Explorations For Spherical Geometry
Exploration: Spherical Geometry
Materials: Smooth inflatable ball, magic marker, wet sponge, string.
Geodesics
1. Pick two points on a sphere. What is the shortest way to get from one to the other?
2. You live on a sphere. If you walk "straight" (or fly), what will your path look like?
3. A piece of string pulled tight between two points is "straight". Try this with a ball.
Describe the "straight" lines on a sphere. These are called geodesics. Draw some
geodesics. Do they work well with all three of the above ideas?
4. How many geodesics are there connecting the North and the South Pole?
Between
In the plane, if three points are on a line then one is always between the other two. Is this
true for on a sphere?
5. Can you give a meaningful definition of "between"?
6. Consider the geodesic that is the Earth’s equator. Pick any two points on the equator.
With your definition, what points on the Earth lie between these two points?
7. Two points directly opposite each other are called antipodal points, for example the
North and South poles of the Earth are antipodal. With your definition of between,
which points on the Earth are between the North and South poles?
Circles
A circle is the curve of all points which are the same distance from a given point, called
the centre. We can use the same definition in spherical geometry.
8. Draw some circles on the sphere, by marking a center point and using a piece of
string to find all points at a fixed distance from it.
9. As the radius gets larger, what happens to the circle? Then what happens? Then
what happens?
10. In the plane, a circle has an inside which is finite and an outside which is infinite. Do
circles have insides and outsides on the sphere? Explain.
Postulates
The five postulates (assumptions) for Euclidean geometry are:
I. There is exactly one straight line joining any two points.
II. Any straight line can be extended forever.
III. There is a circle with any given center and radius.
IV. The plane looks the same at every point.
V. Given a line and a point not on the line, there is exactly one line through the given
point which is parallel to the given line.
11. If you replace “straight line” with “geodesic”, most of these are wrong for spherical
geometry. How would you adapt them to spherical geometry?
12. In plane geometry, two triangles which have equal side lengths are congruent. This is
called Side-Side-Side. Is SSS still true in spherical geometry? What about SideAngle-Side, is SAS still true? ASA? In plane geometry, there's no AAA.. why not?
Is there AAA in spherical geometry?
Exploration: Drawing polygons on the Sphere
We have seen before that a polygon in the plane is defined as follows: A polygon is a
closed figure made by joining line segments, where each line segment intersects exactly
two others. On the sphere we will use a similar definition:
Definition:
A polygon on the sphere is a closed figure made by joining geodesic
segments, where each geodesic segment intersects at most two others.
1. In Euclidean (Planar) Geometry there are 3-, 4-, 5-gons etc., but there are no 1- and 2gons. Are there 1-,2-, 3-, 4-, 5-gons etc. on the sphere? Draw some examples of those
that exist.
2. Focus on the 3-gons for a moment. Are there regular triangles? If so, what are their
angle measures?
3. Draw 4 different triangles (different sizes) and measure the sum of their angles.
4. Based on your experiment in question 3, what can we say about the sum of the angles
in a triangle on the sphere? Give a convincing argument.
5. Are there quadrilaterals on the sphere? If there are, how would you construct one?
6. Are there any squares or rectangles on the sphere? Why or why not?
7. What kind of tessellations can you draw on the sphere? Which of the planar
tessellations may be adapted to the sphere? Which ones can definitely not be adapted
(if any)?
Exploration: Isometries on the Sphere
1. On the sphere below draw an example of a translation.
2. On the sphere below draw an example of a rotation.
Center of rotation; 45 degrees counter clockwise
3. On the sphere below draw an example of a reflection.
4. On the sphere below draw an example of a glide-reflection.
Spherical Tessellations
5. Below, draw examples of tessellations of the sphere. Experiment with 2-gons (also
called lunes), triangles, and other polygonal shapes.
6. For the spherical tessellation to the right, answer the following:
a. What is the highest order
rotation?
b. Are there reflections?
c. Are there glide-reflections?
d. What is the underlying
geometric tessellation?
7. For the spherical tessellation to the right, answer the following:
a. What is the highest order
rotation?
b. Are there reflections?
c. Are there glide-reflections?
d. What is the underlying
geometric tessellation?
