1
2
3
3) a. STATEMENT: The vertical angle theorem indicates that two intersecting lines whose
opposing vertical angles are equal.
GIVEN: Pairs of intersected rows (straight lines) showing rotation (changing in direction)
PROVE THAT: ∠1 ≅ ∠3, Showing they are not the same, but have equal magnitude and
opposite directions
Assume that ∠1 and ∠2 form a linear pair, so by the Supplement Postulate, they
are supplementary.:
m∠1 + m∠2 = 180°
(1)
Assume that ∠2 and ∠3 form a linear pair also:
m∠2 + m∠3 = 180°
( 2)
Compare Equation 1 and Equation 2 using the transitive property of equality.
Subtracting m∠2m∠2 from both sides of both equations, we get,
m∠1 + m∠2 = m∠2 + m∠3
m∠1 = 180° - m∠2 = m∠3
∠1 ≅ ∠3 (∴ PROVED)
You can also use the same method in proving that ∠2 ≅ ∠4.
4
b. As we can see on the clock diagram on number 1, this works on a sphere’s surface. In a
spherical triangle, the interior angles add up to more than 180 degrees. Meaning it is a
reflex angle. Whether the diagram rotates or turns up to 180° and more, it fits perfectly.
Polygonal figures on the sphere all have a total turning angle of 2 Pi minus their area.
c. In this case, I use the dynamic concept of a moving angle or angle as a movement. We can
say that the linear pair, angles 1 and 3, are always congruent in the direction they are
oriented, proving that the angles are congruent as they move in different directions. The
diagrams show that whatever turning point, rotation, and the direction it faces will always
be congruent.
4) a. STATEMENT: The vertical angles theorem states that the steep angles of two
intersecting lines opposed to each other are congruent.
y
x
Y
W
D
A
O
C
B
Z
GIVEN: Two intersecting lines, x, and y are given.
PROVE THAT: ∠C ≅ ∠A and ∠D ≅ ∠B
PROOF: Straight line measures = 180°
X
5
∠XOY + ∠WOY = 180°
∠C + ∠D = 180°
(1)
∠WOY + ∠WOZ = 180°
∠D + ∠A = 180°
( 2)
Both Equation 1 and Equation 2 equal 180. Thus:
∠C + ∠D = ∠D + ∠A
Subtract ∠D from both sides
∠C ≅ ∠A (∴ PROVED)
Next,
∠WOY + ∠WOZ = 180°
∠D + ∠A = 180°
(3)
∠WOZ + ∠ZOX = 180°
∠A + ∠B = 180°
(4)
Both Equation 3 and Equation 4 equal 180. Thus:
∠D + ∠A = ∠A + ∠B
Subtract ∠A from both sides
∠D ≅ ∠B (∴ PROVED)
b. Although a sphere is a curved surface, flat (planar) Euclidean geometry laws are good
approximations of local geometry on a sphere. It is only slightly more than 180 degrees in
total angle on Earth’s surface when the angles are added together. This proves that the
straight line measuring 180° represents and follows a great circle’s curves and straightness,
and yes, it works on a sphere, and we did an activity in class to show this fact. We used the
elastic thread to draw 2 great circles on a a ball, and intersected them to create 90 degrees
angle between them. These intersected circles divide the ball in 4 quarter and form
ongorunt angles equal 90 degrees.
6
c. I employ the Angle as a measure in this case. Each line creates one degree of an angle, and
you can measure angles in degrees. Every line matches up with every other line and creates
a 180° angle. This demonstrates that each line is congruent with the previous one.
5) a. STATEMENT: The vertical angles theorem states that the steep angles of two
intersecting lines opposed to each other are congruent.
GIVEN: A and B intersected straight lines , 1& 2 are opposite angles.
As the diagram above, we notice that a reflection has happened over the line symmetry.
Because of that the shape of space, the pair of opposite angles show isometric symmetry.
It is a mirror reflection over the line of symmetry. Thus, these two angles are congruent
because the reflection keeps the measure of angles the same, each point or space must keep
its position from the line of symmetry.
b. We can use this proof on the surface of a sphere and we did that before in class by taking
a ball, and use elastic strick to make two great circles and intersected them to produce
vertical angles. This work produced a reflection of the line of symmetry, and we made the
two great circles perpendicular to each other to produce 90-degrees angles, so the sphere
7
was divided into four equal pieces, and this implies that the pair of opposite angles are
congruent.
c. I used a geometric shape to prove the vertical angles theory. for this meaning of angle, I
have to show the isometric symmetry by showing the reflection of these two angles to
prove that they are congruent.
1
HOMEWORK 3
Q--1. Helix
Helices are produced by straight lines drawn on planes when they are coiled over the
cylinder, particularly a right-angle circular cylinder, which looks like the curve created by the
screw's curvature. See the figure below:
When you draw a straight diagonal across a square paper and roll it into a cylinder, a helix
will be formed. Thus, a helix in a cylinder is intrinsically a straight line and is, thus, a geodesic.
When a shorter line is drawn on a paper and the paper is rolled into a cylinder, it will still form a
shorter form of the helix.
Cross Section of the Cylinder
When a cylinder is sliced along to its circular base, the cylindrical cross-sectional area is
equivalent to circle's area. Since a circle is a 2D plane, when laid flat, it will be seen as a straight
line. Thus, this line is geodesic because it is intrinsically straight. This circle is also considered as
the great circle of the cylinder. Take a look at the figure below:
2
When a horizontal straight line is drawn on a paper and the paper is rolled into a cylinder,
the straight line will form a circle. However, when a short line is drawn on the paper, it will yield
a circular arc.
Straight Line (The height)
The last straight line in the surface of a cylinder is rather the most obvious one. The line
that transverses the height of the cylinder is a geodesic straight line because any straight line is
considered a geodesic. This line is considered the vertical generator.
3
Q--2. Case 1: the two points lie on the same great circle
If two points lie on the same great circle, then only 1 geodesic segment exists. This geodesic
segment is also known as the circle’s diameter. The cylinder is drawn in different perspectives.
The first one is in an upright perspective, the second one is from the top view, and the third one is
from the front view. Refer to the illustration below:
A
A
B
B
A
B
Case 2: the two points do not lie on the same great circle
If the two points do not lie on the same great circle, only 1 geodesic segment exists as well.
This line creates vertically opposite angles since the two great circles are parallel to each other.
The figure is drawn in the upright view and the front view. Refer to the illustration below:
A
A
B
B
4
Q--3. Reflection Paper
“Defining Supports Geometry”
One fundamental task that is undervalued among mathematicians is the activity of defining,
which is important because it enables the mathematician to make sense of issues and identify
suitable definitions by soliciting and developing new ideas via debate. With the implementation of
the Common Core State Standards for Mathematics, a great deal of emphasis will be put on the
eight Standards for Mathematical Practice that precede the main content of the mathematics
curriculum. This set of eight Practices outlines the attitudes and behaviors that instructors should
encourage in their pupils as they work together to study mathematics. The first is "Make sense of
issues and persist in solving them," the second is "Construct valid arguments and criticize the
thinking of others," and the third is "Make sense of problems and persevere in solving them". It is
possible to incorporate the practice of asking students to construct definitions in each of the
aforementioned activities.
When arithmetic vocabulary terms are introduced in the classroom, either at the beginning
of a unit or throughout a unit, instructors usually require students to record the meanings of the
words in their notebooks. After the terms have been presented, students will next study the ideas
that are contained in the definition. When teaching definitions, a second method involves first
engaging students with the idea and then presenting the vocabulary word as a means to identify
the notion once students have generated meaning for it. As the name implies, this kind of defining
is descriptive since it entails explaining a previously established mathematical concept.
Despite the utilization of this second method in the bulk of teaching in “Defining Supports
Geometry”, the teachers have offered a third way to defining, which they have referred to as the
constructive approach. It entails changing pupils' prior understanding of a word while
simultaneously developing a new concept. The students in eighth-grade class were able to develop
a class definition for interior angles via the use of a series of activities. The teachers have
demonstrated that the use of constructive defining provided students with additional chances to
develop more relevant methods for determining their own formulae for the sum of the interior
angles of a polygon when utilizing constructive defining.
Reading this article had given me the understanding that if the learning curriculum is
children- or student-centered, they will be able to come up with their own ideas and would also be
able to create definitions of various mathematical concepts. This ability of the youngsters to
remove or modify their inputs and definitions when other pupils commented and critiqued their
efforts astounded and astonished me. The fact that they were able to connect prior teachings with
geometrical ideas that were being taught at the time the article was written is also important to
note! Furthermore, since students learned how to properly define the geometrical concepts, they
were able to comprehend the material more effectively than they had before. Therefore, this article
may be utilized as a guide to assist instructors in becoming more effective. This, in turn, will make
students become more intelligent, their critical thinking abilities will be enhanced, and they may
even acquire or improve their 21st century talents as a result.
5
Q--4. SAS works in flat planes. However, SAS does not seem to work in spheres.
C
A
C’
A’
B
B’
CASE 1
In the first figure of Case 1, it is evident that the shortest distance between two points, A and B, is
the geodesic line AB. In the second figure, line BA is the longest distance between the points
mentioned. This line completes the triangle formed by line AC and line BC.
For these triangles, it is known that
∠𝐴𝐶𝐵≅∠𝐴′𝐶′𝐵′
𝐴𝐶≅𝐴′𝐶′
𝐶𝐵≅𝐶′𝐵′
The bases of the first triangle, which is AB and B’A’, are not equal. Therefore, SAS does not apply
on these spherical triangles.
Case 2:
In Case 2, it also shows the shortest and longest distance between points A and B. However,
different triangles are now shaded in this sphere.
6
C’
C
A
A’
B
CASE 2
Just like in Case 1, it is known that
∠𝐴𝐶𝐵≅∠𝐴′𝐶′𝐵′
𝐴𝐶≅𝐴′𝐶′
𝐶𝐵≅𝐶′𝐵′
However, since the bases 𝐴𝐵 ≠ 𝐵′𝐴′, then SAS is not true on spheres.
B’
47/50
HOMEWORK 4
+10
1. Def 1: A triangle on the sphere is small if all angles are less than 180 degrees.
Given two small triangles (small per definition 1) that have side-angle-side congruent, prove
that the two triangles are congruent.
On a sphere, this is a practical method because all vertical angles are equal. If we decide that
the term “triangle” only applies to sides shorter than half the sphere’s radius, SAS will keep
hold of the sphere. Because great circles are used to define angles in small triangles, spherical
trigonometry, which differs from traditional trigonometry in that the sum of the interior angles
of the latter is never less than or greater than 180 degrees, is used.
A
A
90°
𝐴𝐵
40°
𝐴𝐶
90°
𝐴𝐶
50°
𝐴𝐵
C
B
𝐵𝐶
A
90°
𝐴𝐶
50°
40°
𝐴𝐵
C
B
50°
40°
C
𝐵𝐶
B
𝐵𝐶
Figure 1. Two Triangles, 1 form, Same Direction
Figure 2. Triangle on a Scenic Route
The figure on the left has an angle and is less than 180°. Hence, it has two congruent sides
because it links in only one (1) form or turns in the same direction. Now, consider the second
figure over on the right side. It has an angle but is greater than 180° because it took a long path.
