Difference between revisions of "Instructor:Spherical Geometry Exercises Solutions"

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<li>Yes, every point on the sphere has exactly one antipodal points.</li>
 
<li>Yes, every point on the sphere has exactly one antipodal points.</li>
 
<li>Although tessellations of the plane suggest infinity because they can be continued forever, Escher felt the necessity of an edge harmed the effect.  Escher says that as you turn the ball, the neverending series of motifs suggests infinity.  On the other hand, there are only finitely many motifs on the ball.  Which is more compelling?</li>
 
<li>Although tessellations of the plane suggest infinity because they can be continued forever, Escher felt the necessity of an edge harmed the effect.  Escher says that as you turn the ball, the neverending series of motifs suggests infinity.  On the other hand, there are only finitely many motifs on the ball.  Which is more compelling?</li>
 +
<li>There are really two valid choices here:  1) A is between B and C if A is on a geodesic segment joining B and C, or 2) A is between B and C if A is on the short geodesic segment joining B and C.  In both cases, St. Louis is between the poles.  In case 1, the north pole is between the south pole and St. Louis, but not in case 2.</li>
 +
<li>Draw the picture.</li>
 +
<li>Draw the picture.</li>
 +
<li>Draw the picture (it should look a lot like question 4's picture).</li>
 
<li>
 
<li>
 +
{| border="1"
 +
!width="100"|Angles
 +
!width="100"|Defect
 +
!width="100"|Area Fraction
 +
|-
 +
| <div style="height:15px"></div> 90° 90° 90°  || a:90° || b:1/8
 +
|-
 +
| <div style="height:15px"></div> 120° 80° 70°  || c:90°|| d:1/8
 +
|-
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| <div style="height:15px"></div> 72° 72° 72°  || e: 36° || f: 1/20
 +
|-
 +
| <div style="height:15px"></div> 90° 45° g:90° || 45° || h:1/16
 +
|-
 +
| <div style="height:15px"></div> 135° 135° i:90° || j:180°  ||  1/4
 +
|}
 +
</li>
 +
</ol>

Revision as of 15:18, 7 April 2007


  1. Yes, every point on the sphere has exactly one antipodal points.
  2. Although tessellations of the plane suggest infinity because they can be continued forever, Escher felt the necessity of an edge harmed the effect. Escher says that as you turn the ball, the neverending series of motifs suggests infinity. On the other hand, there are only finitely many motifs on the ball. Which is more compelling?
  3. There are really two valid choices here: 1) A is between B and C if A is on a geodesic segment joining B and C, or 2) A is between B and C if A is on the short geodesic segment joining B and C. In both cases, St. Louis is between the poles. In case 1, the north pole is between the south pole and St. Louis, but not in case 2.
  4. Draw the picture.
  5. Draw the picture.
  6. Draw the picture (it should look a lot like question 4's picture).
  7. Angles Defect Area Fraction
    90° 90° 90°
    a:90° b:1/8
    120° 80° 70°
    c:90° d:1/8
    72° 72° 72°
    e: 36° f: 1/20
    90° 45° g:90°
    45° h:1/16
    135° 135° i:90°
    j:180° 1/4