Images by Andrew Crompton http://www.cromp.com/tess/home.html
Exploration: Introduction to Spherical Tessellations
Regular Tessellations by Triangles
Let's build a regular tessellation of the sphere by demanding that 4 equilateral triangles
meet at each vertex.
1. What corner angles will each triangle have?
2. What defect will each triangle have?
3. What fraction of the sphere will each triangle cover?
4. How many such triangles will we need to cover the sphere?
Draw on a ball this regular tessellation of the sphere.
5. Answer questions 1-4 assuming three equilateral triangles meet at a vertex.
6. What are other possibilities for number of triangles meeting at a vertex? Do these
give spherical tessellations?
Kaliedotile
Kaleidotile is a free computer program, available at http://geometrygames.org
Play with Kaleidotile. It's pretty cool. Try changing the basepoint, spinning the picture,
and playing with the view options and symmetry groups. Today, we're mostly interested
in the (2,3,3), (2,3,4), and (2,3,5) symmetry groups, which give spheres.
7. Explain the flat vs. curved option.
8. Use Kaleidotile to find your regular tessellations from part I. What are the names of
the flat versions?
9. Which of the other named tessellations are regular tessellations?
10. On the View menu, you can choose More Symmetry Groups. Try /2, /2, /2. Try
/2, /2, /3. What happens for other choices?
Exploration: Tessellations of the Sphere with Kaleidotile
Assume the base point triangle in Kaleidotile is labeled as indicated above.
1. Complete the following table
Symmetry BaseRegular or semigroup
point
regular tessellation?
(2,3,3)
A
Regular
B
Semi-regular
C
Regular
D
E
F
G
Semi-regular
(2,3,4)
A
B
C
D
E
F
G
(2,3,5)
A
B
C
D
E
F
G
There are 3 regular tessellations of the plane:
Triangles: (3,3,3,3,3,3) Squares: (4,4,4,4)
Vertex type
(3,3,3) - all triangles
(3,6,6) - triangles and hexagons
(3,3,3,3) - all triangles
(4,6,6) - quadrilateral and hexagons
Hexagons: (6,6,6)
There are 8 semi-regular tessellations (Page Error! Bookmark not defined.) of the
plane:
(3,3,3,4,4)
(3,6,3,6)
(3,3,4,3,4)
(3,3,3,3,6)
(4,8,8)
(3,4,6,4)
(3,12,12)
(4,6,12)
2. How many regular tessellations of the sphere did you find? List their vertex types.
3. How many semi-regular tessellations of the sphere did you find? List their vertex
types.
Exploration: Area on the Sphere
All spherical triangles have angles adding up to more than 180°. We called the
amount over 180° the defect of the triangle. This project should help convince you
that a triangle covers a fraction of the sphere equal to
defect
720°
.
Run Kaliedotile.
1. Use the (2,3,3) symmetry group and move the basepoint until you get a tetrahedron.
Set Kaliedotile to show the curved tessellation. Turn on the mirror lines. The sphere
is now covered with congruent (identical) triangles.
a. How many of these triangles are there (it may help to look at the flat view).
b. What fraction of the sphere does one of these cover?
c. What are the corner angles of these triangles?
d. What is the defect of one of these triangles?
e. Does the formula for area fraction in terms of defect check out?
2. Repeat question 1, but use the (2,3,4) group and the cube with mirror lines.
3. Repeat question 1, but use the (2,3,5) group and the icosahedron, with mirror lines
turned off.
The area of a sphere is 4π R2, where R is the radius of the sphere.
4. If we have a sphere of radius 1 inch, what is the area of the sphere? What are the
units?
5. If we have a sphere of radius 5 inches, what is the area of the sphere? What are the
units?
6. Suppose we have a 90°-90°-90° triangle. Sketch one. What fraction of the sphere does
this triangle cover? Suppose it lies on the sphere of radius 5, what it the surface area
of the triangle? What's the area on a sphere of radius R?
7. Suppose we have a 60° lune. Sketch one. What fraction of the sphere does this lune
cover? Suppose it lies on the sphere of radius 1, what is the surface area of the lune?
What's the area on a sphere of radius R?
Exploration: Platonic Solids
1. Look at Escher’s Reptiles (Schatt. pg 113, Magic pg 175). What platonic solid is the
focal point of this picture?
