**Java: for any Java issues see the alternative procedure at the end of Exercise 3.
Time Required: approximately 2-3 hours
Did you know that there is more than one possible geometry, or shape, of the Universe? Possible geometries include, 1) Euclidean (or flat space) or, 2) Non-Euclidean (or curved space.) Your text explains both of these types of geometry to you but in this lesson you will investigate each. Remember this theorem (called the Triangle Sum Theorem). In this theorem the sum of the angles of a triangle is equal to 180 degrees.
Test out this theorem by using a piece of string to form the perimeter of a triangle on a sheet of paper. Because we are using a flat sheet of paper, this investigation is a test of Euclidean Geometry. (Include at least three images of the following triangles you have created in your table.)
1A. Describe your data on the sum of the angles on a flat sheet of paper for 5 different triangles. .
2A. Does this agree with the Triangle Sum Theorem? Why or why not?
Let’s try this investigation in Elliptic Geometry. Specifically, you will measure the angles in a triangle formed on the surface of a sphere, rather than a flat piece of paper. In your table, include at least three images of the triangles you create.
2A. Describe your data on the sum of the angles on a sphere for 5 different triangles.
2B. Does this agree with the Triangle Sum Theorem? Why or why not?
There aren’t too many readily available objects that have hyperbolic geometry, so instead we’ll use a computer model.
3A. Describe your data on the sum of the angles on a hyperbolic surface for 2 different triangles.
3B. Does this agree with the Triangle Sum Theorem? Why or why not?
When all of the data have been recorded, look to see if you can find a relationship between the value of the perimeter of the triangle and the sum of the angles in flat, elliptic, and “saddle-shaped” (hyperbolic) geometries.
4A. Describe your data on the sum of the angles on a flat sheet of paper and their perimeters in Euclidean geometry? Is there a relationship between the perimeters and the angles?
4B. Describe your data on the sum of the angles on a sphere and their perimeters in elliptic geometry? Is there a relationship between the perimeters and the angles?
4C. Describe your data on the sum of the angles on a hyperbolic surface and their perimeters in hyperbolic geometry? Is there a relationship between the perimeters and the angles?
You will need outside sources to answer these questions including the video viewed earlier, the textbook, and/or other website resources.
5A. The large-scale structure of the universe is said to be overall homogenous in nature. Describe what astronomers mean by this.
5B. The large-scale structure of the universe is said to be overall isotropic in nature. Describe what astronomers mean by this.
5C. Describe any real life situation where there exists a homogenous state but one that is not isotropic.
5D.Describe any real life situation where there exists an isotropic state but one that is not homogenous.
5E. According to the latest research (in the last two or so years), which geometry is the universe believed to be flat, spherical or saddle-shaped? What evidence is given for this?
5F. Explain how the universe may be geometrically spherical in shape but appear to be flat.
5G. Research and write a short essay of 2-3 paragraphs (minimum of 150 words) on what each of these three geometries means to our view of the universe. (flat geometry, spherical geometry, and “saddle-shaped” geometry).
NOTE: You must provide a reference list showing the source(s) that you used, including our own textbook, in proper APA citation format.
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