Download Spherical Geometry Activities - Notes

Please excuse any typos and any poor grammar that is in the blue comments/solutions. I only
did minimal editing, as I wanted to get the information out to everyone as fast as possible. This
way you could look at your work while the experience was reasonably fresh in your mind.
Exploration 1:
Objectives:
Let’s consider the point to be the simplest shape on the plane and on the sphere.
• Describe the simplest, shortest path between two points on the plane.
• Describe the simplest, shortest path between two points on the sphere.
• Describe the shape you get when you extend each of these two paths.
• Describe how to measure distance on the plane, and what units of measure you use.
• Describe how to measure distance on the sphere, and what units of measure you use.
Investigation on the Plane:
Draw two different points on the plane and label them A and B. Connect them with the shortest
possible path. Finally, extend your shortest path until it reaches the edge of your paper.
1) What is the name of the geometric object that you created with your shortest path?
A line segment
2) If you could continue to extend your shortest path forever, would the extensions ever meet?
How do you know? What is the name of the geometric object that you created?
No – The plane is not a finite surface, so it continues forever in all directions.
A line
3) Into how many sections do points A and B divide your line? How many of these sections are
finite in length? How many are infinitely long?
The two points partition the line up into three sections. The points are the boundaries
between each section. [Note: Some people will say five – the two rays, the line segment and
the two boundary points, because they count the boundaries in the partitioning.]
One of the sections is finite and the other two are infinite.
4) How do you measure the distance between two points on the plane? Measure the distance
between A and B. Is there another way to measure the distance between two points? (We are
not talking about changing the units of measure here.)
You would measure the length of the line segment connecting the two points. This would
be the equivalent of finding the length of the shortest path between the two points.
There is only one way to measure distance on the plane.
5) How many different lines can you draw through one point on a plane? How many different
lines can you draw through any two points on a plane?
You can draw infinitely many lines through one point on the place; however, you can
only draw one line between two points on the plane.
Make a Guess:
6) What shape will you get when you connect two points on a sphere with the shortest possible
path?
7) What will happen when you extend this path in both directions on a sphere?
Investigation on the Sphere:
Draw two different points on the sphere and label them A and B. Use your spherical ruler to
connect the points with the shortest possible path. Finally, continue drawing the path along the
spherical ruler and extend it as far as possible in both directions.
8) In the last part of your construction, what did you notice? [You have created a great circle
on the sphere. How would you describe it? Why does the great circle correspond to a line on
the plane?]
The extensions will meet up with each other; put another way, if you go along a great
circle then you will eventually return to your starting point. [The great circle is the largest
circle that is possible to make on the sphere. Another way to think about this is to imagine a
plane – giant sheet of paper – cutting through the sphere and passing through the center of
the sphere. The intersection of the sphere and the plane will be a great circle. The great
circle satisfies all five symmetries that a geodesic needs to satisfy.]
9) Into how many arcs do points A and B divide your great circle? How many of these arcs are
finite in length? How many are infinitely long? [Arcs on the sphere correspond to what
geometric object on the plane?]
The points partition the great circle into two parts or arcs. Both of these arcs are finite in
length. None of the arcs will be infinite in length. [The arc on the sphere corresponds to the
line segment on the plane.] {For some people the two points will partition the great circle
into four pieces; the two arcs and the two boundary points.}
10) Use the degree markings on your spherical ruler to measure the length of each arc connecting
A and B. What is the distance between points A and B? Which arc’s measure did you choose
as the distance? Explain why you chose this measure.
To find the length of the arc between we measure the shortest distance between the two
points, i.e., we measure along the “minor” or shorter arc. [While we could use the second
arc, we want to retain the convention of measuring the shortest distance between the two
points from the plane.]
11) How many great circles can you draw through one point on a sphere? How many great
circles can you draw through any two points on a sphere? [Are you sure?]
You can draw infinitely great circles through a single point on the sphere. If the two
points are not antipodal points (opposite points – you will also see these points referred to as
pole points), then there is only one great circle that can be drawn between the two points.
