Math 314: Topics in Geometry

Pythagoras	Euclid	Jean-Victor Poncelet	Janos Bolyai	Felix Klein	David Hilbert

Spring Term, 2005

Instructor:  David Murphy

Each class (starting the second week), two students will present their solutions to problems from the book on a rotating schedule (so you will present a solution approximately once every two weeks). The order will be determined by drawing numbers the first week of class. Ten percent (10%) of your final grade will be based on these presentations together with your in-class participation.

In addition to presenting your proof to the class, you will need to turn-in a written version of your proof to me at the end of that class period after each of your presentations.

Notice:  I have been informed that one of the students who had registered for this course has decided to drop it, so there will be just 10 of us. This student had been assigned the number 2, so I have eliminated it from the list below (as you will likely observe) making the order of presenters

Problem 1:  [1] Prove that the number 2^1/3 is irrational.

Problem 2:  [3] Given two points, there is exactly one line containing the two points. Given three points, there are at most three lines containing exactly two of these three points. There are at most six lines each containing exactly two of a given set of four points. What is the greatest number of lines each containing exactly two of a set of five points? Is there a pattern? Can you predict the maximum number of lines corresponding to a set of n points? Prove your answer.

A group G is a set with a binary operation, denoted *, satisfying the following axioms:

1. Whenever g₁ and g₂ are elements of G, g_{1 *} g₂ is an element of G.
2. If g₁, g₂ and g₃ are elements of G, then ( g_{1 *} g₂ ) _* g₃ = g_{1 *} ( g_{2 *} g₃ ).
3. There is an element e in G such that, for all g in G, e _* g = g = g _* e.
4. For each element g in G, there is an element g' in G such that g _* g' = e = g' _* g.

Problem 1:  [4] Prove that the element e of a group G as described by Axiom 3 is unique. It is commonly called the identity element of the group.

Problem 2:  [5] Let G be a group and fix an element g in G. Prove that the element g' associated to g as in Axiom 4 is unique. The element g' is called the inverse of g and is commonly denoted g^-1.

Problem 1:  [6] Prove that the set of integers, Z, with the binary operation of addition, +, is a group. Is the set of integers with the binary operation of multiplication a group?

Problem 2:  [7] Prove that the set of all 2 x 2 matrices with non-zero determinant is a group with respect to matrix multiplication. (This is called the general linear group of 2 x 2 matrices, and denoted GL(2,R).) Is the set of 2 x 2 matrices with non-zero determinant a group with the binary operation of addition?

An operation of a group G on a set S is a function a : G x S ---> S satisfying the following axioms:

1. For all elements g, h in G and s in S, we have a( g, a( h, s ) ) = a( g _* h, s ).
2. If e is the identity element of G, then a( e, s ) = s for all elements s of S.

Problem 1:  [8] Let V = Rⁿ be the vector space of n-tuples of real numbers and let G be the group of invertible n x n matrices with real entries. Show that a( A, x ) = Ax for A in G and x in V is an operation of G on V. (The group G is called the general linear group of n x n matrices, and denoted GL(n,R).)

Problem 2:  [9] Let V = Rⁿ be the vector space of n-tuples of real numbers and let G = GL(n,R), the group of invertible n x n matrices with real entries. Fix a vector v in V and let

Let ( G, _* ) be a group with identity element e, and let H be a subset of G. Then ( H, _* ) is a subgroup of ( G, _* ) if the following axioms hold:

1. For all h₁ and h₂ in H, h_{1 *} h₂ belongs to H.
2. The identity element e belongs to H.
3. For each h in H, the inverse element h^-1 belongs to H.

Problem 1:  [10] Show that the set of n x n matrices A such that A^T A = I (where A^T denotes the transpose of the matrix A and I denotes the identity matrix) is a subgroup of GL(n,R). (It is called the orthogonal group, and denoted O(n,R).)

Problem 2:  [11] Let V = Rⁿ be the vector space of n-tuples of real numbers. Let < -,- > be the usual inner product of vectors in V,

Remark:  Perhaps to your surprise, the subgroups of GL(n,R) described in Problems 1 and 2 are the same, O(n,R). The description of O(n,R) in Problem 2 illustrates the importance of the matrices in this group, for they preserve the inner product in n-dimensional Euclidean space, Rⁿ. Hence, as we can compute both the length of a vector and the angle between vectors in terms of this dot product, elements of the orthogonal group must preserve distances and angles as well. That is, if x and y are points in Rⁿ, then the distance from x to y is the same as the distance from Ax to Ay and the angle between the vector from the origin to x and the vector from the origin to y is the same as the angle between the vector from the origin to Ax and the vector from the origin to Ay for every A in O(n,R).

