Review Examination II (Spring 2006)
Geometric Structures, Mathematics 244 (Part I)
Department of Mathematics and Computing
York College (CUNY)
Jamaica, New York 11451
1. a. From the point of view of analytical geometry, what are the points of the Euclidean plane?
b. From the point of view of analytical geometry, what are the lines of the Euclidean plane?
c. From the point of view of analytical geometry, what are the points of the taxicab plane?
d. From the point of view of analytical geometry, what are the lines of the taxicab plane?
e. Find the lengths of the sides of the Euclidean triangle:
A = (0, 0), B = (4, 0), C = (2, 10)
a. Find the equations of the sides of this triangle.
b. Find the equations of the perpendicular bisectors of the sides of this triangle.
c. In the Euclidean plane, what is the center of a circumcircle for this triangle?
(Hint: For the point where the perpendicular bisectors of two of the sides of the triangle meet. Why is this the center of the circumcircle?)
c. Find the taxicab points which are equidistant from A and B
d. Find the taxicab points which are equidistant from B and C
e. Find the taxicab points which are equidistant from A and C
f. What point (points) do the sets meeting the conditions in c., d., and e. above intersect at?
g. Draw a Euclidean circle of radius 5 about C and write down its equation.
h. Draw a taxicab circle of radius 4 about C.
i. What point(s) are vertices of an equilateral triangle with AC as one its sides in the taxicab plane?
(Hint: What is the length of AC? The other two sides of the equilateral triangle must have this same length.)
j. Write down the equation of a line through A parallel to BC.
j. Does the taxicab plane obey the modern version of Euclid's 5th postulate?
k. Draw a circle of radius 3 about the point (2, -1) in the max metric (distance function). The max metric takes as the distance between the points (a, b) and (c, d) the maximum of the numbers |a-c| and |b-d|.
2. Draw a picture of the Klein model for the hyperbolic plane. What are the points of this model? What are the lines of this model? What "parallel" axiom does this plane obey? Label the ultra-parallels for the diagram that you draw.
3. a. Find the Hamming distance between pairs (this requires 6 calculations) of binary sequences below:
A = 00000000
B = 00011111
C = 11111000
D = 11100011
b. Can the four strings above be used as an error-correction code? If so, how many errors will the code be able to correct per code word?
d. If string E = 00111011 is added to the list above, can the 5 strings be used as an error correction code? If so how many errors will the code be able to correct per code word?
e. If the string 00011001 is received, and one applies the principle of maximal likelihood decoding, what codeword would this string be decoded to?
4. (The usual convention of putting "bars" over the numbers in this finite arithmetic is not used here.) Since x2 + 2x + 4 = 0 has no root in GF (5), we can construct a finite arithmetic with 25 elements where the numbers have the form a + bΔ where Δ2 = -2Δ - 4 = 3Δ + 1. Remember that in GF (5) "-3" = "2" and "-2" = "3" and "-4" = "1."
In this field find:
a. The sum of 2 + 4Δ and 3 + 2Δ and the product of these two numbers. (Hint: Treat the terms like polynomials but use the fact that the coefficients are in GF(5) and also that Δ2 must be replaced by 4Δ + 1.)
b. Repeat for 1 + 4Δ and 4Δ
c. Repeat for 4 + 4Δ and 3 + 3Δ
d. In a finite affine plane with coordinates from GF(5), list all the points which lie on the lines:
a. x = 4
b. x + 2y = 3.
(Hint: There will be exactly 5 points on each of the lines.)
c. In the finite projective plane associated with the finite affine plane mentioned above, what are the lines associated with the lines x =4 and x + y = 3? What are the points of the projective plane on these associated lines? (Hint: The lines of this projective plane have 6 points on every line.)
d. Find where the lines x + y + 2z = 0 and x + 2y +4z = 0 in the projective plane above meet. (Hint: Try doing this using an appropriate determinant but remember that the entries are not real numbers but members of GF(5).)
e. Find the equation of the line through the points (1,2, 1) and (1,4,4)
f. What is the value of -4 in GF(5)?
g. What is the value of x in GF(5) so that 3x = 1?
h. What is the value of x in GF(5) so that 2x = 3?