College Geometry MATH 355
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This 4 page Class Notes was uploaded by Jeromy Torphy on Monday October 5, 2015. The Class Notes belongs to MATH 355 at California State University - Long Beach taught by Staff in Fall. Since its upload, it has received 10 views. For similar materials see /class/218789/math-355-california-state-university-long-beach in Applied Math And Statistics at California State University - Long Beach.
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Date Created: 10/05/15
Sample project topics Math 355 Fall 2007 Bear in mind The quality of a project is directly related to its having substantive content and a wellde ned7 narrow focus The following are intended to be suggestive Many topics need a narrowing of focus Optimal paths and shapes Explore shortest path problems in 4 point networks in the plane Is the solution for the 3 point network of use here Explore shortest path problems in 4 point networks in three dimensional space Is the solution for a 2D network of use here Reasoning by analogy7 what are the various ways of connecting the points What can you say about the optimum network lengths How far can analogical reasoning take you An ellipse is the set of points whose network distance to two xed points is constant Investigate sets of points in the plane whose network distance to three given points is constant What about the analogous situation in three dimensions Investigate the geometry of soap bubbles or soap lmsia question related to the isoperimetric problem in 3D Given an equilateral triangle and an interior point P7 the sum of the distances from P to the edges is equal to the triangles altitude Establish this claim and show how to use it to solve the threepoint shortest network problem Symmetry polyhedra and tiling Extend the idea of a frieze pattern which is onedimensional to a tiling or wallpaper pattern two dimensional Exploiting considerations of symmetry and angle measure at special points in the tiling7 show how many such patterns there are in the plane An alternative approach is to study the tilings of a torus Use the fact that the angles of a spherical triangle sum to be greater than 7139 to describe all groups of rotational symmetries of the sphere that have a nite number of elements Each such group corresponds to a tiling of the sphereithat is7 a polyhedron Describe these tilings Investigate the soccer ball and its relationship to the icosahedron What polyhedron is anal ogous to the soccer ball but is based on the octahedron Consider the family of truncated polyhedra to which the soccer ball belongs Study the connection of the soccer ball to the newly found form of carbon known as fullerene The rhomblc dodecahedron is based on the octahedron and is like a regular polyhedron Study the construction of this object and describe analogues based on the tetrahedron and icosahedron Study and build a polyhedral kaleidoscope a pyramid of mirrors that s open on each ends whose re ective properties produce the image of a polyhedron Try this for the cube7 octa hedron7 dodecahedron7 and icosahedron Explain the geometry A semi regular polyhedron has faces that are regular polygonsibut not all the same shape as in the regular caseiand that t together in such a way that any vertex is indistinguishable from any other vertex Study and enumerate all such polyhedra o The complete graph on ue uertices is the network of ten edges that connect ve vertices in every way a vertex is not connected to itself Use considerations of polyhedral networks on the sphereior something elseito show that this graph cannot lie in the plane without the edges crossing Study the question of whether a tiling of the plane can consists of polygons each of which has vefold symmetry This is different from the impossible task of tiling the plane with regular pentagons Source Danzer Griinbaum Shephard Can all tiles of a tiling haue ue fold symmetry The American Mathematical Monthly 89 No 8 Oct 1982 pp 568 570583 585 Transformations o Specify reflection in the unit circle as a map 957 y 1 00 2079007 24 How would you de ne reflection in the unit sphere in threedimensional space What would you see when you look into a spherical mirror77 of this type How far behind the mirror do you appear to be Compare this to a at mirror and to one made of a re ective coating on a sphere like the underside of a spoon 0 Describe stereographic projection ofa circle a 1 dimensional sphere or 1 sphere 2 sphere 3 sphere etc Show that stereographic projection of the the 2 sphere sends circles on the sphere to circleslines in the plane Where does the center of the spherical circle go From where does the center of the circle in the plane come Begin by examining the simple case of stereographic projection of a 1 sphere circle onto a 1 plane line This maps 0 spheres on the circle to 0 spheres on the line What s a 0 sphere Answer the analogous