You can’t take the scenic route because it ends up with an angle greater than the bare minimum.
Also, the total of the angles of a triangle is no longer 180 degrees. Angles add to just over 180
degrees in small triangles (because, from the perspective of a very small triangle, the surface of
a sphere is nearly flat). Angles in larger triangles will exceed 180 degrees.
∴ Two small triangles on a sphere that has a side-angle-side are congruent.
2. Def 2: A triangle on the sphere is small if all sides are less than ½ of a great circle.
+10
Given two small triangles (small per definition 2) that have side-angle-side congruent,
prove that the two triangles are congruent.
Y
C
B
𝐴𝐶
𝐴𝐵
𝑍𝑌
X
A
𝑍𝑋
Z
Prove that: ∆ BAC ≅ ∆ YZX
Proof (1):
𝐴𝐵 = 𝑍𝑌
Line AB and Line ZY are equal.
𝐴𝐶 = 𝑍𝑋
Line AC and Line ZX are equal.
∠𝐵𝐴𝐶 = ∠𝑌𝑍𝑋
Angle BAC and Angle YZX are equal.
∴ Triangle ABC and Triangle ZYX are congruent. (∆ ABC ≅ ∆ ZYX)
1
2
B
Proof (2):
C
1
𝐴𝐵 𝑎𝑛𝑑 𝐴𝐶 = 4
1
𝐵𝐶 = 2
1
4
1
4
Y
1
𝑍𝑋 𝑎𝑛𝑑 𝑍𝑌 =
4
1
2
1
𝑋𝑌 =
2
1
4
X
1
4
Z
There is only one path between points (from Z to X, X to Y, and Y to Z – all links in one
form). The smaller sides are less than half the diameter of the great circle. Proving that a small
triangle on a sphere is less than ½ of the great circle.
3. Def 3: A triangle on the sphere is small if it is contained in a hemisphere.
+10
Given two small triangles (small per definition 3) that have side-angle-side congruent, prove
that the two triangles are congruent.
Proof (1):
For instance, we have a ball of tennis and three rubber bands. Pull two rubber bands in
opposite directions and loop them all around the tennis ball to intersect at right angles (imagine
the Equator and Prime Meridian). The rubber bands have created four identical regions on the
surface of the sphere. Divide the four areas of the ball into two unequal halves, like so. Each
rubber band must ensure that the ball is split into two equal hemispheres.
All eight of your triangles have interior angle measures that are more than 180 degrees.
You’ll end up with a triangle whose angles add up to 190 degrees if you play around with the
positioning of the third rubber band.
∴ The triangles on a sphere, when contained on a hemisphere, are small.
Proof (2):
A
∠D
C
B
A
∠D
C
B
Side-Angle-Side (SAS) congruency is observed in 𝐴, 𝐵, and ∠D in Figure 1. Because ∠D is
an interior angle and assuming that 𝐶 is a part of a great circle, we can say that the small triangle
in figure 1 are congruent and fit on a hemisphere. However, the triangle in figure 2 does not fit
into the hemisphere even though it does share the Side-Angle-Side congruency. Because 𝐶 took
the more scenic route, it completed the great circle that connects the triangle’s three points on
the longer run.
Figure 2. Not Congruent Triangle
Figure 1. Congruent Triangle
4. Are all three definitions of small triangles the “same”? For example, is Def 1 the same as Def
+3
2? The answer is yes if every triangle that meets the criteria for Def 1 also meets the criteria
for Def 2. Conversely, every triangle that meets the criteria for Def 2 also meets the criteria for
Def 1. If both things are true, then we say the two definitions are the same (or in more mathy
terms, they are equivalent).
Yes, based on the proof above, all three definitions of small triangles are precisely the same.
Def 1 is equivalent to Def 2 since small triangles that are less than the diameter of the great
circle makes up an angle that is less than 180°. Also, Def 1 is the same as Def 3 since two
triangles on a hemisphere sum up only a little more than 180 degrees because, from its
perspective, the sphere is nearly flat. Lastly, Def 2 is equivalent to Def 3 since small triangles
with less than the diameter of the great circle is small, and triangles contained on a hemisphere
are also small.
5. Which definition of the small triangle do you prefer and why?
+4
For me, all of them are excellent definitions of a small triangle since all of them meet the
criteria of each other. They are somehow connected, and all of them are facts of a small triangle.
These facts define a small triangle’s identity; that’s why they are all essential and share equal
status.
47/50
+12
1. ANGLE-SIDE-ANGLE TRIANGLES ON THE SURFACE OF A SPHERE
As long as one side and the adjoining angles of two triangles are congruent, the two
triangles are congruent, and vice versa. However, the surface of a sphere is different from a flat
surface. That is why ASA is not true for all triangles on the surface of the sphere.
A
B
C
A’
𝛼
CC
𝛽
CC’
B’
Figure 1. ASA in a Spherical Surface
A
A’
B
B’
𝛼
CC
𝛽
CC’
Figure 2. ASA Triangles Side by Side
In these triangles, angle ABC and angle 𝛼 is congruent with angle A’B’C’ and angle 𝛽
while side AB is congruent with side A’B’.
Figure 3. Spherical triangles on a plane
The following diagrams show a counterexample of ASA on sphere. This may be shown
using the parallel transport theorem, which can alternatively be derived from the fuller theorem.
As you spin the sphere, you'll see that the side-angle-side relationship remains the same, as does
the neighboring triangle, since the angle and side length are invariant under the rotation. Moreover,
if the spherical triangles were to be put in a plane, they would look like the illustration in Figure
3. Therefore, these triangles serve as a counterexample against ASA.
Figure 4. Unique ASA Triangle on a Sphere
This is another counterexample of ASA on a sphere. When in a sphere, it creates a unique
triangle and serves as a unique counterexample as well because, although it follows the angle-sideangle theorem, the congruency between triangles like this will not exist because it does not follow
the criteria for triangular congruency. Aside from that, this kind of triangle will create 2 different
triangles even though they are using the same side-angle-side.
2. ASA IS TRUE FOR SMALL TRIANGLES
+13
D
A
E
C
B
F
BACK
FRONT
Figure 5. ASA on Spherical Small Triangles
Using Def 3 which states that a triangle on the sphere is small if it is contained in a
hemisphere, ASA is true for small triangles. This is proven by the triangles in Figure 5 which
shows 2 triangles that are in the same sphere. Additionally, we know that these triangles have their
angles and side congruent, assuming that their angles and sides have the same values. Thus:
∠𝐴𝐶𝐵≅∠𝐷𝐹𝐸 and ∠𝐴𝐵𝐶≅∠𝐷𝐸𝐹
𝐵𝐶≅𝐸𝐹
Aside from that, it is observable that all lengths of each side of the triangles do not
exceed half of the great circle. Therefore, ASA is true for small triangles.
+12
3. ASA IS TRUE FOR ALL TRIANGLES IN A PLANE
Using the Angle-Side-Angle postulate, two triangles are said to be congruent if they have
two angles and an included side that match another triangle's two angles and an included
side. Using the Angle-Side-Angle Postulate, we can show that triangle ABC is identical to triangle
DEF, as shown in Figure 6. Since they are identical with each other, then this proves that ASA is
true for all triangles in a plane.
B
D
C
A
F
E
Figure 6. ASA Triangles on a Plane
46/50
HOMEWORK 6
1. a) Prove that the Isosceles Triangle Theorem (ITT) is true on the plane using SAS.
b) Prove that ITT is true on the plane using reflection in the angle bisector line.
c) Prove that ITT is true on the plane using reflection of the triangle across an arbitrary line
followed by a translation.
2. Google the term “corollary” and write down the meaning that you found.
3. Prove the following corollary of ITT: The angle bisector of the angle between the two congruent
sides of an Isosceles triangle is also a perpendicular bisector of the side opposite the bisected
angle.
[Note: there are two things to prove here – that the angle bisector bisects the opposite side, and
it does so at a right angle].
4. a. Prove ITT is true for all triangles on the sphere. Which of the three proofs from the plane you
want to adopt is up to you, but you need only one proof.
5. a) State the converse of ITT on the plane.
b) Prove the converse of ITT on the plane.
ANSWER:
1. a) We already know that in the Isosceles Triangle Theorem, the angles opposite to the equal
sides of an isosceles triangle are also equal in length and width.
+4
GIVEN: Consider an Isosceles Triangle XYZ with XZ≅YZ
WTP: ∠ ZXY ≅ ∠ ZYX
Z
PROOF:
XZ
X
YZ
W
Y
As shown in the figure, we begin by drawing a bisector of ∠ XZY and naming it ZW. Now,
in ∆ XZW and ∆ YZW, we have the following:
(Given)
XZ≅YZ
∠XZW≅∠YZW
(By Construction / Angles by Bisector)
ZW≅ZW
(Common to both Triangles)
Thus,
∆ XZW ≅ ∆ YZW
(By SAS Congruence)
∴ ∠ ZXY ≅ ∠ ZYX
(By CPCTC)
Angles opposite to the equal sides of an isosceles triangle are also equal. Hence, proved!
+2
b) Using reflection on the line, we must demonstrate that the Isosceles Triangle Theorem holds
R
for all triangles on the plane.
P
Q
P’
Q’
R’
GIVEN: ∆ PRQ is an Isosceles Triangle.
PR≅RQ
(Given)
PR≅P′R′
(By Reflection Property)
RQ≅R′Q′
∴ PR = RQ≅R′Q′ = P′R′
∴ ∠ RP’R’ ≅ ∠ RQ’R’
1
2
1
∠ RP’R’ ≅ 2 ∠ RQ’R’
∴ ∠ RPQ ≅ ∠ RQP
Isosceles Triangle Theorem is valid on the plane using reflection in the angle bisector line.
Note: P’ and P coincide, and Q’ and Q coincide.
+4
c) There are several steps involved in reflecting a triangle over an arbitrary line. Reflecting a
triangle over an arbitrary line is the act of flipping a shape, in this case, a triangle, over a line
that is not on the x or y-axis, as shown in the illustration. It is necessary to locate your b point
in your line, where your point to reflect will be, and which you will create yourself. Using your
starting point, you can complete the equation y= m (x) +b by using the coordinates from the
point. Your X’s will separate from each other at this point as well.
GIVEN: Consider a triangle whose vertices are (2, -4), (2, -2), and (4, -4).
WTP: ∠ CBA ≅ ∠ CAB
PROOF:
A A’
C
BC≅CA
(Given)
BC≅B′C′
(By Reflection Property)
C’
B’
BB
B
CA≅C′A′
∴ BC = CA≅C′A′ = B′C′
∴ ∠ CB’A’ ≅ ∠ CA’B’
1
2
1
∠ CB’A’ ≅ 2 ∠ CA’B’
∴ ∠ CBA ≅ ∠ CAB
Isosceles Triangle Theorem is valid on the plane using the reflection of the triangle across
an arbitrary line followed by a translation.
