Provide a drawing. Mark any line with three points in blue color.

Transcription

1 Math 3181 Name: Dr. Franz Rothe August 18, 2014 All3181\3181_fall14h1.tex Homework has to be turned in this handout. For extra space, use the back pages, or blank pages between. The homework can be done in groups up to three students due August 27/28 1 Homework 0 Problem 1.1 (The four-point incidence geometries). Describe all possible non-isomorphic incidence geometries on a set of four points. For each of them Provide a drawing. Mark any line with three points in blue color. Tell how many lines exist. Tell which parallel property (elliptic, Euclidean, hyperbolic, or neither) does hold. 1

2 Problem 1.2 (The five-point incidence geometries). Find all (four) non-isomorphic incidence geometries with five points. For each of them Provide a drawing. Tell how many lines exist. Tell how many lines have three or more points, and describe the properties of these lines. Tell which parallel property (elliptic, Euclidean, hyperbolic, or neither) does hold. 2

3 Problem 1.3. As far as two-dimensional geometry is concerned, Hilbert s Proposition 1 reduces to one simple statement: any two different lines either intersect in one point, or are parallel. This statement can be rephrased in many formulations. Encircle the statements below which are such equivalent reformulations; and scratch through the statements which are not. Any two different lines which are not parallel, have a unique point of intersection. There exists at most one parallel to a given line through a given point. On any given line lie two or more points. If two lines have two or more points in common, they are equal. Two lines can but need not have some points in common. Two different lines have at most one point in common. Two different points determine a line going through them uniquely. 3

4 Problem 1.4. How many lines does the hand-shake incidence geometry with an arbitrary number n of points have. How many lines does the straight fan with n points have. Problem 1.5. Which parallel property holds for the hand-shake model with 4 points. Which parallel property holds for the hand-shake model with n 5 points. Which parallel property holds for a straight fan with any numbers n 3 of points.. Problem 1.6. From the axioms of incidence (I.1) (I.2) (I.3), one shows easily that there exist two different lines through every point. Complete the following proof for this statement. Proof. By axiom (I.3b), there exist at least? that do? a line. We call them A, B and C. Let any point P be given. We now distinguish two cases: 1. In the case that point P is? from all three points A, B, C, we draw the three lines?. At least? are different since A, B, C do not lie on a line. Thus we are ready. 2. In the case that point P is one of?, we draw the three lines AB, BC and CA. These are three different lines, and two of them go through the given point P. In? P. we have obtained two different lines through the arbitrary point 4

5 Definition 1 (Isomorphism of incidence planes). Two incidence planes are called isomorphic if and only if there exists a bijection between the points of the two planes, and a bijection between the lines of the two planes such that incidence is preserved. Figure 1: Two isomorphic six-point incidence geometries Problem 1.7. Given two incidence geometries, it is not obvious whether they are isomorphic. By corresponding labelling of the points in both geometries, show an isomorphism between the two six-point incidence geometries in the figure on page 5. Color each pair of corresponding sides with the same color, using altogether seven colors. 5

6 Definition 2 (Affine plane). An affine plane is a set of points, and a set of lines, satisfying the axioms: A.1 Every two different points lie on exactly one line. A.2 If point P does not lie on a line l, there exists exactly one line m through the point P that does not intersect l. A.3 There exist three points that do not lie on a line. Axiom (A.2) is just the Euclidean parallel property. Because of the result of the problem below, an affine plane is just an incidence plane with the Euclidean parallel property. Problem 1.8. Convince yourself that in an affine plane, the relation that two lines are either equal or parallel is an equivalence relation. 6

7 Because the construction of the affine plane from the real numbers uses only addition, subtraction, multiplication and division, replacing the real numbers by any finite or infinite field F and doing the same construction once more, leads to an affine plane, too. Thus one gets the following general definition. Definition 3 (The Cartesian plane over a field F). The "points" of the Cartesian plane are ordered pairs (x, y) of elements x, y F. The "lines" of the Cartesian plane of the field F are equations ax + by + c = 0 with coefficients a, b, c from the field, of which a and b are not both zero. A "point lies on a line" if and only if the coordinate pair (x, y) satisfies the equation of the line. One defines the slope of a line in the usual way. Lines are parallel if and only if they have the same slope. To find the parallel to a given line through a given point, one uses the point slope equation of a straight line. This procedure shows that the Euclidean parallel property holds. It is left to the reader to work out all details. Theorem 1 (The Cartesian plane over an arbitrary field). In a Cartesian plane over any field, there exist a unique line between any two points. There exists a unique parallel to a line through a given point. Hilbert s axioms (I.1)(I.2)(I.3a)(I.3b) and the Euclidean parallel postulate (IV*) hold. Hence a Cartesian plane over a field is an affine plane. Problem 1.9. Use your knowledge of College algebra, and give in a few words an explanation why Theorem 1 is true. 7

8 Problem 1.. An illustrative affine incidence geometry is the Cartesian plane Z 3 Z 3. In other word, we use analytic geometry and arithmetic modulo 3 for the coordinates. The points are the ordered pairs (a, b) with a, b Z 3. As in analytic geometry, lines are given by linear equations ax + by + c = 0 with a, b, c Z 3 and a, b are not both zero. One defines the slope of a line in the usual way. Lines are parallel if and only if they have the same slope. To find the parallel to a given line through a given point, one uses the point slope equation of a straight line. This procedure shows that the Euclidean parallel property holds. To better understand this model, provide a drawing. How many lines does the model have? Use clearly different colors for lines with different slopes, but give each set of three parallel lines different shades of nearby the same color. 8