questions about the respective centers of circles Brie y discuss stereographic projection in higher dimensions Source Hilbert and Cohn Vossen Geometry and the Imagination Threedimensional geometry 0 Explore the three dimensional analogue of Pascal7s triangleipascal s pyramid How are the entries generated What special properties do they exhibit Are there connections to geometric structures 0 Classify isometries in three dimensional space How many xed points can a 3D isometry have A xed point argument analogous to the 2D case is an illuminating way to go Con jecture how the classi cation of isometries goes in dimension higher than 3 0 Four mutually tangent spheres create an intriguing con guration Study some of the proper ties that that this structure possesses Source Eppstein Tangent spheres and triangle centers The American Mathematical Monthly 108 No 1 Jan 2001 pp 63 66 0 Give an analytical description of the geometric conditions that produce a rainbow 0 Examine the question of how the shape of an object affects the way it oats In particular how can a symmetrical object oat in an asymmetrical way Source Gilbert How things float The American Mathematical Monthly 98 No 3 Mar 1991 pp 201216 Fourdimensional geometry 0 Build models of the 4 dimensional versions of the tetrahedron and cube In what sense are these hyperpolyhedra regular What don t we see in a 3 dimensional model 0 Investigate the geometry of the a threedimensional sphere How do you describe it in coor dinates How is it built up out of 3D slices How much 4D volume does it enclose What s meant by volume here 0 Explore higher dimensional analogues to the general space slicing problem What acts as a knife What slicing principles should we follow How might we approach the problem Try to solve it Curves surfaces knots and the like 0 Study a knot polynomialithat of Jones or Alexander say Describe the construction of the polynomial and its invariance under the elementary knot Reidemeister moves Use the polynomial to show that the right and left handed trefoil knots are not equivalent Source Messer and Stra in Topology Now 0 Each point p on a curve C that encloses a convex region has a diameteerithe length of the longest chord across C passing through p A curve has constant width when D17 is constant as p varies over C lnvestigate constructions for and properties of curves with constant width Source M Gardner The Unexpected Hanging and Other Mathematical Diuersions Ch 18 0 Explore some of the properties of a Geoboard For instance show the relationship between the euler number the angles and the area of a polygon 0 Investigate the geometry of fractals What s meant by the term Discuss some of the peculiar properties that they can have Explain the notion of fractal dimension u 4 11 1439 Source Falconer Fractal geometry and rr 0 Consider the motion of a billiard ball moving on a rectangular billiard table Equivalently you can treat the path of a light ray that moves inside a two dimensional rectangular rectan gular region that has mirrors along the boundary Discuss the path called the trajectory of the ball as a function of its initial position and direction of motion Are there trajectories that repeatithat is the ball returns to its initial position while travelling with the initial direction Try looking at simple cases where the table is square or m x n mn are integers Circles and ellipses are also interesting cases Source Kinsey and Moore Symmetry Shape and Space 0 Examine the curve on baseball that s made along the stitchesiwhere the two pieces of skin come together Find a way of expressing this curveiin terms of equations or parametrically How do you go about looking for such a description Finding the baseball s symmetries might be useful 0 Can a wobbly four legged table be turned so that it eventually rests on all four legs simulta neously lnvestigate this phenomenon of turning a table on a variety of surfaces Source Bill Baritompa Rainer Lowen Burkard Polster and Marty Ross Mathematical Table Turning Revisited arXivorgabsmathHO0511490 Teaching 0 Discuss the pedagogical value ofthe transformation point ofview in geometry Contrast this way of looking at geometry with the synthetic approachibased upon proof by construction Can the use of vectors or complex numbers provide help to students here At what cost 0 Devise an activity in which students visualize in the broadest sense some geometric property or result To what extent does the activity develop a general way of looking at things 0 One way to approach geometry is through symmetry How can considerations of symmetry be applied at various levels of math teaching from elementary to high school Develop a symmetry related activity or lesson for some grade level What concepts does it involve Where might the students struggle What s the mathematical content ls more than one area of math involved
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