The Four Pillars of Geometry

Question 1

Thales Theorem

A line drawn parallel to one side of a triangle cuts the other two sides ___________.

Answer 1

Thales Theorem

A line drawn parallel to one side of a triangle cuts the other two sides proportionally.

Question 2

Invariance of angles in a circle

If A and B are two points on a circle, then for all points C on one of the arcs connecting them, the angle ACB is _______.

Answer 2

Invariance of angles in a circle

​If A and B are two points on a circle, then for all points C on one of the arcs connecting them, the angle ACB is constant.

Question 3

Angle in a semicircle theorem

If A and B are the ends of a diameter of a circle, and C is any other point on the circle, then angle ACB is a ______ angle.

Answer 3

Angle in a semicircle theorem

If A and B are the ends of a diameter of a circle, and C is any other point on the circle, then angle ACB is a right angle.

Question 4

For linear equations, some / all intersection points involved in a straightedge and compass construction can be found with the operations +, -, x, /, sqrt()

Answer 4

For linear equations, all intersection points involved in a straightedge and compass construction can be found with the operations +, -, x, /, sqrt()

Question 5

A transformation f is called an ______ if it sends any two points P₁ and P₂ to points f(P₁) and f(P₂) the same distance apart.

Thus, an ______ is a function f with the property:

f(P₁) f(P₂) | = | P₁ P₂ |

Answer 5

A transformation f is called an isometry if it sends any two points P₁ and P₂ to points f(P₁) and f(P₂) the same distance apart.

Thus, an isometry is a function f with the property:

f(P₁) f(P₂) | = | P₁ P₂ |

Question 6

A _______ moves each point of the plane the same distance in the same direction.

It sends each point (x,y) to the point ______, where a and b are the change of distance.

Answer 6

A translation moves each point of the plane the same distance in the same direction.

It sends each point (x,y) to the point (x+a, y+b), where a and b are the change of distance.

Question 7

A ______ takes two numbers c and s such that c²+s²=1 where c and s are the numbers that result from cos() and sin() respectively.

It sends the point (x,y) to the point _______.

Answer 7

A rotation takes two numbers c and s such that c²+s²=1 where c and s are the numbers that result from cos() and sin() respectively.

It sends the point (x,y) to the point (cx-sy, sx+cy).

Question 8

Three Reflections Theorem

Any isometry of R²is a combination of one, two, or three ________.

Answer 8

Three Reflections Theorem

Any isometry of R²is a combination of one, two, or three reflections.

Question 9

The role of transformations was first characterized by Felix Kelin in an address he delivered at the University of Erlangen in 1872. His address is known as the ______ ______, which characterizes geometry as the study of _______ _____ and their _______.

Answer 9

The role of transformations was first characterized by Felix Kelin in an address he delivered at the University of Erlangen in 1872. His address is known as the Erlangen Program, which characterizes geometry as the study of transformation groups and their invariants.

Question 10

The concept of distance is introduced in linear algebra through the concept of the inner product u•v of vectors u and v.

If u = (u₁, u₂) and v = (v₁, v₂),

Then u•v = ________

Answer 10

The concept of distance is introduced in linear algebra through the concept of the inner product u•v of vectors u and v.

If u = (u₁, u₂) and v = (v₁, v₂),

Then u•v = u₁v_{1 +} u₂v_{<span>2</span>}

Question 11

The inner product gives us distance because u•u=|u|²

where |u| is the distance of u from the origin 0. It also gives us angles because

u•v = _______

Answer 11

The inner product gives us distance because u•u=|u|²

where |u| is the distance of u from the origin 0. It also gives us angles because

u•v = |u| |v| cos(theta)

Question 12

Complete the 8 properties for something to be considered a vector space:

u+v =

u + (v+w) =

u + 0 =

u + (-u) =

1u =

a(u+v) =

(a+b)u =

a(bu) =

Answer 12

Complete the 8 properties for something to be considered a vector space:

u+v = v + u

u + (v+w) = (u+v) + w

u + 0 = u

u + (-u) = 0

1u = u

a(u+v) = au + av

(a+b)u = au + bu

a(bu) = (ab)u

Question 13

The ____-_____ is preserved as an invariant in a projection. It is a quantity that is associated with four points on a line. If the four points have coordinates p, q, r, s, then their _____-______ is the function of the ordered 4-tuple (p,q,r,s) written as:

_________

Answer 13

The cross-ratio is preserved as an invariant in a projection. It is a quantity that is associated with four points on a line. If the four points have coordinates p, q, r, s, then their cross-ratio is the function of the ordered 4-tuple (p,q,r,s) written as:

(r-p) (s-q) / (r-q) (s-p)

Question 14

It follows immediately from the definition of an isometry that when f and g are isometries, so is their ____ or _____ f•g.

