Geometry :: Notes

1. Incidence Geometry
2. Neutral Geometry
3. Neutral Geometry Theorems
4. Complex Numbers
5. Project Ideas
6. Optional Topics
- 6.1. Hyperbolic Geometry
  - 6.1.1. Introduction
- 6.2. More Mobius Geometry
  - 6.2.1. Cline Types
  - 6.2.2. Spiral Examples

Survey of topics from classical Euclidean geometry, modern Euclidean geometry, projective geometry, transformational geometry and non-Euclidean geometries.

1 Incidence Geometry

For the incidence geometry we will have three undefined terms: point, line, and incident. We may think of incident as lies on. So if \(P\) is a point and \(l\) a line, we can say \(P\) is incident with \(l\) or \(P\) lies on \(l\) to mean the same thing. Incident is preferred since we can say either \(P\) is incident with \(l\) or \(l\) is incident with \(P\).

For every pair of distinct points \(P\) and \(Q\), there exists a unique line \(l\) incident with \(P\) and \(Q\). We say that "\(l\) is the line joining \(P\) and \(Q\)", and denote it by \(\overleftrightarrow{PQ}\).

For every line \(l\), there exist at least two distinct points incident with \(l\).

There exist three distinct points with the property that no line is incident with all three of them.

Three or more points \(A,B,C, \ldots\) are collinear if there exists a line incident with all of them. The points are noncollinear if no such line exists.

Three or more lines \(l,m,n,\ldots\) are concurrent if there exists a point incident with all of them. The lines are nonconcurrent if no such point exists.

Lines \(l\) and \(m\) are parallel if they are distinct and no point is incident with both of them. If \(l\) and \(m\) are parallel we write \(l \parallel m\).

Interpret point to mean one of the letters \(A,B,C\), and interpret line to mean a set of two points. Finally interpret incident to mean "is an element of" (or "contains" if going the other direction). List all lines in this interpretation and show that this interpretation is a model for incidence geometry.

Interpret point to mean one of the letters \(A,B,C\), and interpret line to mean the set of all points. Finally interpret incident to mean "is an element of" as in the Three Point Geometry Example. List all lines in this interpretation. Is this interpretation a model for incidence geometry?

Interpret point to mean one of the letters \(A,B,C,D\), line to mean a set of two points, and incident to mean "is an element of". How many lines are there? Is this interpretation a model for incidence geometry?

Interpret point to mean one of the letters \(A,B,C,D,E\), line to mean a set of two points, and incident to mean "is an element of". How many lines are there? Is this interpretation a model for incidence geometry?

Let \(0 < N \in \mathbb{Z}\). Interpret point to mean one of the numbers \(1,2,\ldots N\), line to mean a set of two points, and incident to mean "is an element of". How many lines are there? Is this interpretation a model for incidence geometry?

Interpret point to be any ordered pair \((x,y)\) of real numbers. Interpret line as the collection of points which satisfy the equation \(ax+by+c = 0\), where \(a,b\) and \(c\) are real numbers and \(a\) and \(b\) are not both 0. Interpret incident as \((x_0,y_0)\) lies on a line if that point satisfies the lines equation. Show that this is a model for incidence geometry. We use \(\bbR^2\) to denote the set of points in this model.

Interpret point to mean an ordered triple \((x,y,z)\) of real numbers so that \(x^2+y^2+z^2 = 1\). This set of points is called the 2-sphere and is denoted by \(\mathbb{S}^2\). A great circle is a circle on the sphere whose radius is equal to that of the sphere. Interpret line to mean a great circle on the sphere and incident to mean "is an element of". Is this interpretation a model for incidence geometry?

If \(l\) and \(m\) are distinct lines that are not parallel, then \(l\) and \(m\) have a unique point in common.

There exist three distinct lines that are not concurrent.

For every line, there is at least one point not incident to it.

For every point, there is at least one line not passing through it.

For every point \(P\), there exists at least two distinct lines through \(P\).

Interpret point to mean a point in the Cartesian Plane and inside the unit circle, that is a pair \((x,y)\) so that \(x^2 + y^2 < 1\). Interpret line to mean the part of a Euclidean line which lies inside the unit circle. Finally interpret lies on (incident) to be the Euclidean meaning, see the Cartesian Plane Example. Show this is a model for the incidence geometry.

Interpret point to mean an element of the set \(\{A,B,C,D,E,F,G\}\) and line to mean an element of the set

\begin{equation*} \{ \{A,B,C\}, \{C,D,E\}, \{E,F,A\}, \{A,G,D\}, \{C,G,F\}, \{E,G,B\}, \{B,D,F\} \}. \end{equation*}

Lies on means element of. Show this is a model for the incidence geometry. Draw a picture of this model.

Two models for the incidence geometry are isomorphic if:

there is a one to one correspondence \(P \leftrightarrow P'\) between the points in the models;
there is a one to one correspondence \(l \leftrightarrow l'\) between the lines in the models;
and \(P\) is incident with \(l\) if and only if \(P'\) is incident with \(l'\).

Such a correspondence between the models is called an isomorphism.

Let the set of lines to be \(\mathcal{L} = \{a,b,c\}\) and the set of points to be the two element subsets of \(\mathcal{L}\). Let incidence be set membership; so the point \(\{a,b\}\) is incident with lines \(a\) and \(b\). Show this is a model for the incidence geometry. It is the dual of the model given in Three Point Geometry Example.

Give an isomorphism between the model in Three Point Geometry and the model in Dual Three Point Geometry.

Give the "dual" of the four point model given in Four Point Example. Is this a model? Does an isomorphism exist between the four point model and its dual?

For every line \(l\) and for every point \(P\) that does not lie on \(l\), there is no line \(m\) such that \(P\) lies on \(m\) and \(m \parallel l\).

For every line \(l\) and for every point \(P\) that does not lie on \(l\), there is exactly one line \(m\) such that \(P\) lies on \(m\) and \(m \parallel l\).

For every line \(l\) and for every point \(P\) that does not lie on \(l\), there are at least two lines \(m\) and \(n\) such that \(P\) lies on both \(m\) and \(n\), \(m \parallel l\) and \(n \parallel l\).

Classify each of the three, four, and five point geometries given in Three Point Geometry Example, Four Point Geometry Example, and Five Point Geometry Example respectively as having either the Elliptic, Euclidean, or Hyperbolic Parallel Property (or none of the properties).