However, if the two points are antipodal points, then there are infinitely many great circles
that can be drawn between the two points on the sphere.
12) Describe a pair of points on a sphere between which it is possible to measure the shortest
distance along more than one arc. [Is there a corresponding scenario on the plane? Why or
why not?]
If the two points are antipodal points, then there are infinitely many ways to measure to
the shortest distance, as there are infinitely many arcs of the same length between the two
points. See the above response. [It is not possible to do this on the sphere.]
13) What is the longest possible distance between two points on a plane? On a sphere?
There is no greatest distance between two points on the plane – two points can be any
distance apart from each other. The greatest distance between two points on the sphere is
180 degrees. This is the longest shortest path that we can create on the sphere.
14) What is the shortest possible distance between two points on a plane? On a sphere?
The shortest possible distance between two points on both the plane and the sphere is
zero. [The points would coincide in this case. We need this to happen in order for the formal
concept of distance, as something called a metric, to work at the theoretical level.]
Compare the Plane and the Sphere:
15) Create a compare/contrast chart to record your observations from this investigation. Your
chart should have two columns, one for the plane and one for the sphere.
Plane:
• The line is infinite.
• If you follow a line on a plane, you will never return to your starting point.
• The line segment is the shortest path between two points.
• There is a unique line through two points on the plane.
• Two points partition the line into three sections – one finite and two that are infinite.
• There is only one distance that you can measure between two points on the plane. This
distance is measured long the line segment connecting the two points together. There is
only one shortest path between two points.
• There is no greatest distance between two points on the plane.
Sphere:
• The great circle is finite.
• If you follow a great circle on a sphere, you will return to your starting point.
• A “minor” is the shortest path between two points.
• If two points on the sphere are not antipodal points, then there is a unique great circle
through the points.
• If two points on the sphere are antipodal points, then there are infinitely many great
circles through the points.
• Two points partition a great circle into two arcs – both of which are finite.
• Distance between two points on the sphere is measured along an arc of a great circle
connecting them. There are two possible choices for the arcs to measure along. The
convention is to use the smaller arc or the shorter arc.
• The greatest distance between two points on the sphere is 180 degrees.
• If two points are not antipodal points, there is only one shortest path between the points.
If the two points are antipodal points, there are infinitely many shortest paths between the
points, but all have the same length.
Exploration 2:
Objectives:
• Investigate how many points two lines can share on the plane.
• Investigate how many points two lines can share on the sphere.
• Explain your observations about parallel lines on the plane and on the sphere.
• Investigate vertical angles (opposite angles) on the plane and on the sphere.
Investigation on the Plane:
1) Draw a line and label it l.
2) Try to construct another line a that has no points in common with line l. If it is possible,
explain your construction method. Otherwise, explain why it is not possible to construct line
a.
This is the case for parallel lines. [Paper folding – fold a perpendicular line to l, and then
fold a line perpendicular to the line you just created. Why does this work?]
3) Try to construct another line b that has exactly one point in common with line l. If it is
possible, explain your construction method. Otherwise, explain why it is not possible to
construct line b.
This is possible to do. There are lots of examples.
4) Try to construct another line c that has exactly two points in common with line l. If it is
possible, explain your construction method. Otherwise, explain why it is not possible to
construct line c.
This is not possible to do because there is only one line that will pass through two points
on the plane.
5) Try to construct another line d that has more than two points in common with line l. If it is
possible, explain your construction method. Otherwise, explain why it is not possible to
construct line d.
The two lines would have to coincide for this to occur.
6) Which of your lines are parallel? Why?
This is covered in prompt 2).
7) Describe all the different ways in which two distinct lines can intersect on the plane.
Two distinct lines can either intersect at exactly one point or not at all (parallel lines). [If
the lines coincide, then they are really the same line.]
Make a Guess:
8) Will your conclusions be the same for great circles on the sphere?
