While geometry means measuring the earth, too often it is presented in an axiomatic way, divorced from reality and experiences. In this segment we will use intuition from experiences with hands on models and we will 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. 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. This process can be frustrating, but that it is the struggle and the process itself that leads to true understanding.
Problem 2 For thousands of years, people argued about the
necessity and validity of Euclid's Parallel Postulate.
One form of this postulate is given as
Through a given point, only one line can be drawn parallel to a given line.
Is this true on the sphere?
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 the north pole, a point on the equator, and a point one-quarter away around the equator. Do the sides satisfy the Pythagorean Theorem? 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 3000 miles in each direction? Can we make a square on a sphere? Explain.
If we slice 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 (or surface area)? Why?
Geometry of our 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 We know that the shape of the earth is close to a round sphere. Could the universe be round too? Does it have any kind of shape?
Create an annotated bibliography with the annotations in your group members' own words providing
The bibliography and annotations must be in a scholarly, professional and consistent format and style of writing, and you will be graded on the depth and clarity of your research and annotations.