Mathematics 138
Format of the final examination
For both Math 138 and Math E-138 the exam is at 2:15 PM on Saturday, May 21 in Boylston 105.
There will be a beverage-and-snack party right after the exam, in Quincy 102 at 5:45.

Part I - 15 multiple-choice items, worth 2 points if answered correctly on the first try, 1 point if on the second try.  These will cover whatever gets missed by the rest of the exam.

Part II - 5 construction problems, worth 3 points each.  These can be answered by drawing lines on a diagram.  Bring a straightedge.  A protractor might be useful if you want to replicate an angle but is not required.  Circles and parabolas will be provided.
Obvious topics:
Arithmetic in affine geometry.
Combining products of reflections in any of several geometries.
Angle bisection and related topics in Galilean and Minkowski geometry.
Adding doublets and related operations.
Poles, polar lines, and perpendicular lines in hyperbolic geometry
Midpoints, midlines, and perpendicular bisectors in hyperbolic geometry.
Constructing an orthonormal basis in hyperbolic geometry.
Reflections in hyperbolic geometry
Diagram for a dual theorem in Galilean or projective geometry.
Diagrams to show that if a singly asymptotic triangle of hyperbolic geometry has finite area, so does a doubly or triply asymptotic triangle.

Part III - 5 "false questions" worth 2 points each.  From a list of statements, select 5 that are false and invent a counterexample for each.

Part IV - 5 "analytical questions" worth 4 points each. 
Obvious topics:
Coordinates in finite affine and projective geometry.
Matrix for a reflection in any of several geometries
Matrix for a shear or cyclic rotation in Galilean geometry.
Matrix for a Lorentz transformation in Minkowski geometry.
Matrix for a horolation or translation in hyperbolic geometry.
Matrix for a rotation in any of several geometries.
Midpoint or perpendicular bisector in Minkowski, spherical, or hyperbolic geometry.
Use of Law of Sines and/or Law of Cosines in any of several geometries.
Right triangles in spherical geometry.
Right triangles and angle of parallelism in hyperbolic geometry.
Calculating poles and intersections in spherical, projective and hyperbolic geometry.
Implementation of "fundamental theorem" of affine/projective geometry

Part V - 5 short proofs worth 5 points each, perhaps ones that you have seen, perhaps not.
Obvious topics:
Arithmetic operations in affine geometry (e.g. commutative laws)
Fixed points and lines of isometries in various geometries
Galilean geometry - duals, angle bisectors, etc.
Theorems with hyperbolas (“circles”) in Minkowski geometry
3-reflection theorems and representation theorems in various geometries.
Desargues and Pappus - projective and affine versions
Fixed points and lines of projective transformations in relation to eigenvectors.
Orthonormal bases and parametric representation of lines in hyperbolic geometry
Connection between cross products and formulas of spherical/hyperbolic trigonometry.
Areas in spherical and hyperbolic geometry.
Congruence conditions in various geometries.

For each of up to 10 proposed placement test questions that you email to bamberg@tiac.net by noon on the day of the final exam, you will receive 1 point, and the percentage on the final exam will be applied what remains of 100 points.  So if you submit 10 questions and get 80% on the final, your score will be 10 + 90*0.8 or 82.