+4
2. A corollary is a theorem of lesser importance that can be easily deduced from a previous, more
significant statement in mathematics and logic (Wikipedia, 2019). With little or no additional
proof, a corollary is a proposition that is inferred immediately from a proven proposition
(Merriam-Webster). To sum it all up, a corollary is a theorem that follows on from another
theorem.
+8
3. When the angle between two congruent sides of an Isosceles triangle is bisected, a
perpendicular bisector of the opposite congruent side of the triangle is formed.
GIVEN: Let us consider ∆ IJK as an Isosceles Triangle.
K
IK
JK
I
J
L
As illustrated, we begin by drawing a bisector of ∠ IKJ and labeling it KL. With this, we
will prove the following:
i) KL also bisects the opposite side AB
:
IL≅JL
ii) KL is a perpendicular bisector
:
∠ ILK ≅ ∠ JLK = 90°
PROOF:
i) In ∆ ILK and ∆ JLK:
IK≅JK
(Given)
(Common to both Triangles)
KL≅KL
And,
∠ IKL ≅ ∠ JKL
(KL is a Bisector of the Angle)
∴ By SAS Congruence Criterion: ∆ ILK ≅ ∆ JLK
In ∆ ILK ≅ ∆ JLK:
IK
IL
(Congruent Angles)
JK≅JL
Since IK≅JK,
IK
IL
IK
IK≅JL
( is the same, we can cancel them both and refer to as 1.)
1≅JL
IL
(Here, we can cancel 1 and transpose JL to the other side.)
∴ JL≅IL
(Angle by Bisector)
IK
Hence, the angle bisector is also a bisector of the side opposite to the bisected angle,
proved!
ii) In ∆ IJK:
∠ J + ∠ I + ∠ K = 180°
(By Angle Sum Property)
Since ∠ I≅∠ J,
∴ ∠ 2A + ∠ C = 180°
In ∆ ILK:
𝐾
∠ I + ∠ L + ∠ 2 = 180°
(By Angle Sum Property)
(∠ C is Bisected)
∴ ∠ 2I + ∠ 2L + ∠ K = 360°
Putting ∠ 2A + ∠ C = 180° above, we get:
180° + ∠ 2L = 360°
∠ 2L = 180°
∴ ∠ L = 90°
Thus, ∠ ILK = ∠ JLK = 90°. Hence, KL is a perpendicular bisector, proved! The angle
bisector between two congruent sides of an isosceles triangle is perpendicular to the
side opposite the bisected angle.
4. GIVEN: ∠ B ≅ ∠ C
+6
WTP: AB≅AC
Construct AD ⊥ BC
A
AB
AC
B
D
C
PROOF:
∠B≅∠C
(Given)
AD ⊥ BC
(Construction)
AD = AD
(By the Reflexive Property)
∆ ABD ≅ ∆ ACD
(AAS)
AB≅AC
(CPCTE, Corresponding Parts of Congruent Triangles are
Equal)
m∠B=m∠C
∴ Isosceles Triangle Theorem is valid for all triangles on a sphere, valid for all triangles on
the plane, and valid for all triangles on a perpendicular bisector.
5. a) Converse of Isosceles Triangle Theorem states that if two angles of a triangle are congruent,
+8
then the sides opposite to these angles are congruent.
b) GIVEN: ∠ L ≅ ∠ M and a perpendicular bisector that does not go through point N.
WTP: LN ≅ NM
N
MN
LN
L
M
O
PROOF:
NO is the bisector of the vertex angle ∠ LNM
∠L≅∠M
(Given)
∠ LNO ≅ ∠ MNO
(Angle Bisector)
NO ⊥ LM
(Construction)
NO ≅ NO
(By the Reflexive Property)
(Half of ∆ LNO and half of ∆ MNO)
Reflection of ∆ LNO ≅ ∆ MNO, only intersect at one point in the plane.
N’
M’
L
N
L’
M
The bisector is not possible, and as a result of the contradiction, the bisector O must
pass through the intersection point N.
Hence,
∆ LNO ≅ ∆ MNO
(AAS Congruent)
∴ LN ≅ NM
(CPCTC)
CONCLUSION:
Since corresponding parts of congruent triangles are congruent, LN≅NM.
Hence, the Isosceles Triangle Theorem is proved!
Chapter 2
STRAIGHTNESS ON SPHERES
...
[I]t will readily be seen how much space lies between the two places themselves on the
circumference of the large circle which is drawn through them around the earth. ... [W]e
grant that it has been demonstrated by mathematics that the surface of the land and water is
in its entirety a sphere, ... and that any plane which passes through the center makes at its
surface, that is, at the surface of the earth and of the sky, great circles, and that the angles of
the planes, which angles are at the center, cut the circumferences of the circles which they
intercept proportionately, ...
— Ptolemy, Geographia (ca. 150 A.D.) Book One, Chapter II
This chapter asks you to investigate the notion of straightness on a sphere, drawing
on the understandings about straightness you developed in Problem 1.1.
EARLY HISTORY OF SPHERICAL GEOMETRY
Observations of heavenly bodies were carried out in ancient Egypt and Babylon, mainly
for astrological purposes and for making a calendar, which was important for organizing
society. Claudius Ptolemy (c. 100–178), in his Almagest, cites Babylonian observations
of eclipses and stars dating back to the 8th century B.C. The Babylonians originated the
notion of dividing a circle into 360 degrees — speculations as to why 360 include that it
Chapter 2 Straightness on Spheres 36
was close to the number of days in a year, it was convenient to use in their hexadecimal
system of counting, and 360 is the number of ways that seven points can be placed on a
circle without regard to orientation (for the ancients there were seven “wandering bodies”
— sun, moon, Mercury, Venus, Mars, Saturn, and Jupiter). But, more important, the
Babylonians developed a coordinate system (essentially the same as what we now call
“spherical coordinates”) for the celestial sphere (the apparent sphere on which the stars,
sun, moon, and planets appear to move) with its pole at the north star. Thus, it is a
misconception to think that the use of coordinates originated with Descartes in the 17 th
century. As a fun fact it can be mentioned the use of spherical coordinates in modern
times. One way to ensure that electronic message is not garbled is to use a geometric way
of packing information called a “spherical code” – that is a way of translating a message
written in binary code, into a point on a high-dimensional sphere by the way of relating
each letter to a coordinate on a sphere.
Figure 2.1 Armillary sphere (1687) showing (from inside out):
earth, celestial sphere, ecliptic, and the horizon
The ancient Greeks became familiar with Babylonian astronomy around 4th century
B.C. Eudoxus (408–355 B.C.) developed the “two-sphere model” for astronomy. In this
model the stars are considered to be on the celestial sphere (which rotates one revolution
a day westward about its pole, the north star) and the sun is on the sphere of the ecliptic,
whose equator is the path of the sun and which is inclined to the equator of the celestial
sphere at an angle that was about 24° in Eudoxus’ time and is about 23½° now. The
sphere of the ecliptic is considered to be attached to the celestial sphere and has an
apparent rotation eastward of one revolution in a year. Both of these spheres appear to
rotate about their poles. See Figure 2.1.
Autolycus, in On the Rotating Spheres (333–300 B.C.), introduced a third sphere
whose pole is the point directly overhead a particular observer and whose equator is the
visible horizon. Thus, the angle between the horizon and the celestial equator is equal to
the angle (measure at the center of the earth) between the observer and the north pole.
Autolycus showed that, for a particular observer, some points (stars) of the celestial
Chapter 2 Straightness on Spheres 37
sphere are “always visible,” some are “always invisible,” and some “rise and set.”
The earliest known mathematical works that mention spherical geometry are
Autolycus’ book just mentioned and Euclid’s Phaenomena [AT: Berggren] (300 B.C.).
Both of these books use theorems from spherical geometry to solve astrological problems
such as What is the length of daylight on a particular date at a particular latitude? Euclid
used throughout definitions and propositions from spherical geometry. The definitions
include A great circle is the intersection of the sphere by a plane through its center and
the intersection of the sphere by a plane not through the center forms a (small) circle that
is parallel to a unique great circle. The assumed propositions include, for example,
Suppose two circles are parallel to the same great circle C but on opposite sides; then the
two circles are equal if and only if they cut off from some other great circle equal arcs on
either side of C. (We will see similar results in Chapter 10.) There are other more
complicated results assumed, including one about the comparison of angles in a spherical
triangle; see [AT: Berggren], page 25. Thus, it is implied by Autolycus’ and Euclid’s
writings that there were previous works on spherical geometry available to their readers.
Hipparchus of Bithynia (190–120 B.C.) took the spherical coordinates of the
Babylonians and applied them to the three spheres (celestial, ecliptic, and horizon). The
solution to navigational and astrological problems (such as When will a particular star
cross my horizon?) necessitated relating the coordinates on one sphere with the
coordinates on the other spheres. This change of coordinates necessitates what we now
call spherical trigonometry, and it appears that it was this astronomical problem with
spherical coordinates that initiated the study of trigonometry. Plane trigonometry,
apparently studied systematically first by Hipparchus, seems to have been originally
developed in order to help with spherical trigonometry, which we will study in Chapter
20.
The first systematic account of spherical geometry was Sphaerica of Theodosius
(around 200 B.C.) It consisted of three books of theorems and construction problems.
Most of the propositions of Sphaerica were extrinsic theorems and constructions about a
sphere as it sits with its center in Euclidean 3-space; but there were also propositions
formulated in terms of the intrinsic geometry on the surface of a sphere without reference
to either its center or 3-space. We will discuss the distinction between intrinsic and
extrinsic later in this chapter.
A more advanced treatise on spherical trigonometry was On the Sphere by Menelaus
(about 100 A.D.) There exist only edited Arabic versions of this work. In the introduction
Menelaus defined a spherical triangle as part of a spherical surface bounded by three arcs
of great circles, each less than a semicircle; and he defined the angles of these triangles.
Menelaus’ treatise expounds geometry on the surface of a sphere in a way analogous to
Euclid’s exposition of plane geometry in his Elements.
Ptolemy (100–178 A.D.) worked in Alexandria and wrote a book on geography,
Geographia (quoted at the beginning of this chapter), and Mathematiki Syntaxis
(Mathematical Collections), which was the result of centuries of knowledge from
Chapter 2 Straightness on Spheres 38
Babylonian astronomers and Greek geometers. It became the standard Western work on
mathematical astronomy for the next 1400 years. The Mathematiki Syntaxis in generally
known as the Almagest, which is a Latin distortion of the book’s name in Arabic that was
derived from one of its Greek names. The Almagest is important because it is the earliest
existing work containing a study of spherical trigonometry, including specific functions,
inverse functions, and the computational study of continuous phenomena.
More aspects of the history of spherical geometry will appear later in this book in the
appropriate places. For more readings (and references to the primary literature) on this
history, see [HI: Katz], Chapter 4, and [HI: Rosenfeld], Chapter 1.
