For each graph in 16, a. Is the graph traversable b. If it

For each graph in Exercises 1–6, a. Is the graph traversable? b. If it is, describe a path that will traverse it.

For each graph in Exercises 1–6, a. Is the graph traversable? b. If it is, describe a path that will traverse it.

For each graph in Exercises 1–6, a. Is the graph traversable? b. If it is, describe a path that will traverse it.

For each graph in Exercises 1–6, a. Is the graph traversable? b. If it is, describe a path that will traverse it.

For each graph in Exercises 1–6, a. Is the graph traversable? b. If it is, describe a path that will traverse it.

For each graph in Exercises 1–6, a. Is the graph traversable? b. If it is, describe a path that will traverse it.

The model shows the way that carbon atoms (red) and hydrogen atoms (blue) link together to form a propane molecule. Use the model to answer Exercises 7–9. Using each of the three red carbon atoms as a vertex and each of the eight blue hydrogen atoms as a vertex, draw a graph for the propane molecule.

The model shows the way that carbon atoms (red) and hydrogen atoms (blue) link together to form a propane molecule. Use the model to answer Exercises 7–9. Determine if each vertex of the graph is even or odd.

The model shows the way that carbon atoms (red) and hydrogen atoms (blue) link together to form a propane molecule. Use the model to answer Exercises 7–9. Is the graph traversable?

The figure below on the left shows the floor plan of a four-room house. By representing rooms as vertices, the outside, E, as a vertex, and doors as edges, the figure below on the right is a graph that models the floor plan. Use these figures to solve Exercises 10–13. Use the floor plan to determine how many doors connect the outside, E, to room A. How is this shown in the graph?

The figure below on the left shows the floor plan of a four-room house. By representing rooms as vertices, the outside, E, as a vertex, and doors as edges, the figure below on the right is a graph that models the floor plan. Use these figures to solve Exercises 10–13. Use the floor plan to determine how many doors connect the outside, E, to room C. How is this shown in the graph?

The figure below on the left shows the floor plan of a four-room house. By representing rooms as vertices, the outside, E, as a vertex, and doors as edges, the figure below on the right is a graph that models the floor plan. Use these figures to solve Exercises 10–13. Is the graph traversable? Explain your answer.

The figure below on the left shows the floor plan of a four-room house. By representing rooms as vertices, the outside, E, as a vertex, and doors as edges, the figure below on the right is a graph that models the floor plan. Use these figures to solve Exercises 10–13. If the graph is traversable, determine a path to walk through every room, using each door only once.

In Exercises 14–17, give the genus of each object.

In Exercises 14–17, give the genus of each object.

In Exercises 14–17, give the genus of each object.

In Exercises 14–17, give the genus of each object.

In Exercises 14–17, which objects are topologically equivalent?

Draw (or find and describe) an object of genus 4 or more.

What does the figure shown illustrate about angles in a triangle when the triangle is drawn on a sphere?

The figure shows a quadrilateral that was drawn on the surface of a sphere. Angles C and D are obtuse angles. What does the figure illustrate about the sum of the measures of the four angles in a quadrilateral in elliptic geometry?

In Exercises 22–24, describe whether each figure shown exhibits self-similarity.

In Exercises 22–24, describe whether each figure shown exhibits self-similarity.

In Exercises 22–24, describe whether each figure shown exhibits self-similarity.

Consult an Internet site devoted to fractals and download two fractal images that imitate natures forms. How is selfsimilarity shown in each image?

What is a graph?

What does it mean if a graph is traversable?

How do you determine whether or not a graph is traversable?

Describe one way in which topology is different than Euclidean geometry.

What is the genus of an object?

Describe how a rectangular solid and a sphere are topologically equivalent.

State the assumption that Euclid made about parallel lines that was altered in both hyperbolic and elliptic geometry.

How does hyperbolic geometry differ from Euclidean geometry?

How does elliptic geometry differ from Euclidean geometry?

What is self-similarity? Describe an object in nature that has this characteristic.

