on the sphere? Why | + | # Which definition cannot possibly work on the sphere? Why? |

− | 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? | + | # 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? |

## Revision as of 17:24, 27 October 2009

**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).

- Draw an example of a 2 gon on a sphere.
- 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.

- Which of these triangles exist in Spherical geometry?
- 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.

- Which ones exist in Spherical geometry?
- 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.

- Which of these definitions cannot possibly work on the sphere?
- 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).

- Which definition cannot possibly work on the sphere? Why?
- 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?