This means transformations A in O(n,R) preserve congruence of figures. So, if P, Q and R are three non-colinear points in R² and if P' = A(P), Q' = A(Q) and R' = A(R), then triangles PQR and P'Q'R' are congruent. This can make proving that two triangles are congruent much easier. We only have to find an orthogonal matrix A sending vertices of the first to corresponding vertices of the second, and we're done. Of course, this isn't quite true as stated, for we also need to allow translations, which are not linear transformations of R² (think about it...), but up to translations, the entire notion of Euclidean congruence that we studied in high school geometry is encoded in the orthogonal group. Are you beginning to see why we're asking these questions about group theory in a geometry class? If not yet, you will soon.

A function f : R² ---> R² is called an isometry of R² if, for any two points a and b in R², the distance from f(a) to f(b) is equal to the distance from a to b.

Problem 1:  [1] Show that the set of all the isometries which fix the origin and map the set { ( 1, 0 ), ( 0, 1 ), ( -1, 0 ), ( 0, -1 ) } onto itself is a group. How many elements are in this group? This group is called the dihedral group D₄.

Problem 2:  [3] Let P and Q be distinct points of the plane R². Suppose f is an isometry of the plane which stabilizes both P and Q (that is, f(P) = P and f(Q) = Q). Let X be any point of the plane distinct from both P and Q. Sketch the circle with center P passing through X and the circle with center Q passing through X. Must these circles be mapped onto themselves by f? Where are the possible images of X in the plane? [Hint: Treat the cases where P, Q, X are collinear and where P, Q, X are not collinear separately.] Describe the transformation f geometrically.

An affine transformation of the plane is a function t : R² ---> R² of the form t(x) = Ax + b, where A is an invertible 2 x 2 matrix and b is a vector in R².

Problem 1:  [4] Find necessary and sufficient conditions that under the affine transformation t(x) = Ax + b, every line L of R² corresponds to a line parallel to itself.

Problem 2:  [5] Find necessary and sufficient conditions that under the affine transformation t(x) = Ax + b, there exists a point P such that t maps every line on P to itself. (We say that a line m is on a point Q is Q is a point on m.)

Let A( a₁ , a₂ ), B( b₁ , b₂ ), and C( c₁ , c₂ ) be three points in the plane, R². The points A, B and C are collinear if and only if the determinant

Problem 1:  [6] Given triangle ABC and the midpoint D of the segment BC, prove that m(ABD) = m(ADC).

Problem 2:  [7] Given triangle ABC and a point D such that CD is parallel to AB, prove that m(ABC) = m(ABD).

The set of affine transformations that preserve the measures of triangles is called the set of equiaffine transformations.

Problem 1:  [8] Prove that any equiaffine transformation may be expressed in the form t(x) = Ax + b where det A = 1.

Problem 2:  [9] Prove that the set of equiaffine transformations forms a group.

An affine plane is an ordered pair ( P, L ), where P is a nonempty set of elements called points and L is a nonempty collection of subsets of P called lines which have the following properties:

1. If P and Q are distinct points, there is a unique line l such that P and Q both belong to l. [This line is denoted l( P, Q ).]
2. If P is a point not contained in the line l, there is a unique line m such that P belongs to m and the intersection of m and l is empty. (In this case, l is said to be parallel to m, written l || m.)
3. There are at least two points on each line; there are at least two lines.

Problem 1:  [10] Verify that the axioms for an affine plane hold in the following example. The points are the number-triples ( 1, 0, 0), ( 0, 1, 0 ), ( 0, 0, 1 ) and ( 1, 1, 1 ). The lines are the point-sets satisfying the equations x = 0, y = 0, z = 0, x = y, x = z, y = z. (Note: this is the smallest affine plane.)

Problem 2:  [11] Let P be the points in the Euclidean plane with one point removed, and let L be the sets of points satisfying linear equations. Show, by giving an example, that the parallel axiom (Axiom 2 above) fails in ( P, L ).

A projective plane is an ordered pair ( P', L' ), where P' is a nonempty set of elements called points, and L' is a nonempty collection of subsets of P' called lines satisfying the following axioms:

1. If P and Q are distinct points, there is a unique line l ' such that P and Q belong to l '. (The line ``through'' P and Q will be written l '( P, Q ).)
2. If l ' and m ' are distinct lines in L', then their intersection is not empty.
3. There are at least three points on each line; there are at least two lines.

Problem 1:  [1] If ( P', L' ) is any projective plane, then L' contains at least three lines.

Problem 2:  [3] If ( P', L' ) is any projective plane, then each point in P' is contained in at least three lines.

Problem 1:  [4] Prove directly from the axioms for a projective plane that there is a bijection between any two lines in a projective plane, regarding each line as a set of points.