Investigation on the Sphere:
9) On the sphere complete steps 1)-5) from above, replacing “line” with “great circle” in each
step. Keep track of which constructions are possible on the sphere.
2) Not possible – see 10)
3) Not possible
4) Two distinct great circles always intersect at two points on the sphere.
5) This would occur if the two great circles coincide.
10) Which of your great circles are parallel? Why?
You can have parallel circles on the sphere, but not parallel great circles. Great circle
always have two points in common. Circles that are not great circles do not satisfy all the
symmetry characteristics.
11) Describe all the different ways in which two distinct great circles can intersect on the sphere.
Two distinct great circles always intersect at two points. [If two great circles coincide,
then they are really the same great circle.]
12) One characterization of parallel lines on the plane is that they are always the same distance
apart. What happens if you try to do this on the sphere? Draw a great circle on your sphere.
Now draw a different figure that is always the same distance from your great circle. [Over]
Describe the resulting figure. Decide if this figure is a great circle. Explain. [Hint: Use
your spherical compass to do the drawings.]
Your spherical compass draws circles centered at the point you place the point of the
compass. [One could think of the distances on the compass as giving a “radius” of the circle
– the distance from the center along the surface of the sphere. The reason for the double
quotes is that there are two radii for circles, but this is not the place for that discussion.] You
will end up with parallel circle, but not parallel great circles – see 10).
Additional Investigations:
13) Pairs of vertical angles (opposite angles) are always congruent on the plane. (Explain why
this is true.) Construct three or four pairs of vertical angles on the sphere and determine if
this result is always true on the sphere. Justify your response.
Vertical angles are congruent on the sphere. To see this, try thinking in terms of your
symmetries. [An actual proof can be done with transformations – symmetries – but needs to
be written carefully to account for properties on the sphere and so you do not assume
something that may or may not be true on the sphere or assuming what you want to prove.]
14) Describe all the ways that three distinct lines can intersect on the plane. Now describe all the
ways that three distinct great circles can intersect on the sphere.
Three distinct lines can either intersect at one point – the lines would be concurrent – or
at three points in the plane. Three distinct great circles can either intersect at two points –
antipodal points [this is the sphere equivalent of concurrent] – or at six points on the sphere.
Compare the Plane and the Sphere:
15) Create a compare/contrast chart to record your observations from this investigation. Your
chart should have two columns, one for the plane and one for the sphere.
Plane:
• Two distinct lines with no points in common are parallel lines.
• Two distinct lines can intersect at exactly one point.
• Two distinct lines can never have more than one point in common.
• Three distinct lines can either intersect at one point or three points.
• Vertical angles are congruent on the plane.
Sphere:
• Two distinct great circles are never parallel to each other.
• Two distinct great circles always have exactly two points of intersection.
• Three distinct great circles either intersect at two points or at six points.
• Vertical angles are congruent on the plane.
• You can have parallel circles on the sphere.
Exploration 3:
Objectives:
Perpendicular lines intersect in a special way on the plane and on the sphere.
• Describe two perpendicular lines on the plane.
• Describe two perpendicular lines on the sphere.
If a line is perpendicular to two other lines at once, it is a common perpendicular to those lines.
• Investigate the common perpendiculars of two lines on the plane.
• Investigate the common perpendiculars of two great circles on the sphere.
Investigation on the Plane:
1) Construct two intersecting lines that divide the plane into regions that are all congruent.
Measure and label all the angles at each point of intersection of these two lines.
The angles should all have measure 90 degrees.
2) The two lines you constructed are perpendicular to each other. (How could you justify this
without using angle measures?) Record any observations you can make about perpendicular
lines on a plane in one column of a compare/contrast table.
(You can justify this by using paper folding to show the angles are congruent, and use
that a straight angle has angle measure of 180 degrees.)
• A pair of perpendicular lines intersects once and creates four right angles.
• A pair of perpendicular lines partitions the plane into four infinite (but congruent)
regions.