Answer 14

It follows immediately from the definition of an isometry that when f and g are isometries, so is their composite or product f•g.

Question 15

It is less obvious that any isometry f has an ______, __, which is also an isometry. To prove this fact we can use the result that any isometry in R² is the product of one, two, or three reflections, and thus:

fr₃r₂r₁ = r₁r₂r₃r₃r₂r₁

= r₁r₂r₂r₁

= r₁r₁

= identity function

Answer 15

It is less obvious that any isometry f has an inverse, f^-1, which is also an isometry. To prove this fact we can use the result that any isometry in R² is the product of one, two, or three reflections, and thus:

fr₃r₂r₁ = r₁r₂r₃r₃r₂r₁

= r₁r₂r₂r₁

= r₁r₁

= identity function

Question 16

The following two properties are characteristic of a group of transformations. A transformation of a set S is a function from S to S, and a collection G of transformations forms a group if it has two properties:

1)

2)

Answer 16

The following two properties are characteristic of a group of transformations. A transformation of a set S is a function from S to S, and a collection G of transformations forms a group if it has two properties:

1) If f and g are in G, then so if fg
2) If f is in G, then so is its inverse f^-1

Question 17

The meaningful concepts of a geometry correspond to properties that are left _____ by a transformation of a group. It is called an ____ of the isometry group. 3 examples of these are:

Answer 17

The meaningful concepts of a geometry correspond to properties that are left unchanged by a transformation of a group. It is called an invariant of the isometry group. 3 examples of these are:

distance

straightness of lines

circularity of circles

Question 18

Unlike in R², in R³ one now has the concept of _______, which distinguishes the ___-___ from the ___-___. We can preserve these transformations of orientation by using products of an even number of reflections of planes.

Answer 18

Unlike in R², in R³ one now has the concept of handedness, which distinguishes the right-hand from the left-hand. We can preserve these transformations of orientation by using products of an even number of reflections of planes.

Question 19

A transformation is called linear if it preserves the following two operations:

_____ = _____

_____ = _____

Answer 19

A transformation is called linear if it preserves the following two operations:

f(u + v) = f(u) + f(v)

f(au) = af(u)

Question 20

A linear transformation (x,y) –> (ax+by, cx+dy) is usually represented by the matrix:

Answer 20

A linear transformation (x,y) –> (ax+by, cx+dy) is usually represented by the matrix:

(a b,

c d)

Question 21

To find where (x,y) in R² is sent by f, one writes it as a column vector (x, y) and multiplies this column on the left by M according the matrix product rule:

Answer 21

To find where (x,y) in R² is sent by f, one writes it as a column vector (x, y) and multiplies this column on the left by M according the matrix product rule:

(a b, c d) (x, y) = (ax+by, cx + dy)

Question 22

The main advantage of the matrix notation is that it gives the product of two linear transformations, first (x, y) –> (a₂x + b₂y, c₂x + d₂y) and then (x, y) –> (a₁x + b₁y, c₁x + d₁y) by the matrix product rule:

Answer 22

The main advantage of the matrix notation is that it gives the product of two linear transformations, first (x, y) –> (a₂x + b₂y, c₂x + d₂y) and then (x, y) –> (a₁x + b₁y, c₁x + d₁y) by the matrix product rule:

(a₁ b₁, c₁ d₁) (a₂ b₂, c₂ d₂) = (a₁a₂ + b₁c₂ a₁b₂ + b₁d₂,

c₁a₂ + d₁c₂, c₁b₂ + c₁b₂ + d₁d₂)

Question 23

Matrix notation also exposes the role of the determinant, det(M) which must be __-____ for the linear transformation to have an inverse.

If M = ( a b, c d ),

then det(M) = ____

Answer 23

Matrix notation also exposes the role of the determinant, det(M) which must be non-zero for the linear transformation to have an inverse.

If M = ( a b, c d ),

then det(M) = ad - bc

Question 24

If the det(M) != 0,

then M^-1 = _________

Answer 24

If the det(M) != 0,

then M^-1​ = ( 1 / det(M) ) * ( d -b, -c a )

Question 25

Linear transformations preserve geometrically natural properties such as straightness and parallelism, but they also preserve origin, which is really not geometrically different from any other point. To abolish the special position of the origin, we allow linear transformations to be composed with translations, thus obtaining what are called _____ transformations. They preserve everything about linear transformations except for \_\_\_\_\_.