PROBLEM 2.1 WHAT IS STRAIGHT ON A SPHERE?
Drawing on the understandings about straightness you developed in Problem 1.1, this
problem asks you to investigate the notion of straightness on a sphere. It is important for
you to realize that, if you are not building a notion of straightness for yourself (for
example, if you are taking ideas from books without thinking deeply about them), then
you will have difficulty building a concept of straightness on surfaces other than a plane.
Only by developing a personal meaning of straightness for yourself does it become part
of your active intuition. We say active intuition to emphasize that intuition is in a process
of constant change and enrichment, that it is not static.
a. Imagine yourself to be a bug crawling around on a sphere. (This bug can
neither fly nor burrow into the sphere.) The bug’s universe is just the surface;
it never leaves it. What is “straight” for this bug? What will the bug see or
experience as straight? How can you convince yourself of this? Use the
properties of straightness (such as symmetries) that you talked about in
Problem 1.1.
b. Show (that is, convince yourself, and give an argument to convince others)
that the great circles on a sphere are straight with respect to the sphere, and
that no other circles on the sphere are straight with respect to the sphere.
SUGGESTIONS
Great circles are those circles that are the intersection of the sphere with a plane through
the center of the sphere. Examples include longitude lines and the equator on the earth.
Any pair of opposite points can be considered as the poles, and thus the equator and
longitudes with respect to any pair of opposite points will be great circles. See Figure 2.2.
Chapter 2 Straightness on Spheres 39
Figure 2.2 Great circles
The first step to understanding this problem is to convince yourself that great circles
are straight lines on a sphere. Think what it is about the great circles that would make the
bug experience them as straight. To better visualize what is happening on a sphere (or
any other surface, for that matter), you must use models. This is a point we cannot stress
enough. The use of models will become increasingly important in later problems,
especially those involving more than one line. You must make lines on a sphere to fully
understand what is straight and why. An orange or an old, worn tennis ball work well as
spheres, and rubber bands make good lines. Also, you can use ribbon or strips of paper.
Try placing these items on the sphere along different curves to see what happens.
Also look at the symmetries from Problem 1.1 to see if they hold for straight lines on
the sphere. The important thing here is to think in terms of the surface of the sphere,
not the solid 3-dimensional ball. Always try to imagine how things would look from the
bug’s point of view. A good example of how this type of thinking works is to look at an
insect called a water strider. The water strider walks on the surface of a pond and has a
very 2-dimensional perception of the world around it — to the water strider, there is no
up or down; its whole world consists of the 2-dimensional plane of the water. The water
strider is very sensitive to motion and vibration on the water’s surface, but it can be
approached from above or below without its knowledge. Hungry birds and fish take
advantage of this fact. This is the type of thinking needed to visualize adequately
properties of straight lines on the sphere. For more discussion of water striders and other
animals with their own varieties of intrinsic observations, see the delightful book The
View from the Oak, by Judith and Herbert Kohl [NA: Kohl and Kohl].
Water striders (Wikimedia Creative Commons)
Chapter 2 Straightness on Spheres 40
DEFINITION.
Paths that are intrinsically straight on a sphere (or other surfaces)
are called geodesics.
This leads us to consider the concept of intrinsic or geodesic curvature versus
extrinsic curvature. We all have what Felix Klein called “naïve intuition” — we speak
without hesitancy of the direction and curvature of a river or a road the same way in
geometry we talk about, for example, curvature of a circle, although the “line” in this
case has certainly considerable width. As an outside observer looking at the sphere in 3space, all paths on the sphere, even the great circles, are curved — that is, they exhibit
extrinsic curvature. But relative to the surface of the sphere (intrinsically), the lines may
be straight and thus have intrinsic curvature zero. See the last section of this chapter,
Intrinsic Curvature. Be sure to understand this difference and to see why all symmetries
(such as reflections) must be carried out intrinsically, or from the bug’s point of view.
It is natural for you to have some difficulty experiencing straightness on surfaces
other than the 2-dimensional plane; it is likely that you will start to look at spheres and
the curves on spheres as 3-dimensional objects. Imagining that you are a 2-dimensional
bug walking on a sphere helps you to shed your limiting extrinsic 3-dimensional vision of
the curves on a sphere and to experience straightness intrinsically. Ask yourself the
following:
What does the bug have to do, when walking on a non-planar surface, in order
to walk in a straight line?
How can the bug check if it is going straight?
Experimentation with models plays an important role here. Working with models
that you create helps you to experience great circles as, in fact, the only straight lines on
the surface of a sphere. Convincing yourself of this notion will involve recognizing that
straightness on the plane and straightness on a sphere have common elements. When you
are comfortable with “great-circle-straightness,” you will be ready to transfer the
symmetries of straight lines on the plane to great circles on a sphere and, later, to
geodesics on other surfaces. Here are some activities that you can try, or visualize, to help
you experience great circles and their intrinsic straightness on a sphere. However, it is
better for you to come up with your own experiences.
Stretch something elastic on a sphere. It will stay in place on a great circle, but
it will not stay on a small circle if the sphere is slippery. Here, the elastic
follows a path that is approximately the shortest because a stretched elastic
always moves so that it will be shorter. This a very useful practical criterion of
straightness.
Roll a ball on a straight chalk line (or straight on a freshly painted floor!). The
chalk (or paint) will mark the line of contact on the sphere, and it will form a
great circle.
Chapter 2 Straightness on Spheres 41
Take a narrow stiff ribbon or strip of paper that does not stretch and lay it
“flat” on a sphere. It will only lie (without folds and creases) along a great
circle. Do you see how this property is related to local symmetry? This is
sometimes called the Ribbon Test. (For further discussion of the Ribbon Test,
see Problems 3.4 and 7.6 of [DG: Henderson].)
The feeling of turning and “non-turning” comes up. Why is it that on a great
circle there is no turning and on a latitude line there is turning? Physically, in
order to avoid turning, the bug has to move its left feet the same distance as its
right feet. On a non-great circle (for example, a latitude line that is not the
equator), the bug has to walk faster with the legs that are on the side closer to
the equator. This same idea can be experienced by taking a small toy car with
its wheels fixed to parallel axes so that, on a plane, it rolls along a straight line.
On a sphere, the car will roll around a great circle; but it will not roll around
other curves.
Also notice that, on a sphere, straight lines are intrinsic circles (points on the
surface a fixed distance along the surface away from a given point on the
surface) — special circles whose circumferences are straight! Note that the
equator is a circle with two intrinsic centers: the north pole and the south pole.
In fact, any circle (such as a latitude circle) on a sphere has two intrinsic
centers.
These activities will provide you with an opportunity to investigate the relationships
between a sphere and the geodesics of that sphere. Along the way, your experiences
should help you to discover how great circles on a sphere have most of the same
symmetries as straight lines on a plane.
You should pause and not read further until you have expressed your thinking
and ideas about this problem.
SYMMETRIES OF GREAT CIRCLES
Reflection-through-itself symmetry: We can see this globally by placing a
hemisphere on a flat mirror. The hemisphere together with the image in the mirror
exactly recreates a whole sphere. Figure 2.3 shows a reflection through the great circle g.
Reflection-perpendicular-to-itself symmetry: A reflection through any great circle
will take any great circle (for example, g´ in Figure 2.3) perpendicular to the original
great circle onto itself.
Chapter 2 Straightness on Spheres 42
g
'
g
Figure 2.3 Reflection-through-itself symmetry – half sphere reflected in the mirror
Half-turn symmetry: A rotation through half of a full revolution about any point P on
a great circle interchanges the part of the great circle on one side of P with the part on the
other side of P. See Figure 2.4.
P
Figure 2.4 Half-turn symmetry
Rigid-motion-along-itself symmetry: For great circles on a sphere, we call this a translation along
the great circle or a rotation around the poles of that great circle. This property of being able to move rigidly along
itself is not unique to great circles because any circle on the sphere will also have the same symmetry. See Figure
2.5.
Figure 2.5 Rigid-motion-along-itself symmetry
Central symmetry or point symmetry: Viewed intrinsically (from the 2-dimensional
bug’s point-of-view), central symmetry through a point P on the sphere sends any point A
to the point at the same great circle distance from P but on the opposite side. See Figure
Chapter 2 Straightness on Spheres 43
2.6.
A
P
A'
A
P
A’
Center
Intrinsically
Extrinsically
Figure 2.6 Central symmetry through P
Extrinsically (viewing the sphere in 3-space) central symmetry through P would send
A to a point off the surface of the sphere as shown in Figure 2.6. The only extrinsic
central symmetry of the sphere (and the only one for great circles on the sphere) is
through the center of the sphere (which is not on the sphere). The transformation that is
intrinsically central symmetry is extrinsically half-turn symmetry (about the diameter
through P). Intrinsically, as on a plane, central symmetry does not differ from half-turn
symmetry with respect to the end result. This distinction between intrinsic and extrinsic is
important to experience at this point.
3-dimensional-rotation symmetry: This symmetry does not hold for great circles
in 3-space; however, it does hold for great circles in a 3-sphere. See Problem 22.5.
You will probably notice that other objects on the sphere, besides great circles, have
some of the symmetries mentioned here. It is important for you to construct such
examples. This will help you to realize that straightness and the symmetries discussed
here are intimately related.
EVERY GEODESIC IS A GREAT CIRCLE
Notice that you were not asked to prove that every geodesic (intrinsic straight line) on the
sphere is a great circle. This is true but more difficult to prove. Many texts simply define
the great circles to be the “straight lines” (geodesics) on the sphere. We have not taken
that approach. We have shown that the great circles are intrinsically straight (geodesics),
and it is clear that two points on the sphere are always joined by a great circle arc, which
shows that there are sufficient great- circle geodesics to do the geometry we wish.
To show that great circles are the only geodesics involves some notions from
differential geometry. In Problem 3.2b of [DG: Henderson] this is proved using special
properties of plane curves. More generally, a geodesic satisfies a differential equation
with the initial condition being a point on the geodesic and the direction of the geodesic
at that point (see Problem 8.4b of [DG: Henderson]). Thus, it follows from the analysis
Chapter 2 Straightness on Spheres 44
theorem on the existence and uniqueness of solutions to differential equations that
THEOREM 2.1. At every point and in every direction on a smooth surface
there is a unique geodesic going from that point in that direction.
From this it follows that all geodesics on a sphere are great circles. Do you see why?
INTRINSIC CURVATURE
You have tried wrapping the sphere with a ribbon and noticed that the ribbon will only lie
flat along a great circle. (If you haven’t experienced this yet, then do it now before you
go on.) Arcs of great circles are the only paths on a sphere’s surface that are tangent to a
straight line on a piece of paper wrapped around the sphere.
If you wrap a piece of paper tangent to the sphere around a latitude circle (see Figure
2.7), then, extrinsically, the paper will form a portion of a cone and the curve on the
paper will be an arc of a circle when the paper is flattened. The intrinsic curvature of a
path on the surface of a sphere can be defined as the curvature (1/radius) that one gets
when one “unwraps” the path onto a plane. For more details, see Chapter 3 of [DG:
Henderson].
