J A N E T M. S H A R P
AND
CORRINE HEIMER
What Happens to
Geometry
on a
Sphere?
JANET SHARP, [email protected], teaches future and practicing K–12 teach-
ers at Iowa State University in Ames, IA 50011. She is interested in developing
mathematics teaching approaches to help students make connections within
mathematics and to other disciplines. CORRINE HEIMER, cheimer@stcecilia.
pvt.k12.ia.us, teaches sixth-grade mathematics and language arts, along with
fourth-, fifth-, and sixth-grade science, at Saint Cecilia Elementary School in
Ames, Iowa. She challenges her students to think creatively.
This article is lovingly dedicated to George Downing, a mathematician, professor, and friend who taught one of the authors to think beyond the plane.
182
W
E HAVE TO SHARE THIS WITH OUR
students! They will love it!” This
statement was all we could think
about after a professional development session dealing with geometry. Spherical geometry challenged our capabilities in geometry but
greatly interested us. Before we could teach our students about spherical geometry, we needed to learn
more about this strange new world ourselves. In this
article, we describe our discoveries and some of the
activities we developed for our sixth-grade students.
MATHEMATICS TEACHING IN THE MIDDLE SCHOOL
Copyright © 2002 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.
This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
Why Study Spherical Geometry?
IN ANCIENT CIVILIZATIONS, GEOMETRY LITER-
ally meant “the science of measuring the land”
(Greenberg 1972, p. 5). The typical study of geometry in modern classrooms works from the assumption that “the land” to be measured is flat. It resembles part of a mathematical plane. We know,
however, that our land is on the earth, which is, basically, a sphere. In fact, Robin (one of our students) decided that spherical geometry might be
used “if you were a meteorologist and you studied
what happens to the earth!”
The Basic Concepts
WE STARTED THE FIRST OF OUR TWO SESSIONS
with a review of vocabulary from plane geometry.
We knew that the students would have to be able
to visualize lines, segments, angles, and polygons
in a plane and easily identify properties and definitions of these concepts before moving their thinking onto the sphere. Armed with washable markers
and large plastic balls, we set to work. The students drew on the balls during our initial classroom discussions to help them visualize concepts
before we moved the discussion to include properties and definitions.
draw several more of their own lines, anywhere on
their spheres. As we walked around, we listened to
students’ conversations about whether or not their
drawings were lines. We asked them how they
knew if they had drawn a line and were confident
that they understood the concept when we heard
such answers as the following:
• “A great circle is a line on the fattest part of the
sphere.”
• “[My drawing goes] in a straight line and not . . .
zigzag.”
• “A line connecting two points.”
• “It is a line because it is on a great circle.”
Students must understand the definition of a great
circle and its relationship to the definition of a line.
Melissa said, “A great circle is the only way you can
draw a line on a sphere, or in other words, a line is a
great circle.” Jessica’s definition was “A great circle
is a line that goes all the way around. There are an
infinite number of them.” After assuring ourselves
that all students understood the idea of a line on a
sphere, we revisited the idea of a segment that had
started the lesson earlier in the day. Together, we
established the following definitions:
A great circle is a straight line and a segment is a
portion of a great circle.
Line and segments on a sphere
We began by trying to translate planar ideas onto a
spherical surface. We first considered a line. Be
forewarned! Students’ mental images of a line drastically change when dealing with spheres. We
strongly encourage teachers to use the plastic balls
in this discussion because, as our student, Alicia,
said, “It was easier to learn about spheres’ [figures]
on a ball than on paper.” To begin the exploration,
we plotted two points on a sphere and stretched a
string between them. One child held the string taut,
pressing it between the two points, while the other
child sketched a line segment along the edge of the
string. Then we extended that segment in one direction, presumably indefinitely (as we might imagine on a plane). The children found that the extension eventually wrapped around and met the other
endpoint! We all held our taut strings along that
line for a quick check. We explained that a great circle is a circle that circumscribes a sphere at its
largest girth. Students realized that on a sphere, a
line is actually a great circle. Moreover, the segment that we had created between the two points
was a portion of that great circle.
We were excited to watch our students stretch
and expand their ideas of a line! We asked them to
A great circle is a path that circumscribes a
sphere at its largest girth.