3) Draw two intersecting lines on the plane. Now try to construct some lines that are
perpendicular to both of the original lines. How many common perpendiculars could you
draw? (Why?)
You can be perpendicular to one of the lines, but not the other. (This illustrates Euclid’s
Fifth Postulate – you are trying to create a transversal with two lines that are not parallel.)
4) Construct two parallel lines on the plane. Now try to construct some lines that are
perpendicular to both of the parallel lines. How many common perpendiculars could you
draw? (Why?)
You can draw infinitely many lines that are perpendicular to both. (The common
perpendicular is acting like a transversal.)
5) Record any observations you can make about common perpendiculars on the plane in the
same column of the compare/contrast table you made above.
• Two intersecting lines do not have a common perpendicular.
• Parallel lines have infinitely many common perpendiculars.
Investigations on the Sphere:
6) Construct two great circles that divide your sphere into regions that are all congruent.
Measure and label all the angles at each point of intersection of these two great circles.
The angle measures should all be 90 degrees.
7) The two great circles you constructed are perpendicular to each other. (How could you
justify this without using angle measures?) Record any observations you can make about
perpendicular lines on a sphere in the second column of your compare/contrast table.
(You can justify this in similar manner as on the plane with symmetries.)
• Two perpendicular great circles intersect twice and form eight right angles.
• Two perpendicular great circles partition the sphere into four finite congruent regions.
8) Draw two intersecting great circles on the sphere. Now try to construct some great circles
that are perpendicular to both of the original great circles. How many common
perpendiculars could you draw? (Why?)
There will be only one common perpendicular line. (This comes properties of lunes – see
the next activity – or a concept called parallel transport, i.e., rotate the great circle that has a
perpendicular great circle about an intersection of the original two great circles along the
common perpendicular; the perpendicularity is retained along the rotation. You could also
treat the common perpendicular as a mirror and see that the two original great circles
intersect the mirror at right angles.)
9) Construct two parallel great circles on the sphere. Now try to construct some great circles
that are perpendicular to both of the parallel great circles. How many common
perpendiculars could you draw? (Why?)
This is not possible to do because great circles are not parallel.
10) Record any observations you can make about common perpendiculars on the sphere in the
second column of your compare/contrast table.
• Two intersecting great circles have exactly one common perpendicular.
• Even two great circles that are perpendicular to each other will only have one common
perpendicular.
Bonus Exploration:
Use your spherical ruler to draw a small triangle on your sphere. Then extend each side of your
triangle until you have drawn an entire great circle. Label the great circles a, b, and c. Now
draw a common perpendicular to the great circles a and b. Label this great circle p. Draw a
common perpendicular to the great circles a and c. Label this great circle q. What do you notice
about the common perpendicular to great circles p and q? Why is this case?
You have to experience this one by actually doing it to fully understand or “see it.” By
the construction method great circles a and p are perpendicular to each other. Also, by
construction great circles a and q are perpendicular to each other. Thus a is perpendicular to
both p and q. However p and q intersect each other and only have one common perpendicular.
Thus the common perpendicular must be a.
Exploration 4:
Objectives:
• Determine if it is possible to create a two-sided polygon.
Investigation on the Plane:
1) Using different lines on the plane draw two rays having the same endpoint. Will the two rays
intersect each other if extended indefinitely? The two rays divide (partition) the plane into
different regions. Describe the size and the shape of the regions.
The two rays will not intersect again. They will partition the plane into two infinite
regions.
2) A polygon is a (simple) closed geometric figure in the plane composed of only line segments.
Explain whether it is possible to create a two-sided polygon on the plane.
This is not possible to do on the plane – if it were possible the two rays would have to
intersect a second time.
Make a Guess:
3) Can you make a two-sided polygon on a sphere?
Investigation on the Sphere:
4) Using different great circles on the sphere draw two arcs having the same endpoint. Extend
the arcs. Do the arcs intersect each other at another point? The two extended arcs partition
the sphere into different regions. Describe the size and shape of the regions.