Figure 2.7 Finding the intrinsic curvature
Chapter 2 Straightness on Spheres 45
Differential geometers often talk about intrinsically straight paths (geodesics) in
terms of the velocity vector of the motion as one travels at a constant speed along that
path. (The velocity vector is tangent to the curve along which the bug walks.) For
example, as you walk along a great circle, the velocity vector to the circle changes
direction, extrinsically, in 3-space where the change in direction is toward the center of
the sphere. “Toward the center” is not a direction that makes sense to a 2-dimensional
bug whose whole universe is the surface of the sphere. Thus, the bug does not experience
the velocity vectors as changing direction at points along the great circle; however, along
non-great circles the velocity vector will be experienced as changing in the direction of
the closest center of the circle. In differential geometry, the rate of change, from the bug’s
point of view, is called the covariant (or intrinsic) deriva- tive. As the bug traverses a
geodesic, the covariant derivative of the velocity vector is zero. This can also be
expressed in terms of parallel transport, which is discussed in Chapters 7, 8, and 10 of
this text. See [DG: Henderson] for discussions of these ideas in differential geometry.
Chapter 3
WHAT IS AN ANGLE?
A (plane) angle is the inclination to one another of two lines in a plane which meet one another
and do not lie in a straight line. — Euclid, Elements, Definition 8
In this chapter you will be thinking about angles. In Problem 3.1 we will investigate
various notions and definitions of angles and what it means for them to be considered to
be the same (congruent). In Problem 3.2 we will prove the important Vertical Angle
Theorem (VAT). It is not necessary to do these parts in order — you may find it easier to
do Problem 3.2 before Problem 3.1 because it may help you think about angles. In a sense,
you should be working on Problems 3.1 and 3.2 at the same time because they are so closely
intertwined. This provides a valuable opportunity to apply and reflect on what you have
learned about straightness in Chapters 1 and 2. This will also be helpful in the further study
of straightness in Chapters 4 and 5; but, if you wish, you may study this chapter after
Chapters 4 and 5.
PROBLEM 3.1 WHAT IS AN ANGLE?
Give some possible definitions of the term “angle.” Do all of these definitions apply
to the plane as well as to spheres? What are the advantages and disadvantages of
each? For each definition, what does it mean for two angles to be congruent? How
can we check?
SUGGESTIONS
Etymologically, “angle” comes through Old English, Old French, Old German,
Latin, and Greek words for “hook.” Textbooks usually give some variant of the definition:
An angle is the union of two rays (or segments) with a common endpoint.
Chapter 3 What Is an Angle? 47
If we start with two straight line segments with a common endpoint and then add
squiggly parts onto the ends of each one, would we say that the angle has changed as a
result? Likewise, look at the angle formed at the lower-left-hand corner of this piece of
paper. Even first grade students will recognize this as an example of an angle. Now, tear
off the corner (at least in your imagination). Is the angle still there, on the piece you tore
off? Now tear away more of the sides of the angles, being careful not to tear through the
corner. The angle is still there at the corner, isn’t it? See Figure 3.1.
Figure 3.1 Where is the angle?
What part of the angle determines how large the angle is, or if it is an angle at all?
What is the angle? Seems it cannot be merely a union of two rays. Here is one of the many
cases where children seem to know more than we do. Paying attention to these insights,
can we get better definitions of “angle”? Do not expect to find one formal definition that
is completely satisfactory; it seems likely that no formal definition can capture all aspects
of our experience of what an angle is.
There are at least three different perspectives from which we can define “angle,” as
follows:
a dynamic notion of an angle – angle as movement;
angle as a measure; and,
angle as geometric shape.
A dynamic notion of angle involves an action: a rotation, a turning point, or a change
in direction between two lines. Angle as measure may be thought of as the length of a
circular arcs or the ratio between areas of circular sectors. Thought of as a geometric shape,
an angle may be seen as the delineation of space by two intersecting lines. Each of these
perspectives carries its methods for checking angle congruency. You can check the
congruency of two dynamic angles by verifying that the actions involved in creating or
replicating them are the same. If you feel that an angle is a measure, then you must verify
that both angles have the same measure. If you describe angles as geometric shapes, then
you describe how one angle can be made to coincide with the other using isometries. Which
of the above definitions has the most meaning for you? Are there any other useful ways of
describing angles?
Note that we sometimes talk about directed angles, or angles with direction. When
considered as directed angles, we say that the angles α and β in Figure 3.2 are not the same
Chapter 3 What Is an Angle? 48
but have equal magnitude and opposite directions (or sense). Note the similarity to the
relationship between line segments and vectors.
Figure 3.2 Directed angles
PROBLEM 3.2 VERTICAL ANGLE THEOREM (VAT)
Figure 3.3 VAT
Prove: Opposite angles formed by two intersecting straight lines are congruent.
[Note: Angles such as α and β are called vertical angles.] What properties of straight
lines and/or the plane are you using in your proof? Does your proof also work on a
sphere? Why? Which definitions from Problem 3.2 are you using in your proof?
Show how you would “move” to make it coincide with . We do not have in mind a
formal two-column proof that used to be in American high school geometry.
Mathematicians in actual practice usually use “proof” to mean “a convincing
communication that answers — Why?” This is the notion of proof we ask you to use. There
are three features of a proof:
It must communicate (the words and drawings need to clearly express what it
is that you want to say — and they must be understandable to your reader
and/or listener.)
It must be convincing (to yourself, to your fellow students, and to your teacher;
preferably it should be convincing to someone who was originally skeptical).
It must answer — Why? (Why is it true? What does it mean? Where did it
come from?)
The goal is understanding. Without understanding we will never be fully satisfied.
With understanding we want to expand that understanding and to communicate it to others.
Chapter 3 What Is an Angle? 49
Symmetries were an important element of your solutions for Problems 1.1 and 2.1.
They will be very useful for this problem as well. It is perfectly valid to think about
measuring angles in this problem, but proofs utilizing line symmetries are generally
simpler. It often helps to think of the vertical angles as whole geometric figures. Also, keep
in mind that there are many different ways of looking at angles, so there are many ways of
proving the vertical angle theorem. Make sure that your notions of angle and angle
congruency in Problem 3.1 are consistent with your proofs in Problem 3.2, and vice versa.
Any of the definitions from Problem 3.1 can, separately or together, help you prove the
Vertical Angle Theorem.
You should pause and not read further until you have expressed your own
thinking and ideas about Problems 3.1 and 3.2.
HINTS FOR THREE DIFFERENT PROOFS
In the following section, we will give hints for three different proofs of the Vertical
Angle Theorem. Note that a particular notion of angle is assumed in each proof. Pick one
of the proofs or find your own different proof that is consistent with a notion of angle and
angle congruence that is most meaningful to you.
1st proof:
Figure 3.4 VAT using angle as measure
Each line creates a 180° angle. Thus, + = + See Figure 3.4. Therefore, we
can conclude that But why is this so? Is it always true that if we subtract a given
angle from two 180° angles then the remaining angles are congruent? See Figure 3.5.
Figure 3.5 Subtracting angles and measures
Chapter 3 What Is an Angle? 50
Numerically, it does not make any difference how we subtract an angle, but
geometrically it makes a big difference. Behold Figure 3.6! Here, really cannot be
considered the same as . Thus, measure does not completely express what we see in the
geometry of this situation. If you wish to salvage this notion of angle as measure, then
you must explain why it is that in this proof of the Vertical Angle Theorem can be
subtracted from both sides of the equation + = + .
Figure 3.6 is not the same as
2nd proof: Consider two overlapping lines and choose any point on them. Rotate
one of the lines, maintaining the point of intersection and making sure that the other line
remains fixed as in Figure 3.7.
Figure 3.7 VAT using angle as rotation
What happens? What notion of angle and angle congruency is at work here?
3rd proof: What symmetries will take onto ? See Figure 3.3 or 3.4. Use the
properties of straight lines you investigated in Chapters 1 and 2.
Chapter 4
STRAIGHTNESS
ON CYLINDERS AND CONES
If a cut were made through a cone parallel to its base, how should we conceive of the two
opposing surfaces which the cut has produced — as equal or as unequal? If they are unequal,
that would imply that a cone is composed of many breaks and protrusions like steps. On the
other hand, if they are equal, that would imply that two adjacent intersection planes are equal,
which would mean that the cone, being made up of equal rather than unequal circles, must have
the same appearance as a cylinder; which is utterly absurd. — Democritus of Abdera (~460 –
~380 B.C.)
This quote shows that cylinders and cones were the subject of mathematical inquiry
before Euclid (~365 – ~300 B.C.). In this chapter we investigate straightness on cones and
cylinders. You should be comfortable with straightness as a local intrinsic notion — this
is the bug’s view. This notion of straightness is also the basis for the notion of geodesics
in differential geometry. Chapters 4 and 5 can be covered in either order, but we think that
the experience with cylinders and cones in Problem 4.1 will help the reader to understand
the hyperbolic plane in Problem 5.1. If the reader is comfortable with straightness as a local
intrinsic notion, then it is also possible to skip Chapter 4 if Chapters 18 and 24 on geometric
manifolds are not going to be covered. However, we suggest that you read the sections at
the end of this chapter — Is “Shortest” Always “Straight”? and Relations to Differential
Geometry — at least enough to find out what Euclid’s Fourth Postulate has to do with
cones and cylinders.
Chapter 4 Straightness on Cylinders and Cones 52
When looking at great circles on the surface of a sphere, we were able (except in the
case of central symmetry) to see all the symmetries of straight lines from global extrinsic
points of view. For example, a great circle extrinsically divides a sphere into two
hemispheres that are mirror images of each other. Thus, on a sphere, it is a natural tendency
to use the more familiar and comfortable extrinsic lens instead of taking the bug’s local
and intrinsic point of view. However, on a cone and cylinder you must use the local,
intrinsic point of view because there is no extrinsic view that will work except in special
cases.
PROBLEM 4.1 STRAIGHTNESS ON CYLINDERS AND CONES
a. What lines are straight with respect to the surface of a cylinder or a cone?
Why? Why not?
b. Examine:
Can geodesics intersect themselves on cylinders and cones?
Can there be more than one geodesic joining two points on cylinders
and cones?
What happens on cones with varying cone angles, including cone angles
greater than 360°? These are discussed starting in the next section.
SUGGESTIONS
Problem 4.1 is similar to Problem 2.1, but this time the surfaces are cylinders and
cones. Make paper models but consider the cone or cylinder as continuing indefinitely with
no top or bottom (except, of course, at the cone point). Again, imagine yourself as a bug
whose whole universe is a cone or cylinder. As the bug crawls around on one of these
surfaces, what will the bug experience as straight? As before, paths that are straight with
respect to a surface are often called the “geodesics” for the surface.