Answer 25

Linear transformations preserve geometrically natural properties such as straightness and parallelism, but they also preserve origin, which is really not geometrically different from any other point. To abolish the special position of the origin, we allow linear transformations to be composed with translations, thus obtaining what are called **affine** transformations. They preserve everything about linear transformations except for ***position***.

Question 26

The unit sphere in R³ consists of all points at unit distance from O, that is all points (x, y, z) satisfying the equation _______ = \_\_ This surface is also called a \_-\_\_\_\_ or __ because its points can be described by two coordinates, ______ and \_\_\_\_\_\_. It is essentially *two-dimensional.*

Answer 26

The unit sphere in R³ consists of all points at unit distance from O, that is all points (x, y, z) satisfying the equation **x² + y² + z² = 1** This surface is also called a **2-sphere** or **S²** because its points can be described by two coordinates, **latitude** and **longitude**. It is essentially *two-dimensional.*

Question 27

There are products of reflections in three planes that are different from products of reflections in one or two planes. One such isometry is called the _____ \_\_\_ which sends each point (x, y, z) to its _____ \_\_\_\_\_ (-x, -y, -z).

Answer 27

There are products of reflections in three planes that are different from products of reflections in one or two planes. One such isometry is called the **antipodal map** which sends each point (x, y, z) to its **antipodal point** (-x, -y, -z).

Question 28

The most elegant and practical way to represent rotations of R³ or S² is with the help of \_\_\_\_\_\_\_. These are a 2x2 matrix of the form q = _______ where a, b, c, d in R² and i² = -1

Answer 28

The most elegant and practical way to represent rotations of R³ or S² is with the help of **quaternions**. These are a 2x2 matrix of the form **q = (a+bi c+di, -c+di a-bi)** where a, b, c, d in R² and i² = -1

Question 29

The determinant of the quaternion q is given by \_\_\_\_\_\_\_\_\_\_

Answer 29

The determinant of the quaternion q is given by ## Footnote **det q = a² + b² + c² + d² = |q|²**

Question 30

A group of rotations is itself a _____ \_\_\_\_\_.

Answer 30

A group of rotations is itself a **geometric object.**

Question 31

We define an *abstract group* to be a set G, which contains a special element 1 and for each g an element g^-1, with a product operation satisfying the following three axioms:

Answer 31

We define an *abstract group* to be a set G, which contains a special element 1 and for each g an element g^-1, with a product operation satisfying the following three axioms: g₁(g₂g₃) = (g₁g₂)g₃ associativity g1 = g identity gg^-1 = 1 inverse

Question 32

When we say that the elements of a certain group G *correspond to* or *behave like* or *can be viewed* as elements of another group G^|, we have in mind a precise relationship called an ______ of G onto G^|. This also means that there is a \_\_\_-\_\_-\_\_\_ _______ between G and G^|, and that this preserves \_\_\_\_\_.

Answer 32

When we say that the elements of a certain group G *correspond to* or *behave like* or *can be viewed* as elements of another group G^|, we have in mind a precise relationship called an **isomorphism** of G onto G^|. This also means that there is a **one-to-one correspondence** between G and G^|​, and that this preserves **products**.

Question 33

Hamilton thought that R³ could be viewed as a number system by some clever choice of multiplication rule. He took a "number system" to be what we now call a ____ together with an absolute value |u| = sqrt( u₁² + u₂²​ + u₃²​ )

Answer 33

Hamilton thought that R³ could be viewed as a number system by some clever choice of multiplication rule. He took a "number system" to be what we now call a **field** together with an absolute value |u| = sqrt( u₁² + u₂²​ + u₃²​ )

Question 34

The simplest surface of constant negative curvature is called the ________ or the \_\_\_\_\_\_.

Answer 34

The simplest surface of constant negative curvature is called the **pseudosphere** or the **tractoid**.

Question 35

Beltrami's first discovery in 1865, established the special role of constant curvature in geometry: The surfaces of constant curvature are precisely those surfaces that can be mapped to the plane in such a way that geodesics map to _____ \_\_\_\_\_.

Answer 35

Beltrami's first discovery in 1865, established the special role of constant curvature in geometry: The surfaces of constant curvature are precisely those surfaces that can be mapped to the plane in such a way that geodesics map to **straight lines**.