The two arcs meet at antipodal points. The two extended arcs determine two finite
regions on the sphere.
5) How would you modify the definition of polygon so that it works on the sphere? Is it
possible to create a two-sided polygon on the sphere? Explain.
Here is one possible definition: A polygon is a closed geometric figure on the sphere
composed of only arcs from great circles.
It is possible to create a two-sided polygon on the sphere. The two arcs interesting at
antipodal points create a closed geometric figure, so the definition is satisfied.
6) A polygon on the sphere that has exactly two angles (and two sides) is called a biangle (or a
lune). The measures of the angles in a biangle cannot be greater than 180 degrees. Mark a
biangle on your sphere.
This would be the smaller of the two regions created as described in 4).
7) Measure the angles and sides of the biangle. What did you learn from this? Will this always
be the case? Explain why or why not.
The two angles have the same measure and the two sides have the same length. This will
always be true in any biangle (or lune). The sides being the same length holds because they
connect two antipodal points so have length of 180 degrees. The angles are a little harder to
see. Think about this from the other perspective of the great circle being the intersection of
the sphere and a plane passing through the center of the sphere. The angle between the sides
is the same as the angle between the two planes that correspond to the two great circles.
8) Summarize the results of this activity in a compare/contrast table where one column contains
your observations from the plane and a second column contains your observations from the
sphere.
Plane:
•
•
•
Two rays with a common endpoint (originating from two different lines) will never meet.
The two rays with a common endpoint will create two infinite regions on the plane.
It is not possible to create a two-sided polygon on the plane.
Sphere:
• Two arcs of great circles will meet at a pair of antipodal points to form a two-sided
polygon.
• The two-sided polygon or biangle has exactly two congruent sides and two congruent
angles.
• The two arcs forming a biangle partition the sphere into two finite biangles!
Bonus Explorations:
A) Draw two great circles on the sphere. How many biangles do your two great circles create?
What relationships do you see between the biangles that you create? Why? [Can you draw
two great circles so that all four biangles are congruent? If so, explain how this is possible.]
You will get four biangles in this case. The opposite biangles are congruent to each other.
To see this refer back to 7) and use the fact that vertical angles are congruent on the sphere.
[If the two great circles are perpendicular to each other, then all four biangles will be
congruent to each other.]
B) A regular polygon is a polygon where all the sides are congruent to each other and all the
angles are congruent to each other. Explain why we would say a biangle (lune) is always a
regular polygon.
This is a re-iteration of 7), but associating it with the concept of regular polygon.
Exploration 5:
Objectives:
• Investigate how many different triangles you can make by connecting three points with
line segments on the plane.
• Investigate how many different triangles you can make by connecting three points with
arcs on the plane.
• Explore the nature of triangles that you construct on the sphere.
Investigations on the Plane:
1) Draw three noncollinear points on the plane. How many different triangles can you construct
that have these three points as vertices? Why? [What happens if the points are collinear?]
You can only create one triangle using the given three noncollinear points on the plane.
This is because any pair of points can only be connected by one line segment. [You need to
remember that a triangle is only the three segments – it does not include the interior region.
If you include the interior region, you are working with a triangular region – a twodimensional object. Triangles are only one-dimensional objects. We do not make this
distinction in every day usage of the word triangle, but in Geometry this distinction is
important and can be a source of misconceptions about properties of triangles – or other onedimensional objects like circles and polygons.]
Make a Guess:
2) Starting with three noncollinear points, how many different ways can you connect these three
points on a sphere to make a triangle?
Investigations on the Sphere:
3) Draw three noncollinear points on your sphere. Using different colored markers show that
your three points do not determine a unique spherical triangle. How many different spherical
triangles can you construct on a sphere using your three chosen points? How do you know
this?
[See comment in 1) to remind yourself what you are looking for when we talk about
triangles. This is especially important when making sense of the concept of triangle on the
sphere.]