As you begin to explore these questions, it is likely that many other related
geometric ideas will arise. Do not let seemingly irrelevant excess geometric baggage worry
you. Often, you will find yourself getting lost in a tangential idea, and that’s
understandable. Ultimately, however, the exploration of related ideas will give you a richer
understanding of the scope and depth of the problem. In order to work through possible
confusion on this problem, try some of the following suggestions others have found helpful.
Each suggestion involves constructing or using models of cones and cylinders.
You may find it helpful to explore cylinders first before beginning to explore
cones. This problem has many aspects but focusing at first on the cylinder will
simplify some things.
If we make a cone or cylinder by rolling up a sheet of paper, will “straight”
stay the same for the bug when we unroll it? Conversely, if we have a straight
Chapter 4 Straightness on Cylinders and Cones 53
line drawn on a sheet of paper and roll it up, will it continue to be experienced
as straight for the bug crawling on the paper? We are assuming here that the
paper will not stretch, and its thickness is negligible.
Lay a stiff ribbon or straight strip of paper on a cylinder or cone. Convince
yourself that it will follow a straight line with respect to the surface. Also,
convince yourself that straight lines on the cylinder or cone, when looked at
locally and intrinsically, have the same symmetries as on the plane.
If you intersect a cylinder by a flat plane and unroll it, what kind of curve do
you get? Is it ever straight? (One way to see this curve is to dip a paper
cylinder into water.)
On a cylinder or cone, can a geodesic ever intersect itself? How many times?
This question is explored in more detail in Problem 4.2, which the interested
reader may turn to now.
Can there be more than one geodesic joining two points on a cylinder or cone?
How many? Is there always at least one? Again, this question is explored in
more detail in Problem 4.2.
There are several important things to keep in mind while working on this problem.
First, you absolutely must make models. If you attempt to visualize lines on a cone or
cylinder, you are bound to make claims that you would easily see are mistaken if you
investigated them on an actual cone or cylinder. Many students find it helpful to make
models using transparent material.
Second, as with the sphere, you must think about lines and triangles on the cone and
cylinder in an intrinsic way — always looking at things from a bug’s point of view. We are
not interested in what’s happening in 3-space, only what you would see and experience if
you were restricted to the surface of a cone or cylinder.
And last, but certainly not least, you must look at cones of different shapes, that is,
cones with varying cone angles.
CONES WITH VARYING CONE ANGLES
Geodesics behave differently on differently shaped cones. So an important variable
is the cone angle. The cone angle is generally defined as the angle measured around the
point of the cone on the surface. Notice that this is an intrinsic description of angle. The
bug could measure a cone angle (in radians) by determining the circumference of an
intrinsic circle with center at the cone point and then dividing that circumference by the
radius of the circle. We can determine the cone angle extrinsically in the following way:
Cut the cone along a generator (a line on the cone through the cone point) and flatten the
cone. The measure of the cone angle is then the angle measure of the flattened planar sector.
Chapter 4 Straightness on Cylinders and Cones 54
Figure 4.1 Making a 180° cone
For example, if we take a piece of paper and bend it so that half of one side meets
up with the other half of the same side, we will have a 180-degree cone (Figure 4.1). A 90º
cone is also easy to make — just use the corner of a paper sheet and bring one side around
to meet the adjacent side.
Also be sure to look at larger cones. One convenient way to do this is to make a
cone with a variable cone angle. This can be accomplished by taking a sheet of paper and
cutting (or tearing) a slit from one edge to the center. (See Figure 4.2.) A rectangular sheet
will work but a circular sheet is easier to picture. Note that it is not necessary that the slit
be straight!
Figure 4.2 A cone with variable cone angle (0 –360°)
You are already familiar with a 360º cone — it’s just a plane. The cone angle can
also be larger than 360º. A common larger cone is the 450º cone. You probably have a cone
like this somewhere on the walls, floor, and ceiling of your room. You can easily make one
by cutting a slit in a piece of paper and inserting a 90º slice (360º + 90º = 450º) as in Figure
4.3.
Figure 4.3 How to make a 450º cone
Chapter 4 Straightness on Cylinders and Cones 55
Two cone angles on a ceiling
You may have trouble believing that this is a cone but remember that just because
it cannot hold ice cream does not mean it is not a cone. If you will look around in the room
you are, perhaps you can locate a corner where five right angles meet – that is 450º cone.
It is important to realize that when you change the shape of the cone like this (that is, either
it is with ruffles or straight lines), you are only changing its extrinsic appearance.
Intrinsically (from the bug’s point of view) there is no difference.
It may be helpful for you to discuss some definitions of a cone, such as the
following: Take any simple (non-intersecting) closed curve a on a sphere and the center P
of the sphere. A cone is the union of the rays that start at P and go through each point on
a. The cone angle is then equal to (length of a)/ (radius of sphere), in radians. Do you see
why?
You can also make a cone with variable angle of more than 180°: Take two sheets
of paper and slit them together to their centers as in Figure 4.4. Tape the right side of the
top slit to the left side of the bottom slit as pictured. Now slide the other sides of the slits.
Try it!
Figure 4.4 Variable cone angle larger than 360°
Chapter 4 Straightness on Cylinders and Cones 56
Experiment by making paper examples of cones like those shown in Figure 4.4.
What happens to the triangles and lines on a 450º cone? Is the shortest path always straight?
Does every pair of points determine a straight line?
Finally, also consider line symmetries on the cone and cylinder. Check to see if the
symmetries you found on the plane will work on these surfaces and remember to think
intrinsically and locally. A special class of geodesics on the cone and cylinder is the
generators. These are the straight lines that go through the cone point on the cone or go
parallel to the axis of the cylinder. These lines have some extrinsic symmetries (can you
see which ones?), but in general, geodesics have only local, intrinsic symmetries. Also, on
the cone, think about the region near the cone point — what is happening there that makes
it different from the rest of the cone?
Few more explorations of geodesics on the cone can
http://www.rdrop.com/~half/Creations/Puzzles/cone.geodesics/index.html
be
found
It is best if you experiment with paper models to find out what geodesics
look like on the cone and cylinder before reading further.
GEODESICS ON CYLINDERS
Let us first look at the three classes of straight lines on a cylinder. When walking on the
surface of a cylinder, a bug might walk along a vertical generator. See Figure 4.5.
Figure 4.5 Vertical generators are straight
It might walk along an intersection of a horizontal plane with the cylinder, what we
will call a great circle. See Figure 4.6
Figure 4.6 Great circles are intrinsically straight
Chapter 4 Straightness on Cylinders and Cones 57
Or, the bug might walk along a spiral or helix of constant slope around the cylinder.
See Figure 4.7 and the photo at the beginning of this chapter depicting lightening damage
to the tree. Watch a squirrel running up the tree!
Figure 4.7 Helixes are intrinsically straight
Helixes can be seen on outside parking garages and in sculptures
Why are these geodesics? How can you convince yourself? And why are these the
only geodesics?
GEODESICS ON CONES
Now let us look at the classes of straight lines on a cone.
Walking along a generator: When looking at straight paths on a cone, you will
be forced to consider straightness at the cone point. You might decide that there is no way
the bug can go straight once it reaches the cone point, and thus a straight path leading up
to the cone point ends there. Or you might decide that the bug can find a continuing path
that has at least some of the symmetries of a straight line. Do you see which path this is?
Or you might decide that the straight continuing path(s?) is the limit of geodesics that just
miss the cone point. See Figure 4.8.
Help!
Figure 4.8 Bug walking straight over the cone point
Chapter 4 Straightness on Cylinders and Cones 58
Walking straight and around: If you use a ribbon on a 90º cone, then you can
see that this cone has a geodesic like the one depicted in Figure 4.9. This particular geodesic
intersects itself. However, check to see that this property depends on the cone angle. In
particular, if the cone angle is more than 180°, then geodesics do not intersect themselves.
And if the cone angle is less than 90°, then geodesics (except for generators) intersect at
least two times. Try it out! Later, in Problem 4.2, we will describe a tool that will help you
determine how the number of self- intersections depends on the cone angle.
Figure 4.9 A geodesic intersecting itself on a 90° cone
PROBLEM 4.2 GLOBAL PROPERTIES OF GEODESICS
Now we will look more closely at long geodesics that wrap around on a cylinder
or cone. Several questions have arisen.
a. How do we determine the different geodesics connecting two points? How many
are there? How does it depend on the cone angle? Is there always at least one
geodesic joining each pair of points? How can we justify our conjectures?
b. How many times can a geodesic on a cylinder or cone intersect itself? How are
the self-intersections related to the cone angle? At what angle does the geodesic
intersect itself? How can we justify these relationships?
SUGGESTIONS
Here we offer the tool of covering spaces, which may help you explore these
questions. The method of coverings is so named because it utilizes layers (or sheets) that
each cover the surface. We will first start with a cylinder because it is easier and then move
on to a cone.
n -SHEETED COVERINGS OF A CYLINDER
To understand how the method of coverings works, imagine taking a paper cylinder
and cutting it axially (along a vertical generator) so that it unrolls into a plane. This is
probably the way you constructed cylinders to study this problem before. The unrolled
sheet (a portion of the plane) is said to be a 1-sheeted covering of the cylinder. See Figure
Chapter 4 Straightness on Cylinders and Cones 59
4.10. If you marked two points on the cylinder, A and B, as indicated in the figure, when
the cylinder is cut and unrolled into the covering, these two points become two points on
the covering (which are labeled by the same letters in the figure). The two points on the
covering are said to be lifts of the points on the cylinder.
cut
A
A
B
B
Figure 4.10 A 1-sheeted covering of a cylinder
Now imagine attaching several of these “sheets” together, end to end. When rolled
up, each sheet will go around the cylinder exactly once — they will each cover the cylinder.
(Rolls of toilet paper or paper towels give a rough idea of coverings of a cylinder.) Also,
each sheet of the covering will have the points A and B in identical locations. You can see
this (assuming the paper thickness is negligible) by rolling up the coverings and making
points by sticking a sharp object through the cylinder. This means that all the A’s are
coverings of the same point on the cylinder and all the B’s are coverings of the same point
on the cylinder. We just have on the covering several representations, or lifts, of each point
on the cylinder. Figure 4.11 depicts a 3-sheeted covering space for a cylinder and six
geodesics joining A to B. (One of them is the most direct path from A to B and the others
spiral once, twice, or three times around the cylinder in one of two directions.)
cut
Figure 4.11 A 3-sheeted covering space for a cylinder
Chapter 4 Straightness on Cylinders and Cones 60
We could also have added more sheets to the covering on either the right or left side.
You can now roll these sheets back into a cylinder and see what the geodesics look like.
Remember to roll sheets up so that each sheet of the covering covers the cylinder exactly
once — all of the vertical lines between the coverings should lie on the same generator of
the cylinder. Note that if you do this with ordinary paper, part or all of some geodesics will
be hidden, even though they are all there. It may be easier to see what’s happening if you
use transparencies.