Classify each of the Cartesian Plane, the Sphere, and the Klein Disk as having either the Elliptic, Euclidean, or Hyperbolic Parallel Property (or none of the properties).

2 Neutral Geometry

The incidence geometry given in Incidence Geometry had only three axioms and as a result has limited complexity. For example, the Euclidean Plane most people are familiar with from high school has a much more complex structure which involves distances and angles. In order to work with more complex structures, such as the Euclidean Plane, we will need a more robust set of axioms. The axioms in this section will give rise to the neutral geometry, and theorems proved in this section will not only be valid for Euclidean Geometry, but also for Elliptical (Spherical) Geometry and Hyperbolic Geometry. Similar to what was done in incidence geometry with Elliptic Axiom, Euclidean Axiom, and ./axiom-incidence-axiom-hyperbolic.html, we will add additional axioms for the above three geometries. It should be noted however that much more can be done with Incidence Geometry than we have, see https://en.wikipedia.org/wiki/Incidence_geometry} for example.

The undefined terms for the neutral geometry are: point, line, distance, half-plane,angle measure, and area.

2.1 Existence Axiom

The collection of all points forms a nonempty set, and there is more than one point in that set.

The set of all points is called the plane and is denoted by \(\mathbb{P}\).

2.2 Incidence Axiom

Every line is a set of points. For every pair of distinct points \(A\) and \(B\) there is exactly one line \(l\) such that \(A \in l\) and \(B \in l\).

For distinct points \(A\) and \(B\), we use the notation \(\overleftrightarrow{AB}\) to denote the line determined by \(A\) and \(B\).

A point \(P\) is said to lie on line \(l\) if \(P \in l\).

A point \(P\) is called an external point for line \(l\) if \(P\) does not lie on \(l\).

Lines \(l\) and \(m\) are parallel if they are distinct and no point is incident with both of them, ie \(l \cap m = \emptyset\). If \(l\) and \(m\) are parallel we write \(l \parallel m\).

For lines \(l\) and \(m\), either \(l=m\), \(l\parallel m\), or \(l \cap m = \{x\}\) for some point \(x \in \mathbb{P}\).

2.3 Distance Axiom

For every pair of points \(P\) and \(Q\) there exists a real number \(PQ\), called the distance from \(P\) to \(Q\). For each line \(l\) there is a one-to-one correspondence from \(l\) to \(\mathbb{R}\) such that if \(P\) and \(Q\) are points on the line that correspond to the real numbers \(x\) and \(y\), respectively, then \(PQ = |x-y|\).

Three or more points \(A,B,C, \ldots\) are collinear if there exists a line incident with all of them. The points are noncollinear if no such line exists.

For distinct points \(A,B,\) and \(C\), we say \(C\) is between \(A\), and \(B\) and write \(A\cdot C\cdot B\), if \(A,B,\) and \(C\) are collinear, and \(AC+CB = AB\).

The segment between two points \(A\) and \(B\) is

\begin{equation*} \overline{AB} = \{A,B\} \cup \{P | A \cdot P \cdot B\}. \end{equation*}

The points \(A\) and \(B\) are called the end points of the segment \(\overline{AB}\). All other points of \(\overline{AB}\) are called interior points.

The ray between two distinct points \(A\) and \(B\) is

\begin{equation*} \overrightarrow{AB} = \overline{AB} \cup \{P | A \cdot B \cdot P\}. \end{equation*}

The point \(A\) is the endpoint of the ray \(\overrightarrow{AB}\).

The length of a segment \(\overline{AB}\) is the distance from \(A\) to \(B\) and is denoted \(AB\).

Two segments \(\overline{AB}\) and \(\overline{CD}\) are said to be congruent if they have the same length, we write \(\overline{AB} \cong \overline{CD}\).

A metric is a function \(d:\mathbb{P}^2 \to \mathbb{R}\) where for all \(P,Q,R \in \mathbb{P}\)

\(d(P,Q) \geq 0\), with equality if and only if \(P = Q\);
\(d(P,Q) = d(Q,P)\);
\(d(P,Q) \leq d(P,R) + d(R,Q)\).

As a function, length is a metric.

\(A\cdot C \cdot B\) if and only if \(B \cdot C \cdot A\).

The Euclidean metric is defined by

\[ d((x_0,y_0),(x_1,y_1)) = \sqrt{(x_1 - x_0)^2 + (y_1 - y_0)^2}, \]

where \((x_0,y_0), (x_1,y_1) \in \mathbb{P}\). Show that the Euclidean metric is in fact a metric. Draw a unit circle in the Euclidean metric. A unit circle is \(\{(x,y) \in \mathbb{P} \,|\, d((0,0),(x,y)) = 1\}\). Hint: For the triangle inequality, see the linear algebra review.

The taxicab metric is defined by

\[ \rho((x_0,y_0),(x_1,y_1)) = |x_1 - x_0| + |y_1 - y_0|, \]

where \((x_0,y_0), (x_1,y_1) \in \mathbb{P}\). Show that the taxicab metric is in fact a metric. Draw a unit circle in the taxicab metric. A unit circle is \(\{(x,y) \in \mathbb{P} \,|\, \rho((0,0),(x,y)) = 1\}\). Hint: See Theorem Length is a Metric.

Investigate the taxicab metric in more detail. For example, what do equilateral triangles look like? What do ellipses look like? Why is it called the taxicab metric? For more information and project ideas see Section 3.2 of the textbook.

Let \(A, B \in \mathbb{S}^2\). Define the spherical distance \(s(A,B)\) to be the length of the shortest arc of a great circle containing \(A\) and \(B\). Show that \(s\) is in fact a metric. What is the greatest distance between two points?

A coordinate function for a line \(l\) is a one-to-one and onto function \(f:l \to \mathbb{R}\) so that for every \(P,Q \in l\), \(PQ = |f(P) - f(Q)|\). The number \(f(P) \in \mathbb{R}\) is called the coordinate of the point \(P\).

Let \(l\) be a line in the Cartesian plane. If \(l\) is not a vertical line, then \(l\) has an equation of the form \(y=mx+b\). In that case define \(f:l \to \mathbb{R}\) by \(f(x,y) = x \sqrt{1+m^2}\). If \(l\) is a vertical line, then it has an equation of the form \(x=a\). In this case define \(f:l \to \mathbb{R}\) by \(f(a,y) = y\). Show that \(f\) is a coordinate function in the Euclidean metric.

