Curvature of Space-Time Tutorial

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Material Required: 

  • Paper
  • String
  • Clear tape
  • Thumbtacks
  • Round object (ball or globe)
  • Computer and internet access
  • Protractor (if you don't have one on hand you can get one here.
  • Digital camera and/or scanner
  • Java

**Java: for any Java issues see the alternative procedure at the end of Exercise 3.

Time Required: approximately 2-3 hours

Pre-exploration Study and Information

I. Introduction

Tutorial

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.

The three possible geometries of space are shown, positive, negative and flat curvature.

 Activity

Exercise 1: Test the Theorem.

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.)

  1. Stretch the string tightly and tape down the corners with clear tape or use thumbtacks in each corner.
  2. Measure the angles of the triangle and record them on the following table.
  3. Measure the perimeter of this triangle and record it. (Perimeter = a + b + c) Take a photo of at least three.
  4. Calculate the sum of the angles of this triangle and record that also.Triangle with sides labeled a, b, & c.
  5. Repeat this process at least 5 times for 5 different triangles.

Euclidean Geometry

Angle 1
Angle 2
Angle 3
Sum of Angles
Perimeter
QUESTIONS:

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?

Exercise 2: Investigate Elliptic Geometry

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.

  1. Use the surface of any spherical object – a tennis ball, globe, etc.  
  2. You may be able to use tacks to hold the string in a triangular shape, so you can measure the angles. Take photos.
  3. Make sure you record at least 5 sets of measurements.

Elliptic Geometry

Angle 1

Angle 2

Angle 3

Sum of Angles

Perimeter

QUESTIONS:

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?

Exercise 3: Investigating Hyperbolic Geometry

There aren’t too many readily available objects that have hyperbolic geometry, so instead we’ll use a computer model.  

  1. Go to Triangles, Angles, and AreaNOTE: If the applet does not work for you perform your measurements on the triangles located at the end of this instruction document. Otherwise, an alternate site you can use is http://www.cs.unm.edu/~Joel/NonEuclid/NonEuclid.html
  2. Read through the information on the site and then at the bottom of the page Click to launch the applet.
  3. Look at the image there. This is a hyperbolic surface.  
  4. You can make your angle measurements on the triangle shown (remember that your lines are special arcs in Hyperbolic Geometry).
  5. To measure the angle formed by two lines, measure the angle formed by the tangents to the arcs at the intersection points (the bow of each line). It might be tough to do this on the computer, but try your best.
  6. In the applet, create one more triangle, again measure the angles.

Hyperbolic Geometry

Angle 1

Angle 2

Angle 3

Sum of Angles

Perimeter

QUESTIONS:

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?

Exercise 4: What is the Relationship?

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.

QUESTIONS:

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?

Exercise 5: Apply to the Universe

You will need outside sources  to answer these questions including the video viewed earlier, the textbook, and/or other website resources.

QUESTIONS

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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