There are 8 different spherical triangles that you can construct with you chosen three
(noncollinear) points. This is because between each pair of points, you can construct two
arcs, i.e., two possible options for sides to a triangle. Since there are three pairs of points, we
would end up with 2x2x2 or 8 possible options for choosing sides for a triangle.
4) If we decide to only use the shorter of the two arcs when we construct spherical triangles,
then how many different triangles can you construct on the sphere? How is this similar or
different from the situation on the plane? [A spherical triangle meeting this additional
condition is called a small (spherical) triangle.]
There will be only one triangle that can be formed under these conditions, as there is only
one option for connecting each pair of points. This would be reminiscent of the situation on
the place.
5) What happens if two of your three points on the sphere are opposite of each other? (These
opposite points are called antipodal points.) Explain why these three points do not determine
unique spherical triangles.
6)
7)
8)
9)
The two antipodal points can be connected with infinitely many different arcs from great
circles. Thus, in this case, there will be infinitely many spherical triangles that can be formed
using your three chosen points.
Explain why it is impossible for a triangle on the plane to have more than one right angle.
What would happen if you tried to make a triangle on the plane with two right angles? What
about on the sphere? Is it possible to make a triangle on the plane with more than one right
angle? Why or why not?
Suppose you did have two right angles in a triangle on the sphere. In this case, you
would have two of the sides of your triangle would be parallel to each other. (This follows
from properties of two lines cut by a transversal.) Thus these two sides would not intersect
each other; however, this contradicts that we started with a triangle (so all three sides must
connect at vertices). Therefore, there cannot be two right angles in a triangle on a plane.
However, on a sphere you can have two right angles in a triangle (even three right
angles!). You can see this if you take the equator as the great circle producing one side of
your triangle. Take one of the vertices to be the North Pole on the sphere (or South Pole).
Any arc extending from the North Pole to the equator will be perpendicular to the equator.
Isosceles triangles on the plane have congruent base angles. Explain why this is true without
using any of the congruent triangle theorems. Now determine if an isosceles triangle on the
sphere will have congruent base angles. Provide a rationale for your conclusion(s).
Idea on the plane: Reflect one half of the isosceles triangle through the angle bisector of
the third angle in the triangle – think mirror reflection. (Some refer to this third angle as a
summit angle.) Paper folding techniques work as well [Note that with paper folding you
have to leave the plane on which the object is drawn and enter the third dimension in order to
do the physical fold – using the mirror ideas allows you to stay on the plane the whole time.]
Yes, isosceles triangles on the sphere will have congruent base angles. If you think in
terms of the reflection symmetry, then you can again reflect half of the isosceles triangle
through the angle bisector of the third angle. With a piece of the spherical transparency you
can also do a “paper folding” technique, but you have to be aware of the caveat noted in the
previous paragraph.
Is it possible to construct an equilateral triangle on the sphere? If so, describe how to do it.
If not, explain why you cannot do it. If you can construct an equilateral triangle, how many
different ones can you construct?
Yes, see above – a triangle with three right angles will be an equilateral triangle. There
are many examples of this. The main point here is that the equilateral triangle can have angle
measures other than 60 degrees. (A spherical triangle with three 60 degree angles in not
possible.)
Repeat prompt number 8) for equiangular triangles on the sphere.
Equiangular triangles (all angles are congruent) are equilateral triangles, so see 8).
Exploration 6:
Objectives:
• Investigate the sum of the measures of the interior angles of a planar triangle.
• Investigate the sum of the measures of the interior angles of a spherical triangle.
Investigations on the Plane:
1) Provide a convincing explanation for why the sum of the measures of the interior angles of a
planar triangle is always constant.
This refers back to the patty paper (paper folding) exercises you did prior to the
combined session at North Central College.
Investigations on the Sphere: (For this activity, focus on the not so strange spherical triangles.)
2) Construct three to four spherical triangles on your sphere. Carefully find the sum of the
measures of the interior angles of the each of your triangles. What do you notice? Why do
you think this is?