Let \(l\) be a line in the Cartesian plane. Define \(f:l \to \mathbb{R}\) by

\begin{equation*} f(x,y) = \begin{cases} x(1 + |m|) & \textrm{ if \(l\) is not a vertical line} \\ y & \textrm{ if \(l\) is a vertical line}. \end{cases} \end{equation*}

Show that \(f\) is a coordinate function in the taxicab metric. See Example Coordinate Functions in the Euclidean Metric for definitions of \(m\) and \(a\).

For every pair of distinct points \(P\) and \(Q\), there is a coordinate function \(f: \overleftrightarrow{PQ} \to \mathbb{R}\) so that \(f(P) = 0\) and \(0 < f(Q)\).

Let \(l\) be a line and \(A,B\), and \(C\) three distinct points on \(l\). Let \(f:l \to \mathbb{R}\) be a coordinate function for \(l\). The point \(C\) is between \(A\) and \(B\) if and only if either \(f(A) < f(C) < f(B)\) or \(f(A) > f(C) > f(B)\).

Let \(A,B,\) and \(C\) be three points such that \(B\) lies on \(\overrightarrow{AC}\). Then \(A \cdot B \cdot C\) if and only if \(AB < AC\).

If \(A,B,\) and \(C\) are three distinct collinear points, then exactly one of them lies between the other two.

Let \(A\) and \(B\) be two distinct points. If \(f\) is a coordinate function for line \(l = \overleftrightarrow{AB}\) such that \(f(A) = 0\) and \(f(B) >0\), then \(\overrightarrow{AB} = \{P \in l\, |\, f(P) \geq 0\}\).

Let \(A\) and \(B\) be two distinct points. The point \(M\) is called a midpoint of \(\overline{AB}\) if \(M\) is between \(A\) and \(B\) and \(AM=MB\).

If \(A\) and \(B\) are distinct points and \(d\) is any nonnegative real number, then there exists a unique point \(C\) such that \(C\) lies on \(\overrightarrow{AB}\) and \(AC=d\).

2.4 Plane Separation Axiom

A set of points \(S\) is convex if for every pair of points \(A,B\in S\), the segment \(\overline{AB} \subset S\).

For every line \(l\), the points that do not lie on \(l\) form two disjoint, nonempty sets \(H_1\) and \(H_2\), called half-planed bounded by \(l\), such that:

each of \(H_1\) and \(H_2\) is convex;
if \(p \in H_1\) and \(Q \in H_2\), then \(\overline {PQ}\) intersects \(l\).

Let \(l\) be a line and let \(A\) be an external point. We use \(H_A\) to denote the half-plane bounded by \(l\) which contains \(A\).

Let \(l\) be a line and \(H_1,H_2\) the two half-planes bounded by \(l\). Let \(A\) and \(B\) be external points. \(A\) and \(B\) are said to be on the same side of \(l\) if they are both in \(H_1\) or both in \(H_2\). Otherwise they are said to be on opposite sides of \(l\).

Let \(l\) be a line with \(A\) and \(B\) points not on \(l\). The points \(A\) and \(B\) are on the same side of \(l\) if and only if \(\overline{AB} \cap l = \emptyset\). The points are on opposite sides of \(l\) if and only if \(\overline{AB} \cap l \neq \emptyset\).

Two rays \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\) which have the same end point are opposite rays if they are not equal, but \(\overleftrightarrow{AB} = \overleftrightarrow{AC}\). Otherwise they are non-opposite.

An angle is the union of two non-opposite rays \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\). The angle is denoted by \(\angle BAC\) or \(\angle CAB\). The point \(A\) is called the vertex, and the rays \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\) are called the sides of the angle.

Let \(A,B\) and \(C\) be points so that \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\) are non-opposite rays. The interior of angle \(\angle BAC\) is

\(\emptyset\) if \(\overrightarrow{AB} = \overrightarrow{AC}\)
\(H_B \cap H_C\) if \(\overrightarrow{AB} \neq \overrightarrow{AC}\), where \(H_B\) is the half-plane determined by \(\overleftrightarrow{AC}\) and \(B\), and \(H_C\) is the half-plane determined by \(\overleftrightarrow{AB}\) and \(C\).

Ray \(\overrightarrow{AD}\) is between rays \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\) if \(D\) is in the interior of \(\angle BAC\).

Let \(l\) be a line, \(A \in l\), and \(B\) an external point of \(l\). If \(C\) is a point on \(\overrightarrow{AB}\) and \(C\neq A\), then \(B\) and \(C\) are on the same side of \(l\).

Let \(A,B\), and \(C\) be three noncollinear points and let \(D\) be a point on the line \(\overleftrightarrow{BC}\). The point \(D\) is between points \(B\) and \(C\) if and only if ray \(\overrightarrow{AD}\) is between rays \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\).

Let \(A,B\), and \(C\) be noncollinear points. The triangle ABC is

\begin{equation*} \triangle ABC = \overline{AB} \cup \overline{BC} \cup \overline{AC}. \end{equation*}

The points \(A,B\) and \(C\) are the vertices of the triangle, while the segments \(\overline{AB}, \overline{BC}\) and \(\overline{AC}\) are its sides.

Let \(\triangle ABC\) be a triangle and let \(l\) be a line such that none of \(A,B\) or \(C\) lies on \(l\). If \(l\) intersects \(\overline{AB}\), then \(l\) intersects either \(\overline{AC}\) or \(\overline{BC}\).

2.5 Protractor Axiom

For every angle \(\angle BAC\) there is a real number \(\mu(\angle BAC)\), called the measure of \(\angle BAC\), such that:

\(0^\circ \leq \mu(\angle BAC) < 180^\circ\) for every angle \(\angle BAC\);
\(\mu(\angle BAC) = 0^\circ\) if and only if \(\overrightarrow{AB} = \overrightarrow{AC}\).
For each real number \(r \in (0,180)\), and for each half-plane \(H\) bounded by \(\overleftrightarrow{AB}\) there exists a unique ray \(\overrightarrow{AE}\) such that \(E\) is in \(H\) and \(\mu(\angle BAE) = r^\circ\).
If the ray \(\overrightarrow{AD}\) is between rays \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\), then \(\mu(\angle BAD) + \mu(\angle DAC) = \mu(\angle BAC)\).

