Dr. Sarah's Geometry of our Earth and Universe

Read through this page carefully and skim through the underlined links (accessible on the web) below.

While geometry means measuring the earth, too often it is presented in an axiomatic way, divorced from reality and experiences. In this next segment we will use intuition from experiences with hands on models and will continue to develop our web searching research skills in order to understand real-world applications of geometry such as the geometry of the earth and universe and applications of geometry to art. Now that you have seen some of the ways that mathematicians do research, you are going to do some research in mathematics the way that mathematicians do. We first think about the problems by ourselves. Then we consult books and journals, and rethink the problem using ideas from other sources to help us. Eventually we might talk to an expert in the field and see if they have ideas to help us. Recall that this process can be frustrating, but that it is the struggle that leads to deep understanding. (For example, Wiles describes doing mathematics as first entering a dark mansion and bumping into furniture. After a while he gets used to the dark and stops bumping into the furniture. Finally, after about 6 months, a light switch comes on and he sees where he has been all along.)

Research Problems

Geometry of the Earth

Problem 0 How could we tell that the earth is round instead of flat without using any technology (ie if we were ancient Greeks)?

Problem 1 One can define a line as the shortest distance path between two points. On curved surfaces such paths are no longer straight when viewed from an extrinsic or external viewpoint (see Problem 2). Nevertheless, these paths do exist on curved surfaces. What is the shortest distance path (staying on the surface of the sphere) between Tallahassee, Florida and Multan, Pakistan on the surface of a perfectly round spherical globe?

Problem 2 A straight line on the surface of a sphere must curve from an extrinsic or external viewpoint, but intrinsically, say for example if we are living in Kansas, we can define what it means to feel like we are walking on a straight path. What is straight (intrinsically) on a sphere? Is the equator an intrinsically straight path? Is the non-equator latitude between Tallahassee, Florida and Multan, Pakistan an intrinsically straight path?

Problem 3 On the surface of a perfectly round beach ball, can the sum of angles of a spherical triangle (a curved triangle formed by three shortest distance paths on the surface of the sphere) ever be greater than 180 degrees? Why?

Problem 4 Is SAS (side-angle-side, which says that if 2 sides of a triangle and the angle between them are congruent to those in a corresponding triangle, then the 2 triangles must be congruent) always true for spherical triangles (a curved triangle formed by three shortest distance paths) on the surface of a perfectly round beach ball? Explain.

Problem 5 Assume that we have a right-angled spherical triangular plot of land (a curved triangle formed by three shortest distance paths on the surface of the sphere that also contains a 90 degree angle) on the surface of a spherical globe between approximately Umanak, Greenland, Goiania, Brazil, and Harare, Zimbabwe, that happens to measure 300 and 400 units on its short sides. Is the measurement of the long side from Greenland to Zimbabwe greater than, less than or equal to 500 units (ie is the Pythagorean Theorem true on the sphere)? Why?

Problem 6 On the surface of a perfectly round beach ball representing the earth, if we head 30 miles West, then 30 miles North, then 30 miles East, and then 30 miles South would we end up back where we started? Why? What about 300 miles in each direction? What about 3000 miles in each direction? Explain.

Problem 7 Is the surface of a sphere 2-dimensional or 3-dimensional? Why?

Extra Credit If we slice one-half of a perfectly round loaf of bread into equal width slices, where width is defined as usual using a straight edge or ruler, which piece has the most crust? Why?

Geometry of the Entire Universe

Problem 8 Is our universe 3-dimensional or is it higher dimensional? Why?

Problem 9 Are there are finitely or infinitely many stars in the universe? Explain.

Problem 10 The geometry that you explored in high school is called Euclidean geometry. For example, you learned about the Euclidean law stating that the shortest distance path between two points is a straight line and about the Euclidean law stating that the sum of the angles in a triangle is 180 degrees. Is our universe Euclidean (ie does it satisfy the laws of Euclidean geometry such as those just mentioned)? How could we tell?

Problem 11 While people thought that the earth was flat for a long time, we know that the shape of the earth is actually a round sphere. What is the shape of space (the universe)?

Dr. Sarah Initial Intuition for Problem 0

Mathematicians first think about problems by ourselves as we explore our initial intuition. While our intuition may not lead us to a "correct" solution, this process of exploration is imperative for our understanding.

Initial Intuition
Problem 0
by Dr. Sarah

Problem 0: How could we tell that the earth is round and not flat without using any technology (ie if we were ancient Greeks)?

For my problem, I am asked how we could know that the earth is round and not flat without using any technology. I will attempt to answer this question by using only my initial intuition.