Establishing a conceptual understanding of line
is paramount in further developing the children’s
expanding geometric ideas and concepts for geometry on the sphere. Our students’ lively discussion
led us to understand the importance of (1) abandoning the notion that a line must be level and (2)
thinking differently about the notion of straightness. On the sphere, when viewed from above, a
segment appears to be flat and straight, but from
the side, the segment clearly follows the curvature
of the sphere. For example, “a roadway is considered to be ‘straight’ (i.e., a line) if you could drive a
car on it without turning left or right (even though
you do go up and down hills). To drive a great circle
on a sphere, you drive straight. To drive a circle on
the plane, you are always turning left (for counterclockwise) or right (for clockwise)” (Hogben 2001).
On the plane, we tend to think of circles as curved
because we can always view them from above and
see the curvature. On a sphere, a great circle is a
straight line, and conversely, on a sphere, a straight
line is a great circle! (See fig. 1.)
V O L . 8 , N O . 4 . DECEMBER 2002
183
PHOTOGRAPHS BY JANET M. SHARP AND CORRINE HEIMER; ALL RIGHTS RESERVED
Fig. 1 Two great circles
latitudes are lines. Latitudes are not great circles.
Hence, those circles are not straight; they are
curved. The students had explored map and globe
ideas in social studies, but they had never thought
about the mathematical potential of the lines of longitude and latitude.
This activity was vital in establishing the concept
of a line, which includes different mental visualizations than the student typically encounters in plane
geometry. We encouraged our students to hold
this unfamiliar image in their minds before they explored shapes and figures on the sphere. As Principles and Standards for School Mathematics points
out, “Students must carefully examine the features
of shapes in order to . . . identify relationships
among the types of shapes” (NCTM 2000, p. 233).
To solidify the concept of a line, we drew some
counterexamples. The students drew circles that
were not great circles on their spheres and compared them with one of their lines (great circles) to
try to distinguish between the two. Many students
noticed that a great circle cut the sphere in half.
Robin referred to this process as “going through
the exact center [of the sphere].” Several students
pointed out that if they sliced the sphere along a
“non-great” circle, the cut would not pass through
the center of the sphere.
This activity allowed us to make the following
statement:
A great circle is the intersection of a sphere with a
plane that passes through the center of the sphere.
What does a ray look like on the sphere?
Fig. 2 Measuring on the globe
To further visualize this idea, we looked to a realworld example. We considered an airplane’s flight
path from Iowa to Moscow. What would be the
most direct route? At first glance, the children
wanted to follow one of the latitudes on the globe.
When we asked them to look again, Katie was
“pretty sure going in a straight line would be
quicker than swerving.” At first, a flight following a
route between 40˚ N latitude and 55˚ N latitude
seemed intuitive. We were all surprised to learn
that the shortest route (a segment) passed over
Greenland and did not follow a latitude. One student, Robert, refused to believe that the shortest
path went across Greenland until he used string to
literally measure the two distances for himself. See
figure 2, which shows Robert and Scott measuring
on the globe.
Students were amazed that all lines of longitude
are great circles, and that except for the equator, no
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MATHEMATICS TEACHING IN THE MIDDLE SCHOOL
The children were quick to accept that rays cannot
be formed on a sphere. Extending a segment in one
direction caused the extension to meet back up with
the other endpoint of the segment. We had already
named that figure a line.
What do parallel lines look like on a sphere?
Carefully maintaining the definition of a line as a
great circle, our students found another anomaly
that they were not so quick to accept. At first, they
believed that drawing parallel lines on the sphere
was possible and tried to draw a pair of them. The
typical first strategy was to draw two nonintersecting figures. A student would sketch a line on his or
her sphere, then plot a point anywhere not on the
line. Next, the student slowly sketched a figure,
right next to the line, in such a way that the figure
maintained an equal distance from the original line.
Certainly, the result was a collection of points that
Fig. 3 A circle and a non-circle
PHOTOGRAPHS BY JANET M. SHARP AND CORRINE HEIMER; ALL RIGHTS RESERVED
was equidistant from the original line, which is a familiar property of parallel lines on a plane. These
two objects appeared to have a necessary property
of parallel lines. However, the children soon recognized that the second figure was not a line because
it was not a great circle! (See fig. 3.) Once again,
we turned to the globe to help illustrate the point.
At first glance, the latitudes appear to be parallel because they are always equidistant from each other.
Remember, however, that except for the equator,
latitudes are not lines in spherical geometry because latitudes are not great circles. On further investigation, the children tentatively suggested that
drawing parallel lines on a sphere may be impossible. We could almost see this realization dawn on
the children one by one. We marveled at how we
could imagine their mental wheels turning.
Just to be sure, we considered another property: On the plane, parallel lines do not intersect.