Two angles \(\angle BAC\) and \(\angle EDF\) are congruent if \(\mu(\angle BAC) = \mu(\angle EDF)\). Angle congruence is denoted by \(\angle BAC \cong \angle EDF\).

Angle \(\angle BAC\) is a(n):

acute angle if \(\mu(\angle BAC) < 90^\circ\),
right angle if \(\mu(\angle BAC) = 90^\circ\),
obtuse angle if \(\mu(\angle BAC) > 90^\circ\).

If \(A,B,C\), and \(D\) are four distinct points such that \(C\) and \(D\) are on the same side of \(\overleftrightarrow{AB}\) and \(D\) is not on \(\overrightarrow{AC}\), then either \(C\) is in the interior of \(\angle BAD\) or \(D\) is in the interior of \(\angle BAC\).

Let \(A,B,C\), and \(D\) be four distinct points such that \(C\) and \(D\) lie on the same side of \(\overleftrightarrow{AB}\). Then \(\mu(\angle BAD) < \mu(\angle BAC)\) if and only if \(\overrightarrow{AD}\) is between rays \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\).

Let \(A,B\), and \(C\) be three noncollinear points. A ray \(\overrightarrow{AD}\) is an angle bisector of \(\angle BAC\) if \(D\) is in the interior of \(\angle BAC\) and \(\mu(\angle BAD) = \mu(\angle DAC)\).

Let \(A\) and \(B\) be two points, \(H\) be one of the half-planes of \(l = \overleftrightarrow{AB}\), and let \(\mathcal{A} = \{\angle BAE \, | \, E \in H\}\). Define \(f:\mathcal{A} \to (0,180)\) by \(f(\angle BAE) = \mu(\angle BAE)\). Then \(f\) is a one-to-one and onto function, and \(\overrightarrow{AF}\) is between \(\overrightarrow{AB}\) and \(\overrightarrow{AE}\) if and only if \(f(\angle BAF)\) is between 0 and \(f(\angle BAE)\).

If \(A,B\), and \(C\) are three noncollinear points, then there exists a unique angle bisector for \(\angle BAC\).

Let \(l\) be a line and \(A\) and \(D\) distinct points on \(l\). If \(B\) and \(E\) are points on opposite sides of \(l\), then \(\overrightarrow{AB} \cap \overrightarrow{DE} = \emptyset\).

If \(\triangle ABC\) is a triangle and \(D\) is a point in the interior of \(\angle BAC\), then there is a point \(G\) such that \(G\) lies on both \(\overrightarrow{AD}\) and \(\overline{BC}\).

A point \(D\) is in the interior of the angle \(\angle BAC\) if and only if \(\overrightarrow{AD}\) intersects the interior of \(\overline{BC}\).

Two angles \(\angle BAD\) and \(\angle DAC\) form a linear pair if \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\) are opposite rays.

Let \(A,B,C,D\), and \(E\) be points. If \(C\cdot A \cdot B\) and \(D\) is in the interior of \(\angle BAE\), then \(E\) is in the interior of \(\angle DAC\).

If angles \(\angle BAD\) and \(\angle DAC\) form a linear pair, then \(\mu(\angle BAD) + \mu(\angle DAC) = 180^\circ\).

Two lines \(l\) and \(m\) are perpendicular if there exists a point \(A\) that lies on both \(l\) and \(m\) and there are points \(B \in l\) and \(C \in m\) so that \(\angle BAC\) is a right angle. This is denoted \(l \perp m\).

If \(l\) is a line and \(P\) is a point on \(l\), then there exists exactly one line \(m\) such that \(P\) lies on \(m\) and \(m \perp l\).

Let \(D\) and \(E\) be distinct points. A perpendicular bisector of \(\overline{DE}\) is a line \(l\) such that the midpoint of \(\overline{DE}\) lies on \(l\) and \(l \perp \overleftrightarrow{DE}\).

If \(D\) and \(E\) are two distinct points, then there exists a unique perpendicular bisector for \(\overline{DE}\).

Angles \(\angle BAC\) and \(\angle DAE\) are called vertical angles if rays \(\overrightarrow{AB}\) and \(\overrightarrow{AE}\) are opposite and rays \(\overrightarrow{AC}\) and \(\overrightarrow{AD}\) are opposite, or if rays \(\overrightarrow{AB}\) and \(\overrightarrow{AD}\) are opposite and \(\overrightarrow{AC}\) and \(\overrightarrow{AE}\) are opposite.

Vertical angles are congruent.

Two triangles are congruent if there is a correspondence between the vertices of the first triangle and the vertices of the second triangle such that corresponding angles are congruent and corresponding sides are congruent.

If \(\triangle ABC\) and \(\triangle DEF\) are congruent, we write \(\triangle ABC \cong \triangle DEF\). We take this to mean that \(A \leftrightarrow D\), \(B \leftrightarrow E\) \(C \leftrightarrow F\); thus \(\overline{AB} \cong \overline{DE}\), \(\overline{BC} \cong \overline{EF}\), \(\overline{AC} \cong \overline{DF}\), \(\angle{ABC} \cong \angle{DEF}\) \(\angle{BCA} \cong \angle{EDF}\), and \(\angle{CAB} \cong \angle{FDE}\).

2.6 Side Angle Side Axiom

If \(\triangle ABC\) and \(\triangle DEF\) are two triangles such that \(\overline{AB} \cong \overline{DE}\), \(\angle ABC \cong \angle DEF\) and \(\overline{BC} \cong \overline{EF}\), then \(\triangle ABC \cong \triangle DEF\).

A triangle is an isosceles triangle if it has a pair of congruent sides. The two angles not between the congruent sides are called base angles.

The base angles of an isosceles triangle are congruent.

The Side angle Side axiom does not hold in the taxicab geometry. As a result, the taxicab metric is not a model for the neutral geometry. (It does satisfy the other five axioms but you need not show this).

3 Neutral Geometry Theorems

The previous chapter replaced Euclid's original first four axioms, as well as all of Euclid's unstated assumptions, we have "patched up" Euclid's errors. We are now ready to start proving theorems from Book I of the Elements. Later we will add a parallel axiom, either Euclidean or Hyperbolic.

Let \(\triangle ABC\) be a triangle. The angles \(\angle CAB, \angle ABC\), and \(\angle BCA\) are called interior angles of the triangle. An angle that forms a linear pair with one of the interior angles is called an exterior angle for the triangle. If the exterior angle forms a linear pair with the interior angle at one vertex, then the other two interior angles at the other two vertices are referred to as remote interior angles (for the original vertex).

