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

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{{Exploration}}
 
{{Exploration}}
 
{{Time|40}}
 
{{Time|40}}
 
{{Objective|
 
{{Objective|
Explore the difference between Euclidean and Spherical Geometry regarding the existence of certain polygons. Understand the importance of definitions.}}
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Explore the existence of certain types of polygon in Euclidean geometry and spherical geometry.
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Understand the importance of definitions.}}
  
  
===A. 2-Gons===
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===A. Biangles===
In Spherical geometry we have  polygons called 2-gons (also called bi-gons, lunes, bi-angles). <br>
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In Spherical geometry, two sided polygons (2-gons) exist.  They are also called biangles, bi-gons, and lunes.
# Draw an example of a 2 gon on a sphere.
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# 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?
 
# Why are there no 2-gons in Eucidean geometry? Which axiom is “responsible” for the difference between the two geometries?
 
  
 
===B. Triangles===
 
===B. Triangles===
 
In Euclidean geometry we can define  
 
In Euclidean geometry we can define  
* A Regular triangle: any 3-gon with congruent sides and angles.  
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* A {{define|regular triangle}}: any 3-gon with congruent sides and angles.
* An Equilateral triangle (1): a triangle with all 3 sides congruent
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At least two definitions of {{define|equilateral triangle}} are possible:
* An Equilateral triangle (2): a triangle with three 60 degree angles
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* ET1: a triangle with all 3 sides congruent.
* An Equiangular triangle: a triangle with three congruent angles.
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* ET2: a triangle with three 60 degree angles.
In Euclidean geometry all 4 of these definitions really describe one and the same polygon.
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Finally, define
# Which of these triangles exist in Spherical geometry?
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* An {{define|equiangular triangle}}: a triangle with three congruent angles.
# Are the ones that exist describing one and the same shape? Or could they be different? Explain.
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In Euclidean geometry all four of these definitions describe the same polygons.
 
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# Which of these triangles exist in spherical geometry?
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# Of the ones that exist, do they define the same shapes? Or could they be different? Explain.
  
 
===C. Squares===
 
===C. Squares===
In Euclidean geometry we can define a square in at least 2 different ways:
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In Euclidean geometry we can define a {{define|square}} in at least two different ways:
* Square (1): A 4 gon with congruent sides and congruent angles
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* S1: A 4-gon with congruent sides and congruent angles.
* Square (2): A 4 gon with congruent sides and all angles measuring 90 degrees.
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* S2: A 4-gon with congruent sides and all angles measuring 90°.
And compare this to:
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Compare this to:
* A regular 4-gon:  A 4 gon with congruent sides and congruent angles
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* 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?  
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# Which of these shapes exist in spherical geometry?  
 
# 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===
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===D. Rectangles===
In Euclidean Geometry we can define a rectangle in several different ways.
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In Euclidean geometry we can define a rectangle in several different ways:
* R1: A 4-gon with all interior angles 90 degrees
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* R1: A 4-gon with all interior angles 90°.
 
* R2: A 4-gon with all interior angles congruent.
 
* R2: A 4-gon with all interior angles congruent.
* R3: A 4-gon with 2 pairs of parallel sides and all angles congruent.
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* R3: A 4-gon with two pairs of parallel sides and all angles congruent.
 
# Which of these definitions cannot possibly work on the sphere?  
 
# 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?
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# 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===
 
===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.
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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
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* 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.
 
* P2: A quadrilateral with both pairs of opposite sides parallel and equal in length.
* P3: A four-sided plane figure with opposite sides parallel.
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* P3: A quadrilateral with opposite sides parallel.
* P4: A four-sided plane figure with opposite sides congruent (theorem).
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* P4: A quadrilateral with opposite sides congruent (theorem).
* P5: A four-sided plane figure with opposite angles congruent (theorem).
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* P5: A quadrilateral with opposite angles congruent (theorem).
* P6: A four-sided plane figure whose diagonals bisect one another (theorem).
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* P6: A quadrilateral 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?
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# 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}}
 
{{handin}}
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[[category:Non-Euclidean Geometry Explorations]]

Latest revision as of 10:25, 2 November 2011


Time-40.svg

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.