This method works because straightness is a local intrinsic property. Thus, lines that
are straight when the coverings are laid out in a plane will still be straight when rolled into
a cylinder. Remember that bending the paper does not change the intrinsic nature of the
surface. Bending only changes the curvature that we see extrinsically. It is important
always to look at the geodesics from the bug’s point of view. The cylinder and its covering
are locally isometric.
Use coverings to investigate Problem 4.2 on the cylinder. The global behavior of
straight lines may be easier to see on the covering.
n-SHEETED (BRANCHED) COVERINGS OF A CONE
Figure 4.12 1-sheeted covering of a 270° cone
Figure 4.12 shows a 1-sheeted covering of a cone. The sheet of paper and the cone
are locally isometric except at the cone point. The cone point is called a branch point of
the covering. We talk about lifts of points on the cone in the same way as on the cylinder.
In Figure 4.12 we depict a 1-sheeted covering of a 270° cone and label two points and their
lifts.
A 4-sheeted covering space for a cone is depicted in Figure 4.13. Each of the rays
drawn from the center of the covering is a lift of a single ray on the cone. Similarly, the
points marked on the covering are the lifts of the points A and B on the cone. In the covering
there are four segments joining a lift of A to different lifts of B. Each of these segments is
the lift of a different geodesic segment joining A to B.
Chapter 4 Straightness on Cylinders and Cones 61
Figure 4.13 4-sheeted covering space for a 89° cone
Think about ways that the bug can use coverings as a tool to expand its exploration
of surface geodesics. Also, think about ways you can use coverings to justify your
observations in an intrinsic way. It is important to be precise; you don’t want the bug to get
lost! Count the number of ways in which you can connect two points with a straight line
and relate those countings with the cone angle. Does the number of straight paths only
depend on the cone angle? Look at the 450° cone and see if it is always possible to connect
any two points with a straight line. Make paper models! It is not possible to get an
equation that relates the cone angle to the number of geodesics joining every pair of points.
However, it is possible to find a formula that works for most pairs. Make covering spaces
for cones of different size angles and refine the guesses you have already made about the
numbers of self-intersections.
In studying the self-intersections of a geodesic l on a cone, it may be helpful for you
to consider the ray R that is perpendicular to the line l. (See Figure 4.14.) Now study one
lift of the geodesic l and its relationship to the lifts of the ray R. Note that the seams between
individual wedges are lifts of R.
Figure 4.14 Self-intersections on a cone with angle
A recent tidbit about coverings: In 1914 Henri Lebesgue (French, 1875-1941) posed a
question: What is the shape with the smallest area that can completely cover a host of other
shapes (which all share a certain trait in common)? The shapes should be such that no two
points are further than one unit apart. In 2014 retired software engineer Philip Gibbs ran
computer simulations on 200 randomly generated shapes with diameter 1. He kept
“trimming corners” of hexagon and found that to be smallest known covering.
(https://www.quantamagazine.org/amateur-mathematician-finds-smallest-universalcover-20181115/)
Chapter 4 Straightness on Cylinders and Cones 62
LOCALLY ISOMETRIC
By now you should realize that when a piece of paper is rolled or bent into a cylinder
or cone, the bug’s local and intrinsic experience of the surface does not change except at
the cone point. Extrinsically, the piece of paper and the cone are different, but in terms of
the local geometry intrinsic to the surface they differ only at the cone point.
Two geometric spaces, G and H, are said to be locally isometric at the points G in
G and H in H if the local intrinsic experience at G is the same as the experience at H. That
is, there are neighborhoods of G and H that are identical in terms of their intrinsic geometric
properties. A cylinder and the plane are locally isometric (at each point) and the plane and
a cone are locally isometric except at the cone point. Two cones are locally isometric at the
cone points only if their cone angles are the same. Because cones and cylinders are locally
isometric with the plane, locally they have the same geometric properties. Later, we will
show that a sphere is not locally isometric with the plane — be on the lookout for a result
that will imply this.
IS “SHORTEST” ALWAYS “STRAIGHT”?
We are often told that “a straight line is the shortest distance between two points,”
but is this really true? As we have already seen on a sphere, two points not opposite each
other are connected by two straight paths (one going one way around a great circle and
one going the other way). Only one of these paths is shortest. The other is also straight, but
not the shortest straight path.
Consider a model of a cone with angle 450°. Notice that such cones appear
commonly in buildings as so-called “outside corners” (see Figure 4.3). It is best, however,
to have a paper model that can be flattened.
Figure 4.15 There is no straight (symmetric) path from A to B
Use your model to investigate which points on the cone can be joined by straight
lines (in the sense of having reflection-in-the-line symmetry). In particular, look at points
such as those labeled A and B in Figure 4.15. Convince yourself that there is no path from
A to B that is straight (in the sense of having reflection-in-the-line symmetry), and for these
points the shortest path goes through the cone point and thus is not straight (in the sense of
having symmetry).
Chapter 4 Straightness on Cylinders and Cones 63
Figure 4.16 The shortest path is not straight (in the sense of symmetry)
Here is another example: Think of a bug crawling on a plane with a tall box sitting
on that plane (refer to Figure 4.16). This combination surface — the plane with the box
sticking out of it — has eight cone points. The four at the top of the box have 270° cone
angles, and the four at the bottom of the box have 450° cone angles (180° on the box and
270° on the plane). What is the shortest path between points X and Y, points on opposite
sides of the box? Is the straight path the shortest? Is the shortest path straight? To check
that the shortest path is not straight, try to see that at the bottom corners of the box the two
sides of the path have different angular measures. (If X and Y are close to the box, then the
angle on the box side of the path measures a little more than 180° and the angle on the
other side measures almost 270°.)
RELATIONS TO DIFFERENTIAL GEOMETRY
We see that sometimes a straight path is not shortest, and the shortest path is not
straight. Does it then make sense to say (as most books do) that in Euclidean geometry a
straight line is the shortest distance between two points? In differential geometry, on
“smooth” surfaces, “straight” and “shortest” are more nearly the same. A smooth surface
is essentially what it sounds like. More precisely, a surface is smooth at a point if, when
you zoom in on the point, the surface becomes indistinguishable from a flat plane. (For
details of this definition, see Problem 4.1 in [DG: Henderson,
https://projecteuclid.org/euclid.bia/1399917369].
See also the last section and especially the endnote in Chapter 1.) Note that a cone
is not smooth at the cone point, but a sphere and a cylinder are both smooth at every point.
The following is a theorem from differential geometry:
THEOREM 4.1: If a surface is smooth, then an intrinsically straight line (geodesic)
on the surface is always the shortest path between “nearby” points. If the surface
is also complete (every geodesic on it can be extended indefinitely), then any two
points can be joined by a geodesic that is the shortest path between them. See [DG:
Henderson], Problems 7.4b and 7.4d.
Consider a planar surface with a hole removed. Check that for points near opposite
sides of the hole, the shortest path (on the planar surface with hole removed) is not straight
Chapter 4 Straightness on Cylinders and Cones 64
because the shortest path must go around the hole. We encourage the reader to discuss how
each of the previous examples and problems is in harmony with this theorem.
Note that the statement “every geodesic on the surface can be extended indefinitely”
is a reasonable interpretation of Euclid’s Second Postulate: Every limited straight line can
be extended indefinitely to a (unique) straight line. Note that the Second Postulate does not
hold on a cone unless you consider geodesics to continue through the cone point.
Also, Euclid defines a right angle as follows: When a straight line intersects another
straight line such that the adjacent angles are equal to one another, then the equal angles
are called right angles. Note that if you consider geodesics to continue through the cone
point, then right angles at a cone point are not equal to right angles at points where the cone
is locally isometric to the plane.
And Euclid goes on to state as his Fourth Postulate: All right angles are equal. Thus,
Euclid’s Second Postulate or Fourth Postulate rules out cones and any surface with isolated
cone points. What is further ruled out by Euclid’s Fourth Postulate would depend on
formulating more precisely just what it says. It is not clear (at least to the authors!) whether
there is something we would want to call a surface that could be said to satisfy Euclid’s
Fourth Postulate and not be a smooth surface. However, we can see that Euclid’s postulate
at least gives part of the meaning of “smooth surface,” because it rules out isolated cone
points.
When we were in high school geometry class, we were confused why Euclid would
have made such a postulate as his Postulate 4 — how could they possibly not be equal? In
this chapter we have discovered that on cones right angles are not all equal.
Chapter 5
STRAIGHTNESS
ON HYPERBOLIC
PLANES
[To son János:] For God’s sake, please give it [work on hyperbolic geometry] up. Fear it no less
than the sensual passion, because it, too, may take up all your time and deprive you of your
health, peace of mind and happiness in life. — Wolfgang Bolyai (1775–1856) [EM: Davis and
Hersh], page 220
We will now study some hyperbolic geometry. As with the cone and cylinder, we must use
an intrinsic point of view on hyperbolic planes. This is especially true because, as we will
see, there is no standard embedding of a complete hyperbolic plane into 3-space.
A SHORT HISTORY OF HYPERBOLIC GEOMETRY
Hyperbolic geometry initially grew out of the Building Structures Strand through the work
of János Bolyai (1802–1860, Hungarian), and N. I. Lobachevsky (1792–1856, Russian).
Hyperbolic geometry is special from a formal axiomatic point of view because it satisfies
all the postulates (axioms) of Euclidean geometry except for the parallel postulate. In
hyperbolic geometry straight lines can converge toward each other without intersecting
(violating Euclid’s Fifth Postulate), and there is more than one straight line through a point
that does not intersect a given line (violating the usual high school parallel postulate, which
states that through any point P not on a given line l there is one and only one line through
P not intersecting l). See Figure 5.1.
Chapter 5 Straightness on Hyperbolic Planes
66
Figure 5.1 Two geodesics through a point not intersecting a given geodesic
The reader can explore more details of the axiomatic nature of hyperbolic geometry in
Chapter 10. Note that the 450° cone also violates the two parallel postulates mentioned
above. Thus the 450° cone has some of the properties of the hyperbolic plane.
Hyperbolic geometry has turned out to be useful in various branches of higher
mathematics. For example, in the classical theory of modular functions, algebraic
geometry, differential geometry, complex variables, and dynamic systems. Hyperbolic
geometry is used in biology and medicine, cosmology, physics, quantum computing,
chemistry, architecture. The geometry of binocular visual space appears experimentally to
be best represented by hyperbolic geometry (see [HY: Zage]). In addition, hyperbolic
geometry was considered as one of the possible geometries for our three-dimensional
physical universe — we will explore this connection more in Chapters 18 and 24.