The measure of an exterior angle for a triangle is strictly greater than the measure of either remote interior angles.

You need not prove this theorem.

For every line \(l\) and for every point \(P\), there exists a unique line \(m\) such that \(P\) lies on \(m\) and \(m \perp l\).

You need not prove this theorem.

If one interior angle of a triangle is a right angle or is an obtuse angle, then both the other interior angles are acute.

If \(\triangle ABC\) and \(\triangle DEF\) are two triangles such that \(\angle CAB \cong \angle FDE, \overline{AB} \cong \overline{DE}\), and \(\angle ABC \cong \angle DEF\), then \(\triangle ABC \cong \triangle DEF\).

If \(\triangle ABC\) is a triangle such that \(\angle ABC \cong \angle ACB\), then \(\overline{AB} \cong \overline{AC}\).

If \(\triangle ABC\) and \(\triangle DEF\) are two triangles such that \(\angle ABC \cong \angle DEF, \angle BCA \cong \angle EFD\), and \(\overline{AC} \cong \overline{DF}\) then \(\triangle ABC \cong \triangle DEF\).

A triangle is a right triangle if one of its interior angles is a right angle. The side opposite the right angle is called the hypotenuse and the two sides adjacent to the right angle are called legs.

If the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the two triangles are congruent.

If \(\triangle ABC\) is a triangle and \(\overline{DE}\) is a segment congruent to \(\overline{AB}\), and \(H\) is a half-plane bounded by \(\overleftrightarrow{DE}\), then there is a unique point \(F \in H\) such that \(\triangle DEF \cong \triangle ABC\).

If \(\triangle ABC\) and \(\triangle DEF\) are two triangles such that \(\overline{AB} \cong \overline{DE}\), \(\overline{BC} \cong \overline{EF}\), and \(\overline{CA} \cong \overline{FD}\), the \(\triangle ABC \cong \triangle DEF\).

You need not prove this theorem.

A scalene triangle is a triangle where each of the three sides has a different length.

Let \(\triangle ABC\) be a triangle. Then \(AB > BC\) if and only if \(\mu(\angle ACB) > \mu(\angle BAC)\).

If \(\triangle ABC\) and \(\triangle DEF\) are two triangles such that \(AB = DE, AC = DF\), and \(\mu(\angle BAC) < \mu(\angle EDF)\), then \(BC < EF\).

You need not prove this theorem.

Let \(l\) be a line and \(P\) a point. A perpendicular from \(P\) to \(l\) is the line \(m\) from Theorem Existence and Uniqueness of Perpendiculars. The intersection of \(l\) and \(m\) is called the foot of the perpendicular.

Let \(l\) be a line, \(P\) an external point, and \(F\) the foot of the perpendicular from \(l\) to \(P\). If \(R\) is any other point on \(l\) distinct from \(F\), then \(PR > PF\).

The distance from a point \(P\) to a line \(l\), denoted \(d(P,l)\), is the distance from \(P\) to the foot of the perpendicular from \(P\) to \(l\).

Let \(A,B\), and \(C\) be three non-collinear points and let \(P\) be a point on the interior of \(\angle BAC\). Then \(P\) lies on the angle bisector of \(\angle BAC\) if and only if \(d(P, \overleftrightarrow{AB}) = d(p, \overleftrightarrow{AC})\).

Let \(A\) and \(B\) be distinct points. A point \(P\) lies on the perpendicular bisector of \(\overline{AB}\) if and only if \(PA = PB\).

4 Complex Numbers

4.1 Basic Definitions

The imaginary unit is \(i\) and satisfies \(i^2 = -1\).

A complex number is a number \(z=(x,y)\) in the cartesian plane. The horizonal axis measures the real part of \(z\) (\(\Re(z) = x\)), and the vertical axis measures the imaginary part of \(z\) (\(\Im(z) = y\)).

The cartesian form of a complex number \(z = (x,y)\) is \(z=x+yi\).

Addition and subtraction of complex numbers \(z=a+bi\) and \(w=c+di\) is defined by

\begin{equation*} z \pm w = (a + c) \pm (b+d)i. \end{equation*}

Multiplication of complex numbers \(z=a+bi\) and \(w=c+di\) is defined by

\begin{equation*} zw = (ac-bd) + (ad+bc)i. \end{equation*}

The absolute value or modulus of a complex number \(z=a+bi\) is

\begin{equation*} |z| = \sqrt{a^2 + b^2}. \end{equation*}

The distance between \(z=a+bi\) and \(w=c+di\) is given by

\begin{equation*} |z-w| = \sqrt{(a-c)^2 + (b-d)^2}. \end{equation*}

The conjugate of \(z=a+bi\) id defined to be \(\overline{z} = a-bi\).

The quotient of \(z=a+bi\) and \(0 \neq w=c+di\) is

\begin{equation*} \frac{z}{w} = \frac{a+bi}{c+di} = \frac{ac+bd}{c^2+d^2} + \frac{bc-ad}{c^2+d^2}i. \end{equation*}

Euler's Formula is

\begin{equation*} e^{i\theta} = \cos \theta + i \sin\theta. \end{equation*}

The polar form of a complex number \(z=x+yi\) is

\begin{equation*} z = r\cos \theta + ir\sin \theta = re^{i\theta}, \end{equation*}

where \(r = |z|\) and \(\theta\) is the angle with initial side along the positive horizontal axis, and terminal side through the point \((x,y)\).

The argument of the complex number \(z=re^{i\theta}\) is \(\arg(z) = \theta\).

\((\cos \theta + i \sin \theta)^n = \cos(n\theta) + i \sin(n\theta)\)

4.1.1 Basic Complex Numbers Problems

Write \(4(1+2i)-2(5-i)\) in cartesian form.

Write \(\displaystyle \frac{2}{3+i}\) in cartesian form.

Write \((1+i)(1-i)\) in cartesian form.

Write \((1+i)^3\) in cartesian form.

Write \(e^{1+i\pi}, e^{i\pi/2}\) and \(e^{e^{ln(\pi + i\pi/2)}}\) in polar and cartesian form.

Verify the identity \(e^{i \pi} + 1 =0\).