2. Look at Escher’s Crystal (Magic pg. 102). It is built out of two intersecting platonic
solids. Which two are they?
3. Look at Salvador Dali’s The Sacrament of the Last Supper (1955). What platonic
solid forms the window?
4. Fill in the following table:
Shape
# of vertices
Tetrahedron
Cube
Octahedron
Dodecahedron
Icosahedron
Can you find any patterns in this table?
# of edges
#of faces
Explorations For Hyperbolic Geometry
Exploration: Hyperbolic Paper
Equipment needed: Triangle paper, scissors, tape.
You can make Euclidean space by gluing equilateral triangles together so that six touch at
each vertex (this is just the usual tessellation by triangles).
You made an icosahedron by gluing equilateral triangles together so that five touch at
each vertex. This corresponds to a tessellation of the sphere, and makes a pretty good
model for an actual curved sphere. Gluing four, three, or two triangles also makes a
sphere, of sorts.
Using triangle paper, cut and tape triangles together so that seven triangles meet at every
vertex. Your group might want to make more than one of these models.
Some tips:
 • The less taping you do, the better. Try to tape on chunks of connected triangles at a
time, folding along the edges for flexibility.
 • To start, it's helpful to cut out an entire hexagon of six triangles, then make a slot to
the center.
Keep building until you've got at least one vertex which is surrounded by two rings of
triangles.
Each triangle in your model is flat, so geodesics are the usual straight lines. When a line
crosses a fold in the model, flatten out the fold and continue the line in the usual straight
way.
Try drawing a long straight line or two on your model. Use light pencil, or things will get
cluttered later.
1. Start two "parallel" lines as shown:
Continue the two lines in both directions. What happens? Do they stay the same
distance from each other?
2. Draw a triangle on the hyperbolic paper. How does its angle sum compare to 180°?
3. What happens to the angles of triangles when the triangles get larger?
Exploration: Escher’s Circle Limit Series
Recall that in Spherical Geometry we determined that shortest distance was measured by
great circles. We called the distance minimizing curves geodesics, and these geodesics
play the same role as lines do in Euclidean Geometry.
"Circle Limit III" by M.C.Escher
In Circle Limit III, the white lines are meant to represent straight lines in this new
geometry. These curves are also called geodesics. Segments of these geodesics will form
the sides of polygons.
1. What type of polygons do you see in this figure?
NOTE: In this picture we should interpret all fish as having the same size. A smaller
figure does not mean that it is physically smaller. Think of it as being farther away.
2. Looking at the angles in a triangle, what will the angle sum be? Equal to, greater than
or less than 180o? Why?
"Circle Limit I" by M.C.Escher
3. What is the highest degree of rotation?
4. Use a colored marker, and draw in some geodesics (the hyperbolic version of a
straight line). The Easiest way of doing this is to follow the spines of the fish. What is
the underlying geometric tessellation?
"Circle Limit II" by M.C.Escher
5. What is the highest degree of rotation?
6. Draw geodesics in this figure. What is the underlying geometric tessellation?
"Circle Limit IV" by M.C.Escher
7. What is the highest degree of rotation? What other degrees of rotation are present?
8. Draw geodesics in this figure. What is the underlying geometric tessellation?
9. Draw a geodesic NOT passing through the center point.
10. How many geodesics can you draw through the center point, so that the new geodesic
does not meet the geodesic you picked? Another way to ask the same question: How
many geodesics pass through the point so that the new geodesic is parallel to the first
geodesic?
Exploration: Isometries of the Hyperbolic Plane
We have seen that a disk may model the hyperbolic plane. Technically, the disk will not
contain its boundary. In this geometry, the geodesics (curves which give you the shortest
distance between two points) are line segments passing through the center, and semicircles meeting the boundary at a right angle.
We have seen that the hyperbolic plane contains 3-, 4-, 5-, … , n -gons. There are no 1
and 2- gons in this geometry. There are however ideal n- gons: these are n-gons whose
vertices lie on the boundary, which we call the circle at infinity.
