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

From EscherMath
Jump to navigationJump to search
(New page: {{Exploration}} {{Time|30}} {{Objective| Explore the difference between Euclidean and Spherical Geometry regarding the existence of certain polygons. Understand the importance of definiti...)
 
Line 1: Line 1:
 
{{Exploration}}
 
{{Exploration}}
{{Time|30}}
+
{{Time|40}}
 
{{Objective|
 
{{Objective|
 
Explore the difference between Euclidean and Spherical Geometry regarding the existence of  certain polygons. Understand the importance of definitions.}}
 
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). <br>
 +
# 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?
  
1. In Spherical geometry we have  polygons called 2-gons (also called bi-gons, lunes, bi-angles).
 
A, Draw an example of a 2 gon on a sphere.
 
B. Why are there no 2-gons in Eucidean geometry? Which axiom is “responsible” for the difference between the two geometries?
 
  
 
+
===B. Triangles===
2. In Euclidean geometry we can define  
+
In Euclidean geometry we can define  
A Regular triangle: any 3-gon with congruent sides and angles.  
+
* 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 (1): a triangle with all 3 sides congruent
An Equilateral triangle (2): a triangle with three 60 degree angles
+
* An Equilateral triangle (2): a triangle with three 60 degree angles
An Equiangular triangle: a triangle with three congruent 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.
 
In Euclidean geometry all 4 of these definitions really describe one and the same polygon.
Which of these triangles exist in Spherical geometry?
+
# 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.
+
# Are the ones that exist describing one and the same shape? Or could they be different? Explain.
  
  
3. In Euclidean geometry we can define a square in at least 2 different ways:
+
===C. Squares===
Square (1): A 4 gon with congruent sides and congruent angles
+
In Euclidean geometry we can define a square in at least 2 different ways:
Square (2): A 4 gon with congruent sides and all angles measuring 90 degrees.
+
* 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:
 
And compare this to:
A regular 4-gon:  A 4 gon with congruent sides and congruent angles
+
* A regular 4-gon:  A 4 gon with congruent sides and congruent angles
 
In Euclidean geometry these are one and the same thing.
 
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?
+
# Which ones exist in Spherical geometry?  
 +
# Would you say squares exist on the sphere? Why or why not?
  
 
4. A rectangle  
 
4. A rectangle  

Revision as of 17:21, 27 October 2009


Time-40.svg

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?

4. A rectangle 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?

5. 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?