Write \(-1+i\sqrt{3}, 4i\), and \(5-5i\sqrt{3}\) in polar form

4.2 Transformations

A transformation is a one-to-one function whose domain and codomain are equal.

A geometric transformation is a translation if it is of the form \(z \mapsto z+b\) for some constant \(b \in \mathbb{C}\).

A geometric transformation is a rotation if it is of the form \(z \mapsto e^{i\theta} z\) for some \(\theta \in \mathbb{R}\).

A geometric transformation is homothetic if it is of the form \(z \mapsto kz\) for some \(0 < k \in \mathbb{R}\). If \(k<1\) then the transformation is called a shrinking, and if \(k>\) the transformation is called a stretching.

Classify each of the following geometric transformations as either translation, rotation, or homothetic:

\(z \mapsto z+2\);
\(z \mapsto 5z\);
\(z \mapsto z + i -3\);
\(z \mapsto iz\).

An isometry of a metric space is a function \(f\) which preserves distances.

A fixed point of a transformation \(T\) is a point so that \(T(x) = x\).

A a geometric transformation is a reflection if it is an isometry whose set of fixed points is a line. This is also called a mirror reflection.

Reflection about the horizontal axis is given by the transformation \(z \mapsto \overline{z}\). Reflection about the line through the origin forming an angle \(\alpha\) with the positive horizontal axis is given by the transformation \(z \mapsto e^{2i\alpha} \overline{z}\).

The geometric transformation \(z \mapsto 1/z\) (for \(z \neq 0\)) is called inversion.

A transformation is euclidean if it is a rotation, translation, or reflection. It is special euclidean if it is a rotation or translation.

A transformation is conformal if it preserves angles.

A mobius transformation is a transformation of the form

\begin{equation*} z \mapsto \frac{az+b}{cz+d}. \end{equation*}

4.2.1 Graphical Transformations

We will now show transformations graphically. In what follows, the blue dots are the set

\begin{equation*} \{z : z=x+iy;\ x,y \in \mathbb{Z};\ -5 \leq x,y \leq 5\}, \end{equation*}

and the red dots are the result of applying said the transformation to the blue dots

The transformation \(z \mapsto z + (2-3i)\).

The transformation \(z \mapsto e^{i\pi/4}z\).

The transformation \(z \mapsto e^{-i\pi/6}z\).

The transformation \(z \mapsto 2z\).

The transformation \(z \mapsto z/3\).

The transformation \(z \mapsto 1/z\). Not the change in scale on this example.

4.2.2 Geometric Transformations

Let \(p,q,\) and \(r\) be complex numbers. Explain why

\begin{equation*} \angle pqr = \arg\left(\frac{r-q}{p-q}\right). \end{equation*}

Here the angle measure is the normal angle measure in radians (as opposed to the one given in the Neutral Protractor Axiom requiring the measure to be between 0 and \(180^\circ\)). Hint: Consider first q=0 and use polar form.

Let \(f(z) = iz+2\) and \(g(z) = -iz+5\). Sketch the composition transformation \(w = f(g(z))\). What type of transformation is this?

Prove that the composition of two [translations, rotations, homothetics] is again a [translation, rotation, homothetic].

Let \(a,b \in \mathbb{C}\). Show that \(w = az+b\) is a composition of rotations, homothetics, and translations.

Prove that the composition of two conformal transformations is conformal.

Find a transformation of the complex plane that takes the unit disk \(\{z : |z| < 1\}\) to the disk \(\{z: |z-5| < 3\}\).

Find transformations for these rotations:

by \(\pi/4\) about the point \(i\);
by \(\pi/2\) about the point \(2+i\);
by \(-\pi/3\) about 3.

4.3 Transformation Groups

Let \(S\) be a nonempty set. A transformation group is a collection \(G\) of transformations on \(S\) so that:

the identity transformation is in \(G\);
the transformations in \(G\) are invertible and their inverses are in \(G\);
\(G\) is closed under composition.

Let \(S\) be a nonempty set and \(G\) a transformation group on \(S\). The pair \((S,G)\) is called a geometry or a model of a geometry, the set \(S\) is the underlying space of the geometry, and the set \(G\) is the transformation group of the geometry.

Let \((S,G)\) be a geometry. If \(A \subset S\), then we say that \(A\) is a figure or that \(A\) is a figure of the geometry.

Two figures \(A\) and \(B\) of a geometry \((S,G)\) are congruent if there is a transformation \(T\in G\) so that \(T(A) = B\). Notice that \(T(A) = \{Tz : z \in A\}\).

The geometry \((\mathbb{C}, E_s)\), where \(E_s\) is the collection of all special euclidean transformations (rotations and translations) and their compositions, is a geometry called the special euclidean geometry. \(E_s\) can also be described as the collection of transformations of the form \(z \mapsto e^{i\theta}z + b\) for \(\theta \in \mathbb{R}\) and \(b \in \mathbb{C}\).

The geometry \((\mathbb{C},E)\), where \(E\) is the collection of all euclidean transformations (rotations, translations, and reflections) and their compositions, is a geometry called the euclidean geometry.

Let \((S,G)\) be a geometry and \(D\) a set of figures in the geometry.

The set \(D\) is invariant if for every figure \(A \in D\) and every transformation \(T \in G\), \(T(A) \in D\).

A function \(f\) defined on \(D\) is invariant if for every figure \(B\in D\) and every transformation \(T \in G\), \(f(B) = f(T(B))\).

Let \(\triangle ABC\) be a triangle. The perimeter of the triangle is \(AB+BC+AB\).

Let \(D\) be the collection of all triangles in \((\mathbb{C},E)\).

Define \(p:D \to \mathbb{R}\) by \(p(\triangle)\) is the perimeter of \(\triangle\). Then \(p\) is an invariant function of \((\mathbb{C},E)\).

Define \(p':D \to \mathbb{R}\) by \(p'(\triangle)\) is the sum of the distance from each vertex of \(triangle\) to the origin. Then \(p'\) is not an invariant function of \((\mathbb{C},E)\).

Two geometries \((S_1,G_1)\) and \((S_2,G_2)\) are isomorphic if there is a continuous invertible function \(f:S_1 \to S_2\) so that

for every transformation \(T_1 \in G_1\), there is a corresponding transformation \(T_2 \in G_2\) so that \(T_1 = f^{-1} \circ T_2 \circ f\), (equivalently \(f \circ T_1 = T_2 \circ f\)), and
for every \(T_2' \in G_2\), there is a corresponding \(T_1' \in G_1\) so that \(T_2' = f \circ T_1' \circ f^{-1} \) (equivalently \(f \circ T'_2 = T'_1 \circ f\)).