On the sphere, all lines (great circles) eventually
intersect. Clearly, this exploration was not proof
because it was based on properties rather than definitions, but it satisfied the students’ developing
conjecture that parallel lines as defined for a plane
could not be drawn on a sphere. In addition, students had previously discovered that any two longitudes on the globe intersected at two points: the
North and South Poles. The students had to depend on their new visualizations about lines to analyze their drawings. They studied their figures to
decide that no two lines on the sphere had those
familiar properties of parallel lines. Annie observed, “It is impossible to have parallel lines on
the sphere; they will touch eventually, but on a
plane, they will never touch.” This sort of analysis
lays the foundation for the formation of informal
deductions, a vital skill in geometric thinking,
whether on the plane or the sphere.
Fig. 4 John’s demonstration of 90-degree angles
Students’ Efforts to Draw Squares
What do perpendicular lines look like on a sphere?
EXPERIENCE WITH PARALLEL LINES AND 90-
The children were eager to return to the sphere and
explore whether or not drawing two segments to
form a 90-degree angle was possible. In figure 4,
John shows what the point of intersection between a
line and a perpendicular segment looks like on a
sphere. John extended his two segments into a pair of
intersecting lines and measured the resulting four angles on the other side of the sphere. All angles were
90 degrees. Finally, we had found a plane idea that
had a corresponding existence on a sphere! Perpendicular lines exist on a sphere, but perpendicular lines
(like all lines on the sphere) intersect at two points,
and both intersections are perpendicular. Once again,
on the sphere, something extraordinary had happened to our previous planar ideas about geometry.
degree angles caused our students to wonder, “What
about squares?” We challenged them to consider
drawing a square, and they valiantly set to work. At
first, they relied on their well-known definition of a
plane square (four equal sides and four 90-degree angles) and their ability to use visualization to “draw geometric objects with specified properties such as side
lengths or angle measures” (NCTM 2000, p. 232).
Students started by trying to draw a shape with
four equal sides and four 90-degree angles (see fig.
5a). One by one, they determined that squares
would be impossible to draw. When they got to point
D, where they would have to draw the fourth side,
they found that they had to make a choice. If they
drew a fourth segment to connect point D to point A,
the segment would be too short, and neither the
V O L . 8 , N O . 4 . DECEMBER 2002
185
b
c
d
PHOTOGRAPHS BY JANET M. SHARP AND CORRINE HEIMER;
ALL RIGHTS RESERVED
a
Fig. 5 Attempting to make a square
third nor the fourth angles would be 90 degrees (see
fig. 5b). If they drew a fourth segment to create a
third 90-degree angle, it would not intersect the first
segment at a fourth 90-degree angle or at point A
(see fig. 5c). If they drew a segment that was the
same length as the first three sides, it would not join
with point A (see fig. 5d). We encourage readers to
try this same process with a ball and a marker.
In the end, our students determined that squares,
as we had defined them, did not exist on a sphere.
They decided that the best they could do was to draw
a “trapezoid-shaped” quadrilateral. We were excited to
see that only moments after they began their attempts
to draw the fourth sides of their squares, students
made the informal deduction that a square could not
exist on a sphere because it would lack necessary
properties (only three equal sides and/or only three
90-degree angles). We were impressed with their acceptance of this anomaly. They had determined that
the existence of a square depends on the definitions in
geometry: A square would be impossible on a sphere
because it would lack parallel sides. The students’
mathematical maturity appeared to be growing.
We were not disappointed. We simply reminded
students that a triangle’s sides, which are segments,
would fall on great circles. Undaunted, our students
washed off their spheres and drew again. This time,
the students drew true (spherical) triangles with
sides (segments) correctly following great circles.
They commented that these triangles “bulged” out
along their sides. (This bulging appearance is related to the fact that the sides fall along the curvature of the sphere.) Justin looked at his triangle and
observed, “It expands!” We did not worry that we
challenged the students’ existing images of triangles. As they moved from thinking of a triangle as a
whole figure to looking at its definition, they
learned to accept a seemingly contradictory image.
This process enabled them to take yet another step
in their growing understanding of the important
role of definitions in geometric thinking.