Problem 2:  [5] Assuming Axioms 1 and 3 for projective planes, prove that Axiom 2 is equivalent to the following denial of the parallel axiom:

Problem 1:  [6] Suppose that ( P, L ) is an affine plane. For each pencil F of parallel lines, define a symbol X_F. The symbols X_F are called points at infinity. Let P' be the union of the set of points of the affine plane, P, with the set of points at infinity, { X_F : F is a pencil of parallel lines in L }. For each line l in L, define l ' to be the union of l with the point at infinity X_F, where F is the unique pencil of parallel lines containing l. Let l_# be the set of all of the points at infinity, l_# = { X_F : F is a pencil of parallel lines in L }. Finally, define L' to be the union of the sets { l ' : l is a line in L } and { l_# }. Prove that the pair ( P', L' ) is a projective plane.

Problem 2:  [7] Given a projective plane ( P', L' ) and a line m' in L', define P = { P in P' : P is not in m' } = P' \ { m' }; for each line l' not equal to m', define l = { P in l' : P is not a point of m' }. Let L = { l : l' is a line of L', l' is not equal to m' }. Prove that the pair ( P, L ) is an affine plane.

Problem 1:  [8] Prove that in every projective plane four points can be found, no three of which are collinear.

Problem 2:  [9] Prove that in every projective plane four lines can be found, no three of which are concurrent.

Each projective plane ( P', L' ) has a dual, namely the abstract system whose points are the lines of the original plane and whose lines are the points of the original plane. In ( L', P' ), l' is ``contained in'' P' (or is incident with P') if and only if P is a point on l' in the plane ( P', L' ).

The projective planes ( P', L' ) and ( Q', K' ) are said to be isomorphic when there is a bijection f : P' ---> Q' such that whenever A is a point on the line l'( B, C ) in ( P', L' ), then f ( A ) is a point on the line l' ( f ( B ), f ( C ) ) in ( Q', K' ). The mapping f is called a collineation.

A projective plane ( P', L' ) is self-dual if it is isomorphic with its dual plane, ( L', P' ).

The Fano plane is the seven-point, seven-line geometry given by

Problem 1:  [10] Define a collineation f from the Fano plane to itself such that f ( A ) = B.

Problem 2:  [11] Prove that the Fano plane is self-dual by defining a collineation which maps A to { CDE }.

Problem 1:  [1] If P₁, P₂, Q₁, Q₂, Q₃ are five collinear points, prove that

Problem 2:  [3] If P₁ = [ x₁, y₁, z₁ ], P₂ = [ x₂, y₂, z₂ ], P₃ = [ x₃, y₃, z₃ ], and P₄ = [ x₄, y₄, z₄ ] are four Points on a Line a x + b y + c z = 0, prove that

If A, B, C, D are four collinear points such that their cross-ratio ( ABCD ) is -1, then ABCD is said to form a harmonic tetrad. By Theorem 3.5.2, it is clear that if ( ABCD ) = -1, then ( ABDC ) = -1. Thus, in any harmonic tetrad, the last two points play a symmetric role with respect to the first two points, and for this reason either is said to be the harmonic conjugate of the other with respect to the first two points.

Problem 1:  [4] If A and B are chosen as base points on the line l which they determine, then the harmonic conjugate of a general point C: ( a, b ) on l with respect to A and B is the point D whose parameters are ( a, -b ). (As in the book, the parameters a, b for C are the coefficients of a and b so that c = aa + bb, where A = [ a ], B = [ b ] and C = [ c ] are homogeneous coordinates for these collinear points.)

Problem 2:  [5] If A and B are distinct finite points on an embedding plane p in R³, show that the midpoint of the segment joining A and B is the harmonic conjugate of the ideal point for the line AB with respect to A and B.

Let C be an arbitrary fixed point in RP², let l be an arbitrary fixed line not passing through C, let Y be a general point, and let C' be the intersection of l and the line CY. Then the transformation which maps the general point Y into its harmonic conjugate with respect to C and C' is known as a harmonic homology. The point C and the line l are the center and axis of the homology, respectively.

Problem 1:  [6] Find the equation of the harmonic homology with center C = [ 1, 0, 1 ] and axis l : { x + z = 0 }.

Problem 2:  [7] Find the equation of the harmonic homology with center C = [ 1, -1, 0 ] and axis l : { y + z = 0 }.

A projective conic is the set of Points in RP² whose homogeneous coordinates satisfy an equation of the form

Problem 1:  [8] If G : { x^T A x = 0 } is a conic for which det( A ) is not zero, then every Point of RP² has a well-defined polar with respect to G.

Problem 2:  [9] It is possible for two distinct Points to have the same polar with respect to a conic G : { x^T A x = 0 } if and only if det( A ) = 0.