The function \(f\) is called an isomorphism. This is equivalent to having the following diagram commute.

The closed unit disk is \(D = \{z \in \mathbb{C} : |z| \leq 1\}\). The open unit disk is \(D^o = \{z \in \mathbb{C} : |z| < 1\}\).

Define \(f : \mathbb{C} \to D^o\) by

\begin{equation*} f(z) = \frac{z}{\sqrt{1+|z|^2}}. \end{equation*}

Then \(f\) is a continuous invertible function which is an isomorphism between models of the euclidean geometry.

The graph of \(\sin(10x)\) is shown in blue below, graphed in the usual model of euclidean geometry. Next, the graph is moved by the function \(f\) given above into the unit disk model of euclidean geometry, and is shown in red.

Some other figures (graphs of functions in this case) are shown both in the standard euclidean plane and in the open disk model.

Let \(\mathcal{T}\) be the set of all translations of \(\mathbb{C}\), so \(\mathcal{T}\) is the set of all transformations of the form \(z \mapsto z+b\) for some \(b \in \mathbb{C}\). The pair \((\mathbb{C}, \mathcal{T})\) is called the translational geometry and is geometry.

4.3.1 Transformation Groups Problems

Let \(\mathcal{GL}\) (\(\mathcal{GL}\) stands for general linear) be all transformations of \(\mathbb{C}\) of the form \(z \mapsto az + b\) for \(a,b \in \mathbb{C}\) and \(a \neq 0\). Show that \(\mathcal{GL}\) is a transformation group (that is show that \(\mathcal{GL}\) contains the identity, show inverses exist and are in \(\mathcal{GL}\), and show compositions are in \(\mathcal{GL}\)).

Convince yourself that \(E_s\) is a transformation group.

Let \(\mathcal{R}\) be all the rotations of \(\mathbb{C}\). Show that \((\mathbb{C},\mathcal{R})\) is a geometry. Hint: show that \(\mathcal{R}\) is a transformation group.

Let \(\mathcal{L}_H\) be the set of all horizontal lines. Is \(\mathcal{L}_H\) an invariant set in:

\((\mathbb{C},\mathcal{R})\)?
\((\mathbb{C},\mathcal{T})\)?
\((\mathbb{C},\mathcal{GL})\)?
\((\mathbb{C},E_s)\)?
\((\mathbb{C},E)\)?

Let \(\mathcal{L}_O\) be the set of all lines through the origin. Is \(\mathcal{L}_O\) an invariant set in:

\((\mathbb{C},\mathcal{R})\)?
\((\mathbb{C},\mathcal{T})\)?
\((\mathbb{C},\mathcal{GL})\)?
\((\mathbb{C},E_s)\)?
\((\mathbb{C},E)\)?

Is \((\mathbb{C}, \{z \mapsto e^{i\theta}z + b : \theta \in \mathbb{R}, b \in \mathbb{C}\})\) a geometry? Have we seen this geometry before?

Let \(\triangle_1\) be the triangle with vertices \(-1+0i, 0+i, 1+0i\) and \(\triangle_2\) be the triangle with vertices \(-3+0i, 0+3i,3+0i\). For which of the following is \(\triangle_1 \cong \triangle_2\)?

\((\mathbb{C},\mathcal{R})\)
\((\mathbb{C},\mathcal{T})\)
\((\mathbb{C},\mathcal{GL})\)
\((\mathbb{C},E_s)\)
\((\mathbb{C},E)\)

4.4 Mobius Geometry

\(\mathbb{C}^+ = \mathbb{C} \cup \{\infty\}\) is called the extended complex plane or the Riemann Sphere.

Let \(\mathcal{M}\) be the set of all Mobius transformations, that is \(\mathcal{M}\) is the set of all transformations of the form \[ z \mapsto \frac{az+b}{cz+d}, \] where \(a,b,c,d \in \mathbb{C}\) with \(ad-bc \neq 0\). The pair \((\mathbb{C}^+,\mathcal{M})\) is a model of Mobius Geometry.

If \(T \in \mathcal{M}\) is a a Mobius transformation which is not the identity, then \(T\) has either one or two fixed points.

If a Mobius transformation has three fixed points, it is the identity.

The cross ratio of four complex numbers \(z_0, z_1, z_2,\) and \(z_3\) is \[ (z_0,z_1,z_2,z_3) = \frac{z_0-z_2}{z_0-z_3} \frac{z_1-z_3}{z_1-z_2}. \]

Let \(z_1,z_2,z_3 \in \mathbb{C}^+\) be three distinct points, and \(w_1,w_2,w_3 \in \mathbb{C}^+\) be three distinct points. There is a mobius transformation so that \(z_1 \mapsto w_1, z_2 \mapsto w_2\), and \(z_3 \mapsto w_3\).

A cline is a circle or a line.

A mobius transformation takes clines to clines.

4.4.1 Mobius Geometry Problems

Show that the identity \(z \mapsto z\) is a Mobius transformation.

Show that for \(T,S \in \mathcal{M}\), \(T^{-1} \in \mathcal{M}\) and \(T \circ S \in \mathcal{M}\).

Find a Mobius transformation \(T\) so that \(1 \mapsto 2, 2 \mapsto 3\), and \(3 \mapsto -1\).

Find a Mobius transformation that takes the unit circle \(|z| =1\) to the line \(x+y=1\).

For the following problems, let \(T,S \in \mathcal{M}\) with \[ Tz = \frac{az+b}{cz+d}\\ Sz = \frac{ez+f}{gz+h}, \] \(ad-bc \neq 0\) and \(eh-fg \neq 0\). Also let \[ M = \begin{bmatrix} a & b \\ c & d \end{bmatrix}, \] and \[ N = \begin{bmatrix} e & f \\ g & h \end{bmatrix}. \]

What is \(\det(M)\)? How does this relate to \(T\)?

What is \(M^{-1}\)? How does this relate to \(T\)?

What is \(MN\)? How does this relate to \(T\) and \(S\)?

5 Project Ideas

Taxicab Project

Investigate the taxicab metric in more detail, see Section 3.2.

Image Creation

Draw digital images for use in the notes.

