Curvature of space-time tutorial | AST102

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.

Get 20% Discount on This Paper
Pages (550 words)
Approximate price: -

Try it now!

Get 20% Discount on This Paper

We'll send you the first draft for approval by at
Total price:
$0.00

How it works?

Follow these simple steps to get your paper done

Place your order

Fill in the order form and provide all details of your assignment.

Proceed with the payment

Choose the payment system that suits you most.

Receive the final file

Once your paper is ready, we will email it to you.

Our Services

Custom Writings Help is a Quality-Oriented Company in Online Writing as a result of hiring exceptional professionals to execute clients' tasks.

Essays

Research Papers

At Custom Writings Help,We understand the struggle of research paper writing, and that is why at Custom WritingS Help, we are all out to help you. We pride ourselves on having a team of clinical writers. The stringent and rigorous vetting process ensures that only the 'BEST' Writers are chosen for the job. We have highly qualified Ph.D. and MA writers working with us; we equally offer these experienced writers specific bonuses and incentives to make them deliver highly original, unique, and informative content at reasonably low prices.

Admissions

Thesis Writing Service

Worlwide, Many Masters Students are struggling with Thesis Completion. A thesis is likely to be the longest and most challenging piece of work a student has ever completed. However, unlike essays and other assignments, a student can pick a particular interest topic and work on their initiative. Fortunately, we are there for you. At Custom Writings Help, you are assured of an authentic, imaginative, informative, linguistically great, and advantageous thesis that adheres to all your needs. So, why continue considering different writers when you have discovered the best in the field?

Editing

Custom Papers

Not a single student can avoid writing custom papers. However, a total lack of experience, skills, and time makes it very hard to produce a superb writing piece. Therefore, if you are seeking professional help, turn to us. Our specialized and experienced writers compose a variety of model papers, including custom essays, college term papers, research papers, book reports, MBA essays, executive summaries, dissertations, Ph.D. theses, admission essays, and research proposals for college and university students at any level.

Coursework

Essay Writing

Most of the students disregard the critical principles of essay writing and compose papers below sensible guidelines. Therefore, with Custom Writings Help, one should not worry about his/her essay. Our Writers compose informative and engaging content on all complexities and topics. We write meaningful and smart essays while prioritizing all aspects that bring about a good grade, such as impeccable grammar, proper structure, zero-plagiarism, and conformance to guidelines.

Coursework

Coourse Work Writing

Don't let the seemingly never-ending onslaught of writing assignments get you down. If you are looking where to get course work assistance online, the writers at Custom Writings Help are here to assist you with all of your writing needs. We undertake to unique delivery of papers that meet the professor's requirements. The content is proofread, edited, and checked plagiarism before submission to customers. No matter how big or small your work is, we will deliver on time. Try US Now! !

Coursework

Dissertation Writing Service

High-Quality Dissertation Writing Services are rare. They require Ph.D. academicians – not easily found. However, are an exception. The years, time, and resources we have invested in the dissertation world has given us a competitive advantage over others. Choose to come to Custom Writings Help; You will find perfect Ph.D. consultants who have written hundreds of dissertations theses ready to help you. Let our dissertation-writing services help you craft your dissertation, for you are assured we will give you the results.