Later, some of our students took this fact to heart
and excitedly drew three-sided polygons that would
challenge anyone’s idea of a triangle. Figure 6, for
example, shows Andy’s odd triangle. Andy’s figure is
a triangle because it is consistent with our definition,
PHOTOGRAPH BY JANET M. SHARP AND CORRINE HEIMER; ALL RIGHTS RESERVED
Students’ Efforts to Draw Triangles
ONE OF THE MOST FASCINATING CHALLENGES
we presented to the students was to invite them to consider their mental images of a triangle and their understanding of how to draw a triangle. After a brief review
of the definition of a triangle as being a three-sided
polygon, the students appeared to be ready to draw triangles on their spheres. At first, students just quickly
sketched triangles. Most of these early sketches
“looked” exactly like a triangle from the plane. We
were not surprised that some of the students pasted
their existing mental images from the plane onto the
sphere, rather than analyze a triangle and consider its
individual properties. We know students build new
knowledge on what they already understand. With
their growing sense of spherical geometry’s figures,
we anticipated a smooth exploration of triangles.
186
MATHEMATICS TEACHING IN THE MIDDLE SCHOOL
Fig. 6 Andy’s triangle
PHOTOGRAPH BY JANET M. SHARP AND CORRINE HEIMER;
ALL RIGHTS RESERVED
ments were accurate, they were not surprised by a
third triangle’s angle sum of 256 degrees. One of the
students wondered what the greatest sum would be.
Someone else questioned if a 180-degree triangle
could even be drawn. Alison stated, “I thought it was
interesting that a triangle measured more than 180
degrees. At first, I thought I measured wrong!” We
were pleased about the high levels of these
thoughts. As an extension, we challenged students
to consider the angle sum of a quadrilateral. They
quickly decided that the angle sum would be greater
than 360 degrees because the quadrilateral is composed of two triangles, each of which has an angle
sum that is greater than 180 degrees.
Fig. 7 A lune
which did not include a requirement for endpoints to
be joined along a shortest-distance pathway. The students stood firm in the conviction that their individual figures were, indeed, triangles if their figures
matched the definition. We believe that their confidence was an example of our students’ blossoming
understanding of the nature of geometry, particularly
regarding the importance and role of definitions.
Their ability to abandon a holistic visualization of triangles in favor of a collection of figures that followed
a definition was remarkably mathematician-like.
The students made amazing discoveries when
they measured the three angles in their triangles.
All sums were more than 180 degrees! One triangle’s angle sum was 292 degrees! A different triangle’s angle sum was 190 degrees. At first, the children wondered if they had measured incorrectly,
but after studying to be certain that their measurePolygons
Our Reflections
THE STUDENTS WERE AMAZED AT WHAT THEY
had discovered and were quite proud of themselves. “This is college mathematics!” Ben said. We
believe our students most enjoyed learning about
concepts in spherical geometry that drastically contradicted ideas from its planar counterpart. The students could see that they could draw some of the
special polygons on the sphere but were even more
excited to draw a two-sided polygon (see fig. 7),
which is impossible in plane geometry! They giggled when we introduced them to the term lune to
name this unique polygon. Rachel said, “It’s really
cool that you can make a two-sided figure.” When
comparing the sphere with the plane, Caleb added,
“You can only make it [a lune] on a sphere.”
We used the worksheet in figure 8 to help students reflect on their discoveries. During reflection
A polygon is a closed figure with edges that are formed of line segments joined
at their endpoints.
1. Draw a square on your sphere. What happened?
2. What is the least number of sides a polygon could have on a plane?
3. Is there a way to draw a two-sided polygon (a lune) on a sphere? On the plane?
Triangles
A triangle is a polygon with exactly three sides.
4. Plot three points on your sphere and label them X, Y, Z.
5. Draw a segment to connect X and Y, a segment to connect X and Z, and one to connect Y and Z.
6. What shape did you create?
7. Draw the funniest looking triangle you can! What do you notice?
Fig. 8 A spherical geometry worksheet
V O L . 8 , N O . 4 . DECEMBER 2002
187
time, our students continued to show
the wonderful curiosity of children. In
her journal, Tessa wrote, “I think it
would be fun to do this with a pyramid.” How interesting that she is moving her thinking to a different solid!
When we give children new worlds to
explore and new challenges to consider, their thinking can be stretched
well beyond current limits. We believe
that our students demonstrated fine
mathematical maturity during the
lessons and we encourage other
teachers to try similar ideas.
References
Greenberg, Marvin Jay. Euclidean and
Non-Euclidean Geometries. San Francisco, Calif.: W. H. Freeman & Co.,
1972.
Hogben, Leslie. Personal communication,
6 February 2001.
National Council of Teachers of Mathematics (NCTM). Principles and Standards for School Mathematics. Reston,
Va.: NCTM, 2000. 188
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