Euclidean Constructions

Use a software package to construct proofs. Some geometry software:

Regular Polygons and Tiling

Investigate tilings of the plane, see Section 4.4. See also the wikipages (more involved projects):

Fractals

Investigate fractals, see the end of Section 4.6.

Area and Volume

Investigate area and volume, from Euclid's point of view, see Section 4.6.

Lecture Notes

Create lecture notes for high school geometry, see Appendix B.

Quaternions

We extend the reals to get complex numbers. We extend the complex numbers to get the quaternions, we extend the quaternions to get the octonians. This project is to investigate the quaternions. See https://www.youtube.com/watch?v=d4EgbgTm0Bg.

Geometric times tables

Investigate/create something like https://www.youtube.com/watch?v=qhbuKbxJsk8.

Toothpick patterns

A project about toothpicks. See https://www.youtube.com/watch?v=_UtCli1SgjI and http://oeis.org/A139250/a139250.anim.html.

Turtle graphics

Use a programming language to make some nice pictures. Google "logo programming" or "turtle graphics". See https://mathnotes.cc/research/applied-topology/turtle-geometry/turtle-graphics/index.html.

Turtle Polynomials

Related to turtle graphics, but solve polynomials. See https://www.youtube.com/watch?v=IUC-8P0zXe8.

L Systems

Similar to turtle graphics, useful in modeling biological systems. See https://en.wikipedia.org/wiki/L-system.

Other

If you have another idea, then talk with me.

6 Optional Topics

6.1 Hyperbolic Geometry

6.1.1 Introduction

Let \(z_0,z_1,z_2\) and \(z_3\) be four distinct point is \(\mathbb{C}\), and \(T\) a Mobius transformation. Then \((z_0,z_1,z_2,z_3) = (Tz_0,Tz_1,Tz_2,Tz_3)\).

The cross ratio \((z_0,z_1,z_2,z_3)\) is real if and only if \(z_0,z_1,z_2\), and \(z_3\) all line on a cline.

Let \(C\) be a cline containing three distinct points \(z_1,z_2\), and \(z_3\). Two points \(z\) and \(z^\star\) are symmetric with respect to \(C\) if \((z^\star,z_1,z_2,z_3) = \overline{(z,z_1,z_2,z_3)}\).

If \(T\) is a Mobius transformation and \(T\) maps the unit disk \(D\) onto itself, then \(T\) is of the form

\[ Tz = e^{i\theta}\frac{z-z_0}{1-\overline{z_0}z} \]

for some \(\theta \in \mathbb{R}\) and some \(z_0 \in D\).

Let \(\mathcal{H}\) be the set of Mobius transformations mapping the unit disk \(D\) onto itself, that is

\[ \mathcal{H} = \left\{z \mapsto e^{i\theta}\frac{z-z_0}{1-\overline{z_0}z} : \theta \in \mathbb{R}, z_0 \in D\right\}. \]

The pair \((D, \mathcal{H})\) is a model for hyperbolic geometry. When talking of hyperbolic geometry, \(D\) is called the hyperbolic plane, and \(\mathcal{H}\) is called the hyperbolic transformation group.

A hyperbolic strait line is a Euclidean circle of line in \(\mathbb{C}\) that intersects the unit circle at right angles.

In hyperbolic geometry, all hyperbolic strait lines are congruent. Two points in the hyperbolic plane determine a unique hyperbolic start line.

Point on the unit circle are called ideal points.

Two hyperbolic lines are parallel if the don't intersect in \(D\), but share and ideal point.

Two hyperbolc lines are hyperparallel if they don't intersect in \(D\), and don't have an ideal point in common.

6.2 More Mobius Geometry

6.2.1 Cline Types

Let \(A\) and \(B\) be points in \(\mathbb{C}^+\).

A type I cline of \(A\) and \(B\) is a cline containing both \(A\) and \(B\).

A cline \(c\) is a type II cline of \(A\) and \(B\) if \(A\) and \(B\) are symmetric with respect to \(c\).

The standard cartesian form of \((x,y)\)-pairs is useful for graphing many objects. For other objects, the polar form of \((r,\theta)\)-pairs is more useful. The usual polar form has two special points, 0 and \(\infty\), and gives one a coordinate system based on these two points.. In this example, we will use the usual polar form to generate a coordinate system about the two points \(P = -4 -2i\) and \(Q = 5+3i\) by applying a Mobius transformation which takes 0 to \(P\) and \(\infty\) to \(Q\).

First we show the usual polar coordinate system. The angular lines differ by \(\pi/12\), start at \(\theta = 0\), and are shown in blue. The circles start at \(r=1\), differ by 1, and are shown in red. Also plotted in black here are the points \(P\) and \(Q\). We can think of the blue lines as circles through 0 and infinity, and hence are tye I clines for the points 0 and infinity. We can think of the red circles as circles about 0, but also as circles about infinity; they are type II clines for the points 0 and infinity.

The result of applying the Mobius transformation taking 0 to \(P\) and \(\infty\) to \(Q\) is shown below. Again the blue are type I clines, and the red are type II clines, this time of the points \(P\) and \(Q\). There are actually more red clines about the point \(P\) but they are not shown (since there aren't many clines around \(\infty\) in the above graphic, the red line corresponds to a circle of radius 1 above).

6.2.2 Spiral Examples

We will examine the transformation \(T: z \mapsto (11/10)e^{\pi/10i}z\). This transformation scales by 1.1, and rotates by \(\pi/10\). It is interesting to look at the iterated application of \(T\) to a starting fixed number. What we mean by this is to look at \(\{T^0z_0, T^1z_0, T^2z_0 \ldots T^{100}z_0\}\), where \(T^0 z_0 = z_0\) and \(T^nz_0 = T(T^{n-1}z_0)\) for \(n\geq 1\). The graph for \(z_0 = 1\) is shown below.

The second plot shows \(\{S^0z_1, S^1z_1, S^2z_1 \ldots S^{100}z_1\}\), where \(S: z \mapsto RTR^{-1}z\) and \(R: z \mapsto (z-1)/(z+1)\), with \(z_1 = -96/100\). Notice that \(R(0) = -1\) and \(R(\infty) = 1\). Thus \(S\) takes maps -1 to 0, applies the transformation \(T\) and then maps everything back to -1. The result is a spiral with source \(z=-1\) and sink \(z=1\).