Next we want to investigate the behavior of isometries in hyperbolic space. We have seen
that the isometries on the sphere behave as expected. Translations move figures around,
and rotations rotate shapes around. Reflections over geodesics exist, as well as glidereflections. There were no major surprises there. Now we need to think about
translations, rotations, reflections and glide-reflections in the hyperbolic plane.
You should think carefully about the following questions. Your answers should be at
least a paragraph for each, and include drawings to illustrate the ideas.. Some answers
may require more than a paragraph. Be thorough in answering the questions.
1. For each of the four hyperbolic tessellations Escher created, determine and sketch the
underlying geometric tessellation.
2. For each tessellation determine what the translations look like. Describe carefully
what hyperbolic translations look like.
3. For each tessellation determine what the rotations look like. Describe carefully what
hyperbolic rotations look like.
4. For each tessellation determine what the reflections look like. Describe carefully what
hyperbolic reflections look like.
Exploration: Computer Assisted Hyperbolic Geometry
NonEuclid is a Java program, written by Joel Castellanos, for doing geometry in
hyperbolic space. You can run it from his web page, at:
http://cs.unm.edu/~joel/NonEuclid
When you have the web page showing, click the big button to run NonEuclid Applet.
It should display a large black circle - that's the Poincaré disk.
Like Geometer's Sketchpad, NonEuclid lets you draw points, lines, and circles. The
menus "File" (#2), "Edit" (#2), "View", "Constructions", "Measurements", and "Help" are
for the applet.. these are to the right of the normal menus for the browser.
When you start, you should be in "Draw Line Segment" mode, as shown in the upper left
corner. Draw some line segments.
1. What do lines that go through the center of the disk look like?
To clear your drawing, use New on the File menu. Now draw a triangle. On the Edit
menu, choose "Move Point". Now you can drag the corners of your triangle and see how
it changes.
On the Measurements menu, select "Measure Triangle", and click the three corners of
your triangle. You should see the side and angle measurements of the triangle in the box
on the left. Go back to "Move Point" mode.
Move your triangle around to get a feel for the lengths of its sides and the sum of its
angles.
2. Draw a triangle which appears large but really has sides of length under 4.
Draw a triangle which appears small but really has sides of length over 10. (The
"micro-move" feature may help: hold down Shift while you move the vertices).
3. Try to make the angle sum 180°. What do you have to do?
4. Try to make all three sides of the triangle large. What happens to the angle sum?
Draw an "Infinite Line" (on the Constructions menu). Now use "Reflect" to make the
reflection of your triangle across your infinite line. Move the line around, and notice the
position and size of the reflection. The reflection is the same size and shape as the
original, it just appears different.
If you have time, try the “Hyperbolic Applet” at
http://www.math.umn.edu/~garrett/a02/H2.html.
Watch the hyperbolic animations at:
http://www.josleys.com/animationsindex3.htm
(#2,4, and 5 are the best)
Exploration: Ideal Hyperbolic Tessellations
1. Draw an example of a 4-gon (vertices in the interior of hyperbolic space) and an ideal
4-gon in hyperbolic space. Label your drawings.
2. Draw an example of a 5-gon (vertices in the interior of hyperbolic space) and an ideal
5-gon in hyperbolic space. Label your drawings.
3. Draw an example of a 6-gon (vertices in the interior of hyperbolic space) and an ideal
6-gon in hyperbolic space. Label your drawings.
Creating an ideal tessellation:
4. Create an ideal tessllation with the following steps:
Step 1: draw a regular, ideal 4-gon.
Step 2: Draw in both diagonals so that the polygon consists of 4 triangles. Color the
triangles alternately black and white.
Step 3: Between two adjacent vertices put 2 equally spaced vertices. Construct
another 4-gon with the 4 vertices you thus created.
Step 4: Draw in both diagonals so that the polygon consists of 4 triangles. Color the
triangles alternately black and white.
Now repeat Step 3 and 4
5. Create an ideal tessllation with the following steps:
Step 1: draw a regular, ideal 4-gon.
Step 2: Connect opposite midpoints so that the polygon consists of 4 quadrilateral.
Color the quadrilaterals alternately black and white.
Step 3: Between two adjacent vertices put 2 equally spaced vertices. Construct
another 4-gon with the 4 vertices you thus created.
Step 4: Divide the quadrilateral into 4 smaller ones, as in step 2. Color the
quadrilaterals alternately black and white.