As I first thought about this problem, it occurred to me that if we traveled around the earth and fell off of it while we were traveling, then we would know that the earth was not round. On the other hand, if we never fell off while traveling, then we could not tell whether the earth was round, flat or some other shape. It could still be flat but perhaps our travels had just not taken us to the edge. Historically, I think that people thought that the earth was indeed flat, and that a ship could fall off the edge. I then realized that this approach would not solve the question, because it would never allow us to determine that the earth is actually round.

I next thought about trying to find a definitive method to tell if the earth was round and not flat. If we could travel all the way around the earth, being assured that we were traveling in the same direction all the time, then this would differentiate our living on a round earth from living on a flat earth. Yet, we are not allowed to use any technology to help us, so a compass would not be allowed. Given this, I'm not sure how we could know that we were traveling in the same direction. Hence, I decided that while this was a good idea, I could not make the method work without technology.

Finally, I gave up on the idea of traveling to reach a specific destination, and started to think about the constellations. If we travel to different places on the earth, we see differences in the stars. For example, constellations look very different in the northern hemisphere than in the southern hemisphere. Also, even within the northern hemisphere, the north star is in different positions in the sky. This would not occur if the earth were flat and would indicate that the earth was round.

This concludes my initial intuition on Problem 0.

Dr. Sarah's Research for Problem 0

After developing initial intuition on a problem, we consult books and journals, and rethink the problem using ideas from other sources in order to help us.

On the web, using google.com, I searched for
+round +earth +flat
but too many pages resulted. Next I searched for
+round +earth +flat +"how can we tell"
which returned about 46 pages. After skimming through them, I decided that the following would be the most relevant . Read these - you are responsible for the ideas in them!
Round Earth feels flat
Why are planets round?
Was Columbus the first person to say the Earth is round?
How we know the Earth is round
How Do We Know the Earth is Round?

STEP 1 - Initial Intuition ONLY - book and web searches not allowed - due today

Call Dr. Sarah over in order to be assigned a problem number. For now, you may use your intuition, Dr. Sarah's globes, and talk to your partner, but you may not use anything else. Open up Microsoft Word and type the problem number, the problem itself, your name(s), and your preliminary response to your question. If you and your partner have conflicting ideas, then present them both. The clarity, depth, and logic of your explorations are important - but whether you obtain the "right" answer is not important. SAVE OFTEN. When you are finished, copy your work from Word and paste it into a bulletin board posting on WebCT to the forum containing just you and I. Check to make sure it posted correctly by clicking on Update Listing, selecting the topic (forum) containing just you and I and reading your message. Then quit the web browser, restart it and log into your partner's WebCT (on the same computer so that you can paste in your work to the forum containing them and Dr. Sarah). At the end of lab today you will present your initial ideas to the rest of the class.

STEP 2 - due Thur Nov 21 at the beginning of class (no lates allowed) - Research Report Counts as 100% of This Major Writing Assignment - Use Your Web Searching Skills to do Internet and Book Research and/or Physical Experimentation to Try and Answer Your Question

Consulting people or using anyone's notes (from other semesters) for help (other than your partner or Dr. Sarah) is NOT ALLOWED and will be considered as PLAGIARISM. I am happy to help you think of experiments and words to search for, but you should try and do so on your own first (see my search techniques for Problem 0 for a reminder of efficient searching methods). I will not tell you whether you are correct or not for part 2, since the point is for you to do your own research.

You and your partner should work together to prepare one report that is due at the beginning of class on Tuesday Nov 21. The report counts as 100% of this major writing assignment grade.

  • Your report should have a cover page, a table of contents, the report, a list of numerous useful and relevant references, comments on how you used each reference, and copies of the references themselves. (You will turn in the first page of any web references that you use, along with the web page addresses, and a photocopy of any book info that you use, along with the book titles and authors.)
  • Your report should include the phrases that led you to your successful search. (For my example below, that would have been +round +earth +flat +"how can we tell" ). If you could not find relevant web pages, even with my help, briefly explain what you used to search, and why there were no relevant pages.
  • Your major writing assignment grade will be based on the quality of the web and/or book references that you find and/or experiments that you conduct, along with the clarity and depth of your answer. Having the "right" answer is not of prime importance as it is often the case that at this stage, mathematicians will still have incorrect ideas - recall from the Fermat video that Shimura made "good mistakes". The idea here is to deeply explore your question with help from web searching and/or experiments, and then to clearly communicate your research. Be sure to follow the writing checklist guidelines.

  • Your classroom presentation (which will count just as participation) will consist of a brief summary of your answer to your question.
  • You and your partner should arrange to bring 1 ball (that you won't mind drawing on) to class - the ball should already inflated and should be between 6 inches and a foot in diameter. Also, if you have masking tape, bring that too.

    STEP 3: Get Intuition from an Expert (Dr. Sarah)

    Eventually mathematicians talk to experts in the field in order to get help and/or feedback from them.

    This will occur in class after your reports are collected and you have completed your presentations.