In many books hyperbolic geometry and non-Euclidean geometry are treated as
being synonymous, but as we have seen there are other non-Euclidean geometries,
especially spherical geometry. It is also not accurate to say (as many books do) that nonEuclidean geometry was discovered about 200 years ago. As we discussed in Chapter 2,
spherical geometry (which is clearly not Euclidean) was in existence and studied (within
the Navigation/Stargazing Strand) by at least the ancient Babylonians, Indians, and Greeks
more than 2000 years ago. For more detailed discussion of the history and applications of
hyperbolic geometry see [Taimina, Crocheting Adventures with the Hyperbolic Planes, 2nd
ed., 2018; ch.5 and 9]
Most texts and popular books introduce hyperbolic geometry either axiomatically
or via “models” of the hyperbolic geometry in the Euclidean plane. These models are like
our familiar map projections of the surface of the earth. Like these maps of the earth’s
surface, intrinsic straight lines on the hyperbolic plane are not, in general, straight in the
model (map) and the model, in general, distorts distances and angles. We will return to the
subject of projection and models in Chapter 17. These “models” grew out of the Art/Pattern
Strand.
In this chapter we will introduce the geometry of the hyperbolic plane as the intrinsic
geometry of a particular surface in 3-space, in much the same way that we introduced
spherical geometry by looking at the intrinsic geometry of the sphere in 3-space. This is
Chapter 5 Straightness on Hyperbolic Planes
67
more in the flavor of the Navigation/Stargazing Strand. Such a surface is called an
isometric embedding of the hyperbolic plane into 3-space. We will construct such a surface
in the next section. Nevertheless, many texts and popular books say that David Hilbert
(1862–1943, German) proved in 1901 that it is not possible to have an isometric embedding
of the hyperbolic plane onto a closed subset of Euclidean 3-space. These authors miss what
Hilbert actually proved. In fact, Hilbert [HY: Hilbert] proved that there is no real analytic
isometry (that is, no isometry defined by real-valued functions that have convergent power
series). In 1902 Holmgren improved Hilbert’s theorem showing that given a smooth
embedding of a piece of the hyperbolic plane in three-dimensional space, the embedding
cannot be extended isometrically and smoothly beyond the finite distance d. Unfortunately,
d depends on the local embedding, and there is not a uniform bound for the size of the
“largest” piece of the hyperbolic plane that can be isometrically embedded in 3-space.
Hilbert’s theorem was also improved by Amsler in 1955, who showed that every
sufficiently smooth immersion of the hyperbolic plane into 3-space has a singular
“edge,”i.e., a one-dimensional submanifold beyond which the embedding is no longer
smooth. In 1964, N. V. Efimov [HY: Efimov] extended Hilbert’s result by proving that
there is no isometric embedding defined by functions whose first and second derivatives
are continuous. Without giving an explicit construction, N. Kuiper [HY: Kuiper] showed
in 1955 that there is a differentiable isometric embedding onto a closed subset of 3-space.
The first hyperbolic plane model made by E. Beltrami in 1868 and David’s model made more than 100 years later
The construction used here was shown to David by William Thurston (b.1946-2012,
American) in 1978; and it is not defined by equations at all, because it has no definite
embedding in Euclidean space. The idea for this construction is also included in [DG:
Thurston], pages 49 and 50, and is discussed in [DG: Henderson], page 31. In Problem 5.3
we will show that our isometric model is locally isometric to a certain smooth surface of
revolution called the pseudosphere, which is well known to locally have hyperbolic
geometry. Later, in Chapter 17, we will explore the various (non-isometric) models of the
hyperbolic plane (these models are the way that hyperbolic geometry is presented in most
texts) and prove that these models and the isometric constructions here produce the same
geometry.
Chapter 5 Straightness on Hyperbolic Planes
68
DESCRIPTION OF ANNULAR HYPERBOLIC PLANES
In Appendix A we describe the details for five different isometric constructions of
hyperbolic planes (or approximations to hyperbolic planes) as surfaces in 3-space. It is very
important that you actually perform at least one of these constructions. The act of
constructing the surface will give you a feel for hyperbolic planes that is difficult to get
any other way. We will focus our discussions in the text on the description of the hyperbolic
plane from annuli that was proposed by W. Thurston.
Figure 5.2 Annular strips for making an annular hyperbolic plane
A paper model of the hyperbolic plane may be constructed as follows: Cut out many
identical annular (“annulus” is the region between two concentric circles) strips as in Figure
5.2.(See template in Appendix). Attach the strips together by taping the inner circle of one
to the outer circle of the other. It is crucial that all the annular strips have the same inner
radius and the same outer radius, but the lengths of the annular strips do not matter. You
can also cut an annular strip shorter or extend an annular strip by taping two strips together
along their straight ends. The resulting surface is of course only an approximation of the
desired surface. The actual hyperbolic plane is obtained by letting → 0 while holding the
radius fixed. Note that since the surface is constructed (as → 0) the same everywhere it
is homogeneous (that is, intrinsically and geometrically, every point has a neighborhood
that is isometric to a neighborhood of any other point). We will call the results of this
construction the annular hyperbolic plane. We strongly suggest that the reader take the
time to cut out carefully several such annuli and tape them together as indicated.
Daina discovered a process for crocheting the annular hyperbolic plane as described in
Appendix A. The result is pictured in Figures 5.1 and 5.3 and other photos in this book.
Chapter 5 Straightness on Hyperbolic Planes
69
Figure 5.3 Daina’s first crocheted annular hyperbolic plane (1997)
There is also a polyhedral construction of the hyperbolic plane that is not directly related
to the annular constructions but is easier for students (and teachers!) to construct. This
construction (invented by David’s son Keith Henderson) is called the hyperbolic soccer
ball. See Appendix for the details of the constructions (and templates) and Figure 5.4 for a
picture. It also has a nice appearance if you make the heptagons a different color from the
hexagons. As with any polyhedral construction we cannot get closer and closer
approximations to the hyperbolic plane. There is also no apparent way to see the annuli.
Figure 5.4 Keith Henderson with his hyperbolic soccer ball
HYPERBOLIC PLANES OF DIFFERENT RADII (CURVATURE)
Note that the construction of a hyperbolic plane is dependent on (the radius of the annuli),
which we will call the radius of the hyperbolic plane. As in the case of spheres, we get
different hyperbolic planes depending on the value of . In Figures 5.5–5.7 there are
crocheted hyperbolic planes with radii approximately 4 cm, 8 cm, and 16 cm. The pictures
were all taken from approximately the same perspective and in each picture, there is a
centimeter rule to indicate the scale.
Chapter 5 Straightness on Hyperbolic Planes
Figures 5.5-5.7
70
Hyperbolic planes with 4 cm, 8 cm and 16 cm
Note that as increases, a hyperbolic plane becomes flatter and flatter (has less and less
curvature). Both spheres and hyperbolic planes, as goes to infinity, become
indistinguishable from the ordinary flat (Euclidean) plane. Thus, the plane can be called a
sphere (or hyperbolic plane) with infinite radius. In Chapter 7, we will define the Gaussian
Curvature and show that it is equal to 1/ 2 for a sphere and −1/ 2 for a hyperbolic plane.
PROBLEM 5.1 WHAT IS STRAIGHT IN A HYPERBOLIC PLANE?
a. On a hyperbolic plane, consider the curves that run radially across each annular
strip. Argue that these curves are intrinsically straight. Also, show that any two of
them are asymptotic, in the sense that they converge toward each other but do not
intersect.
Look for the local intrinsic symmetries of each annular strip and then global symmetries in
the whole hyperbolic plane. Make sure you give a convincing argument why the symmetry
holds in the limit as → 0.
Chapter 5 Straightness on Hyperbolic Planes
71
We shall say that two geodesics that converge in this way are asymptotic geodesics.
Note that there are no geodesics (straight lines) on the plane that are asymptotic.
b. Find other geodesics on your physical hyperbolic surface. Use the properties of
straightness (such as symmetries) you talked about in Problems 1.1, 2.1, and 4.1.
Try holding two points between the index fingers and thumbs on your two hands. Now pull
gently — a geodesic segment with its reflection symmetry should appear between the two
points. If your surface is durable enough, try folding the surface along a geodesic. Also,
you may use a ribbon to test for geodesics.
c. What properties do you notice for geodesics on a hyperbolic plane? How are they
the same as geodesics on the plane or spheres, and how are they different from
geodesics on the plane and spheres?
Explore properties of geodesics involving intersecting, uniqueness, and symmetries.
Convince yourself as much as possible using your model — full proofs for some of the
properties will have to wait until Chapter 17.
PROBLEM 5.2 COORDINATE SYSTEM ON ANNULAR
HYPERBOLIC PLANE
First, we will define coordinates on the annular hyperbolic plane that will help us to study
it in Chapter 17. Let be the fixed inner radius of the annuli and let H be the
approximation of the annular hyperbolic plane constructed from annuli of radius and
thickness . On H pick the inner curve of any annulus, calling it the base curve; and on this
curve pick any point as the origin O and pick a positive direction on this curve. We can
now construct an (intrinsic) coordinate system x : R2 → H by defining x (0, 0) = O,
x (w, s) to be the point on the base curve at a distance w from O, and x (w, s) to be the
point at a distance s from x (w, 0) along the radial geodesic through x (w, 0), where the
positive direction is chosen to be in the direction from outer to inner curve of each annulus.
Such coordinates are often called geodesic rectangular coordinates. See Figure 5.8.
Chapter 5 Straightness on Hyperbolic Planes
72
Figure 5.8 Geodesic rectangular coordinates on annular hyperbolic plane
a. Show that the coordinate map x is one-to-one and onto from the whole of R2 onto
the whole of the annular hyperbolic plane. What maps to the annular strips, and
what maps to the radial geodesics?
b. Let and be two of the radial geodesics described in part a. If the distance
between and along the base curve is w, then show that the distance between
them at a distance s = n from the base curve is, on the paper hyperbolic model,
𝜌 𝑛
𝜌 𝑠/𝛿
𝑤(
) = 𝑤(
)
𝜌+𝛿
𝜌+𝛿
Now take the limit as → 0 to show that the distance between and on the
annular hyperbolic plane is w exp(−s/).
Thus, the coordinate chart x preserves (does not distort) distances along the (vertical)
second coordinate curves but at x(a, b) the distances along the first coordinate curve are
distorted by the factor of exp(−b/) when compared to the distances in R2.
Chapter 5 Straightness on Hyperbolic Planes
73
PROBLEM 5.3 THE PSEUDOSPHERE IS HYPERBOLIC
Show that locally the annular hyperbolic plane is isometric to portions of a (smooth)
surface defined by revolving the graph of a continuously differentiable function of z
about the z-axis. This is the surface usually called the pseudosphere.
OUTLINE OF PROOF
1. Argue that each point on the annular hyperbolic plane is like any other point. (Think
of the annular construction. About a point consider a neighborhood that keeps its
size as the width of the annular strips, , shrinks to zero.)
2. Start with one of the annular strips and complete it to a full annulus in a plane. Then
construct a surface of revolution by attaching to the inside edge of this annulus other
annular strips as described in the construction of the annular hyperbolic plane. (See
Figure 5.9.) Note that the second and subsequent annuli form truncated cones.
Finally, imagine the width of the annular strips, , shrinking to zero.
3. Derive a differential equation representing the coordinates of a point on the surface
using the geometry inherent in Figure 5.9. If f(r) is the height (z-coordinate) of the
surface at a distance of r from the z-axis, t...

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