Let G : { x^T A x = 0 } be a fixed conic. The polar p of a Point P = [ p ] with respect to G is indeterminate if the expression p^T A x = 0 is identically zero; i.e., if and only if the coefficients of x, y and z in the equation of p are all zero. Any Point whose polar Line with respect to G is indeterminate is said to be a vertex of G

Problem 1:  [10] If a conic has a vertex, V, then V lies on the conic.

Problem 2:  [11] If a conic G has a vertex, V, then the polar of every Point P in RP² with respect to G passes through V.

Euclid's Postulates and Euclidean Geometry:  You can read Euclid's Elements online, which consists of 13 books. This web page has the complete translation of Euclid's work by Sir Thomas L. Heath as well as extra historical information about the text, Euclid, and the benefit of a Geometry Applet that allows you to experiment with Euclidean geometry and see the theorems in practice. For our sakes, we will be most interested in Book I of Euclid's Elements up through Proposition 30 (so all 23 definitions, 5 postulates, 5 common notions and the first 30 propositions). For your convenience, here are Euclid's Five Postulates:

1. To draw a straight line from any point to any other point.
2. To produce a finite straight line continuously in a straight line.
3. To describe a circle with any center and radius.
4. That all right angles are equal to one another.
5. That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than the two right angles.

Playfair's Axiom:  Through a given point not on a given line can be drawn exactly one line not intersecting the given line.

The Italian mathematician Gerolamo Saccheri (1667-1733) tried to prove that Euclid's Fifth Postulate was a consequence of the previous four by contradiction. He assumed the contrary statement,

The Hyperbolic Axiom:  Through a given point, not on a given line, at least two lines can be drawn that do not intersect the given line.

After much work, Saccheri concluded he had produced a contradiction. He had not, but what he had done was to prove many of the early theorems of hyperbolic geometry.

From now on, assume the Hyperbolic Axiom. Let P be a point and l a line not containing P as described in the hypothesis of the Hyperbolic Axiom. From P, we can construct a perpendicular to l at Q (by Proposition 12 of Euclid's Elements, whose proof never invokes the Fifth Postulate). Now construct a line m through P perpendicular to the line PQ (by Proposition 11 of Euclid's Elements). Let S be another point on m and construct the line QS. Then the points of QS can be partitioned into two sets:

• A = { X : X is on the line segment QS and the line PX intersects the line l }
• B = { Y : Y is on the line segment QS but the line PY does not intersect l }

Problem 1:  [1] Show that the sets A and B are nonempty and have the property that no point of either lies between two points of the other.

Problem 2:  [3] By Problem 1, the sets A and B satisfy the hypothesis of

Dedekind's Axiom of Continuity:  For every partition of points on a line (or line segment) into nonempty sets such that no point of either lies between two points of the other, there is a point T of one set with the property that it lies between every other point of that set and every point of the other set.

Determine which of the sets A or B contains the point T guaranteed by Dedekind's Axiom and prove your answer. What is the geometric significance of the line PT?

Sensed Parallels:  Let P be a given point not on a given line, l. The first line through P relative to a counterclockwise rotation from PQ (Q is the point where the perpendicular to l through P meets l) which does not intersect l is called the right-sensed parallel to l through P. The first line through P relative to a clockwise rotation from PQ which does not intersect l is the left-sensed parallel to l through P. Any other line through P that does not meet l is said to be ultraparallel to l through P.

Problem 1:  [4] If a straight line m is the right-sensed parallel to a line l through a point P, then for every point Q on m the line m is the right-sensed parallel to l through Q. (Thus we may say that m is the right-sensed parallel to l without specifying through which point.)

Problem 2:  [5] Two lines with a common perpendicular are ultraparallel.

Problem 1:  [6] If a line m is the right-sensed parallel to line l through P, Q is the point on l so that PQ is perpendicular to l at Q, and R is the point on m so that QR is perpendicular to m at R, then R is on the right side of PQ.

Problem 2:  [7] If a line m is the right-sensed parallel to line l, then l is the right-sensed parallel to m. (Thus the relation of sensed parallelism is symmetric.)

Problem 1:  [8] If m is right-sensed parallel to n, P and S are points on m (S to the right of P), and R is a point on n, then any line entering the angle RPS will intersect n at a point T on the right side of R.

Problem 2:  [9] If two lines m and n are both right-sensed parallel to a third line l, then they are right-sensed parallel to one another. (Thus the relation of sensed parallelism is also transitive.)

The figure consisting of two sensed parallel lines and a transversal intersecting the lines at A and B is referred to as an asymptotic triangle. If W is the ideal Point determined by the sensed parallels, we refer to this asymptotic triangle as triangle ABW.

Problem 1:  [10] If a line passes within asymptotic triangle ABW through one of its vertices (including W), it will intersect the opposite side.

Problem 2:  [11] If a straight line intersects one of the sides of an asymptotic triangle ABW but does not pass through a vertex (including W), it will intersect exactly one of the other two sides.

Problems with their solutions