Now repeat Step 3 and 4
6. Construct an ideal tessellation based on an ideal 6-gon. Use some consistent coloring
scheme to “decorate” the tessellation.
Exploration: Tessellations of the Hyperbolic Plane
1. Below, sketch an ideal tessellation for a triangle, a quadrilateral, a pentagon, and a
hexagon.
Triangle
Quadrilateral
Pentagon
Hexagon
2. Generalizing from the work so far, what ideal polygons tessellate the hyperbolic
plane? How many regular tessellations of the hyperbolic plane do we get?
*** These are not regular tessellations - they’re ideal.
3. Below, sketch a tessellation for a triangle with quadrilaterals surrounding it, a
quadrilateral surrounded by triangles, a pentagon with triangles, and a hexagon with
quadrilaterals.
Triangle surrounded by quadrilaterals
Quadrilateral surrounded by triangles
Pentagon surrounded by triangles
Hexagon surrounded by quadrilaterals
4. Generalizing from the work so far, how many semi-regular tessellations of the
hyperbolic plane do we get?
*** These are not semi-regular, they’re ideal
5. Now create a tessellation with polygons that are not ideal:
The following website has hyperbolic tessellations:
http://www.josleys.com/creatures38.htm
6. What polygons appear in these tessellations? List the degrees of rotational symmetry
you see (and record which tessellation has that degree of rotational symmetry).
Example: hyp.01 consists of triangles, and has 6-fold symmetry at the center. It
possibly also has 4-fold symmetry.
Hyperbolic 01 Triangles
6-fold symmetry at the center; possibly also has 4fold symmetry.
Hyperbolic 02
Hyperbolic 03
Hyperbolic 07
Hyperbolic 10
Hyperbolic 11
Hyperbolic 12
Hyperbolic 16
Hyperbolic 21
Hyperbolic 28
Hyperbolic 29
Hyperbolic 35
Exploration: Hyperbolic Tessellations
Go to David Joyce’s hyperbolic tessellations web page:
http://aleph0.clarku.edu/~djoyce/poincare
and explore the regular tessellations of the Poincare disk.
1. In the Schläfli symbol {n,k}, what do n and k represent?
2. For these hyperbolic tessellations by triangles, what are the corner angles of each
triangle? What is the defect of each triangle?
a. {3,7}
b. {3,8}
c. {3,10}
d. {3,12}
Does there appear to be a relationship between area and defect?
3. Experiment with the (2,3,7) symmetry group in Kaliedotile.
4. Look at Salvador Dali’s Santiago el Grande (1957). Is the background hyperbolic?
The table below has one entry for each Schläfli symbol {n,k}. In each box, put an S, E,
or H depending on whether the corresponding regular tessellation is Spherical, Euclidean
(flat) or Hyperbolic.
k
2
3
4
5
6
7
8
n
2
3
4
5
6
7
8
Exploration: New Hyperbolic Tessellations From Old
There are a variety of different methods for constructing new tessellations from old. We
can find the midpoint of each side and connect adjacent points. We can find the dual, just
as in Euclidean geometry: Place a vertex inside each polygon, and connect two points if
and only if there is an edge between them.
1. In the two tessellations below, find the midpoint of all of the sides. Connect adjacent
points. What polygons is the new tessellation made out off?
2. Find the dual of the two tessellations below. That is to say: Place a vertex inside each
polygon, and connect two points if and only if there is an edge between them. What
polygons is the new tessellation made out off?
Exploration: Comparison Between the 3 Geometries
Compare and contrast the three geometries based on the following topics:
Euclidean
Spherical
Hyperbolic
Geometry
Geometry
Geometry
Geodesics
(What do they look like?)
Parallel Lines
(How many, if any?)
Polygons
(Do we get new ones? Which
ones don’t exist?)
Sum of the angles in a triangle.
Regular tessellations
(How many different ones are
there?)
Semi-regular tessellations.
(How many different ones are
there?)
Isometries
(Which ones have we
encountered?)
Area of a triangle.
(What is the formula?)
Escher’s work based on each
geometry.
(How extensively did he work
on each of the geometries?)
Other Artists?
(Did other artists work in any
of these geometries?)