Difference between revisions of "Polygons in Spherical and Euclidean Geometry Exploration"
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===B. Triangles=== | ===B. Triangles=== | ||
In Euclidean geometry we can define | In Euclidean geometry we can define | ||
| − | * A | + | * A regular triangle: any 3-gon with congruent sides and angles. |
| − | * An | + | * An equilateral triangle (1): a triangle with all 3 sides congruent |
| − | * An | + | * An equilateral triangle (2): a triangle with three 60 degree angles |
| − | * An | + | * An {{define|equiangular triangle}}: a triangle with three congruent angles. |
| − | In Euclidean geometry all 4 of these definitions | + | In Euclidean geometry all 4 of these definitions describe the same polygons. |
| − | # Which of these triangles exist in | + | # Which of these triangles exist in spherical geometry? |
| − | # | + | # Of the ones that exist, do they define the same shapes? Or could they be different? Explain. |
| − | |||
===C. Squares=== | ===C. Squares=== | ||
Revision as of 23:14, 28 October 2010
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 describe the same polygons.
- Which of these triangles exist in spherical geometry?
- 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 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?
Handin: A sheet with answers to all questions.