Some suggest that nature produces its many forms through a combination of both iteration and a touch of randomness. Describe what this means

Find an Internet site devoted to fractals. Use the site to write a paper on a specific use of fractals

Did you know that laid end to end, the veins, arteries, and capillaries of your body would reach over 40,000 miles? However, your vascular system occupies a very small fraction of your bodys volume. Describe a self-similar object in nature, such as your vascular system, with enormous length but relatively small volume

Describe a difference between the shapes of man-made objects and objects that occur in nature.

Explain what this short poem by Jonathan Swift has to do with fractal images: So Nat’ralists observe, A Flea Hath Smaller Fleas that on him prey and these have smaller Fleas to bite’em And so proceed, ad infinitum

In Tom Stoppards play Arcadia, the characters talk about mathematics, including ideas from fractal geometry. Read the play and write a paper on one of the ideas that the characters discuss that is related to something presented in this section

In Exercises 42–45, determine whether each statement makes sense or does not make sense, and explain your reasoning. The most effective way to determine if a graph is traversable is by trial and error.

In Exercises 42–45, determine whether each statement makes sense or does not make sense, and explain your reasoning. Objects that appear to be quite different can be topologically equivalent.

Non-Euclidean geometries can be used to prove that the sum of the measures of a triangle’s angles is \(180^{\circ}\). Text Transcription: 180^circ

Make Sense? In 4245, determine whether each statement makes sense or does not make sense, and explain your reasoningEuclidean geometry serves as an excellent model for describing the diverse forms that arise in nature.

This activity is suggested for two or three people. Research some of the practical applications of topology. Present the results of your research in a seminar to the entire class. You may also want to include a discussion of some of topologys more unusal figures, such as the Klein bottle and the Mbius strip.

Research non-Euclidean geometry and plan a seminar based on your group’s research. Each group member should research one of the following five areas: a. Present an overview of the history of the people who developed non-Euclidean geometry. Who first used the term and why did he never publish his work? b. Present an overview of the connection between Saccheri quadrilaterals and non-Euclidean geometry. Describe the work of Girolamo Saccheri. c. Describe how Albert Einstein applied the ideas of Gauss and Riemann. Discuss the notion of curved space and a fourth dimension. d. Present examples of the work of M. C. Escher that provide ways of visualizing hyperbolic and elliptic geometry. e. Describe how non-Euclidean geometry changed the direction of subsequent research in mathematics. After all research has been completed, the group should plan the order in which each group member will speak. Each person should plan on taking about five minutes for his or her portion of the presentation.

Albert Einstein’s theory of general relativity is concerned with the structure, or the geometry, of the universe. In order to describe the universe, Einstein discovered that he needed four variables: three variables to locate an object in space and a fourth variable describing time. This system is known as space-time. Because we are three-dimensional beings, how can we imagine four dimensions? One interesting approach to visualizing four dimensions is to consider an analogy of a two-dimensional being struggling to understand three dimensions. This approach first appeared in a book called Flatland by Edwin Abbott, written around 1884. Flatland describes an entire civilization of beings who are two dimensional, living on a flat plane, unaware of the existence of anything outside their universe. A house in Flatland would look like a blueprint or a line drawing to us. If we were to draw a closed circle around Flatlanders, they would be imprisoned in a cell with no way to see out or escape because there is no way to move up and over the circle. For a two-dimensional being moving only on a plane, the idea of up would be incomprehensible. We could explain that up means moving in a new direction, perpendicular to the two dimensions they know, but it would be similar to telling us we can move in the fourth dimension by traveling perpendicular to our three dimensions. Group members should obtain copies of or excerpts from Edwin Abbott’s Flatland. We especially recommend The Annotated Flatland, Perseus Publishing, 2002, with fascinating commentary by mathematician and author Ian Stewart. Once all group members have read the story, the following questions are offered for group discussion. a. How does the sphere, the visitor from the third dimension, reflect the same narrow perspective as the Flatlanders? b. What are some of the sociological problems raised in the story? c. What happens when we have a certain way of seeing the world that is challenged by coming into contact with something quite different? Be as specific as possible, citing either personal examples or historical examples. d. How are A. Square’s difficulties in visualizing three dimensions similar to those of a three-dimensional dweller trying to visualize four dimensions? e. How does the author reflect the overt sexism of his time? f. What “upward not northward” ideas do you hold that, if shared, would result in criticism, rejection, or a fate similar to that of the narrator of Flatland?