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Thinking Mathematically

Thinking Mathematically. Beyond Euclidean Geometry. The Geometry of Graphs. A vertex is a point. An arc is a line segment or curve that starts and ends at a vertex.

cameron-gaines
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Thinking Mathematically

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  1. Thinking Mathematically Beyond Euclidean Geometry

  2. The Geometry of Graphs A vertex is a point. An arc is a line segment or curve that starts and ends at a vertex. Vertices and arcs form a graph. A vertex with an odd number of attached arcs is an odd vertex. A vertex with an even number of attached arcs is an even vertex. A graph is traversable if it can be traced without removing the pencil from the paper and without tracing an arc more than once.

  3. Rules of Traversability • A graph with all even vertices is traversable. One can start at any vertex and end where one began. • A graph with two odd vertices is traversable. One must start at either of the odd vertices and finish at the other. • A graph with more than two odd vertices is not traversable.

  4. Example To Traverse or Not to Traverse Consider the graph in the figure. • Is this graph traversable? • If it is, describe a path that will traverse it.

  5. Solution • Begin by determining if each vertex is even or odd. • Because this graph has two odd vertices, by Euler’s second rule, it is traversable. • Using Euler’s second rule, we start at one of the odd vertices and finish at the other. You might go across the arc and then travel around the triangle.

  6. Example Traversing the Rooms of a House The floor plan of a four-room house is shown. The rooms are labeled A,B,C, and D. The outside of the house is labeled E. The openings represent doors. • Draw a graph that corresponds to the floor plan. • Is the graph traversable?

  7. Solution a. A graph has been created with A,B,C,D,and E as vertices. Each door is an arc so two doors connect the outside to room C while only one door connects the outside to each of the other rooms. Then there is one door between A and B, A and C, C and D, and B and D.

  8. Solution cont. b. There are only two odd vertices, B and D. This means that the graph is traversable.

  9. Topology • A branch of modern geometry called topology looks at shapes in a completely new way. • In topology, objects are classified according to the number of holes in them, called their genus. • Objects with the same genus are topologically equivalent.

  10. Non-Euclidean Geometries • Hyperbolic geometry is based on the assumption that given a point not on a line, there are an infinite number of lines that can be drawn through the point parallel to the given line. • Elliptic geometry is based on the assumption that there are no parallel lines. Geometry is on a sphere, and the sum of the measures of the angles of a triangle is greater than 180º. • Through the use of computers, a new geometry of natural shapes, called fractal geometry, has arrived. The process of repeating a rule again and again to create a self-similar fractal is called iteration.

  11. Thinking Mathematically Beyond Euclidean Geometry

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