The sum of the measures of the interior angles of a triangle adds up to different numbers
of degrees, depending on the size of the triangle. Your sums should be between 180 degrees
and 540 degrees. [Great circles are able to intersect each other in more ways than lines can
intersect on the plane.]
3) What do you think is the smallest possible sum of angle measures for a spherical triangle?
Find an example to illustrate your conjecture.
You can get a triangle where the angle sum is 180 degrees; however, this is what we can
a degenerate case – it can happen but is not what we would normally consider a triangle. It is
here to provide a distinct boundary and to make more advanced ideas and characterizations
easier to discuss. Here is how it could happen: All three vertices are on the same great circle
and one side lies on top of the other two sides. This produces two angles with measures of 0
degrees and one angle with measure of 180 degrees.
4) What do you think is the largest possible sum of angle measures for a spherical triangle?
Find an example to illustrate your conjecture.
You can get a triangle where the angle sum is 540 degrees; however this is also a
degenerate triangle. Here is how it could happen: All three vertices are on the same great
circle, and partition the great circle up into three arcs – all of which are used to make the
triangle. Thus you would end up with three 180-degree angles.
Additional Explorations:
A) If possible construct a rectangle on the sphere. If this is possible, construct one. If this is not
possible to do, explain why one cannot construct a rectangle on a sphere.
This is not possible to do because you cannot create parallel sides that are great circles.
Being able to construct a rectangle separates Euclidean Geometry from Non-Euclidean
Geometries. Rectangles will only exist in Euclidean Geometry!
B) Construct a great circle l and mark points A and B on l such that A and B are not antipodal
points. Construct two congruent arcs AD and BC such that each is perpendicular to l.
Finally, construct arc DC. [Remember that arcs are parts of great circles!] The arc AB is
called the base, and arc DC is called the summit. What can you determine about the summit
angles and the diagonals of this quadrilateral? Compare and contrast the properties of this
quadrilateral with those of a rectangle on a plane.
This polygon should be possible to create on the sphere, but may be tricky to get
constructed with the tools we have. The summit angles will be congruent, but are not right
angles. The diagonals are congruent as well. [These can be proven, but are beyond what we
can do at this point.] This polygon is analogous to a rectangle. If a summit angle could be
made as a 90-degree angle, then you would have a rectangle.
C) Investigate the sums of the interior angles of quadrilaterals on the sphere? What do you
notice? What are the bounds on this sum?
Here the sums will vary between 360 degrees and 720 degrees. Including degenerate
quadrilaterals will yield the 360 and 720 values.
Resources:
You can the Styrofoam spheres at a craft store or Wal-Mart; just make sure that your have rubber
bands that are big enough. I would recommend not going smaller than a 3-inch diameter – 2.5inch diameter at the smallest.
You can also use old tennis balls for spheres. You just need to make sure that most of the “fuzz”
has been worn off of the ball – they need to be just rough enough to hold the rubber band when
you have a great circle and smooth enough for the rubber band to come off when you only have a
circle. (When the PE Department or YMCA or tennis club is ready to throw the balls out is
when they are ready to be used in a Math classroom.)
Here is a link to a free ware piece of software that allows you to complete sketches on the screen
similar to Geometer’s Sketchpad, but is for Spherical Geometry. The software is called
Spherical Easel.
http://merganser.math.gvsu.edu/easel/index.html
Here is a reference to a source of blackline masters of lessons for doing spherical geometry in the
middle school or high school setting – with connections to Geography and other real-world
settings. Many of the explorations above were modified from this source.
Lenart, Istvan (1996). Non-Euclidean Adventures on the Lenart Sphere: Investigations in
Planar and Spherical Geometry. Key Curriculum Press, Berkley, CA.
Note: Key Curriculum Press has sold off several of its products to other companies or dropped
them from distribution; this was one of them. The book is available through Amazon. (Or
ordering directly from the author. This comes from his home country of Hungary. Do a Google
search on the author’s name.)