Three Lines And Look For The Symmetries example essay topic

When it comes to Euclidean Geometry, Spherical Geometry and Hyperbolic Geometry there are many similarities and differences among them. For example, what may be true for Euclidean Geometry may not be true for Spherical or Hyperbolic Geometry. Many instances exist where something is true for one or two geometries but not the other geometry. However, sometimes a property is true for all three geometries. These points bring us to the purpose of this paper.

This paper is an opportunity for me to demonstrate my growing understanding about Euclidean Geometry, Spherical Geometry, and Hyperbolic Geometry. The first issue that I will focus on is the definition of a straight line on all of these surfaces. For a Euclidean plane the definition of a "straight line" is a line that can be traced by a point that travels at a constant direction. When I say constant direction I mean that any portion of this line can move along the rest of this line without leaving it. In other words, a "straight line" is a line with zero curvature or zero deviation. Zero curvature can be determined by using the following symmetries.

These symmetries include: reflection-in-the-line symmetry, reflection-perpendicular-to-the-line symmetry, half-turn symmetry, rigid-motion-along-itself symmetry, central symmetry or point symmetry, and similarity or self-similarity "quasi symmetry". So, if a line on a Euclidean plane satisfies all of the above conditions we can say it is a straight line. I have included my homework assignment of my definition of a straight line for a Euclidean plane so that one can see why I have stated this to be my definition. My definition for a straight line on a sphere is very similar to that on a Euclidean Plane with a few minor adjustments. My definition of a straight line on a sphere is one that satisfies the following Symmetries. These symmetries include: reflection-through-itself symmetry, reflection-perpendicular-to-itself symmetry, half-turn symmetry, rigid-motion-along-itself symmetry, and central symmetry.

If we find that a line on a sphere satisfies all of the above condition, then that line is straight on a sphere. I have included my homework assignment for straightness on a sphere so that one can see why a straight line on a sphere must satisfy these conditions. Finally, I need to give my definition of a straight line on a hyperbolic plane. My definition of a straight line on a hyperbolic plane must satisfy the following symmetries. These symmetries include: reflection-in-the-line symmetry, reflection-perpendicular-to-the-line symmetry, half-turn symmetry, rigid-motion-along-itself symmetry, central-symmetry, and self-symmetry. If a line on a hyperbolic plane satisfies these conditions then we can say that it is straight.

I have included my homework of my definition of a straight line on a hyperbolic plane so that one can see why these conditions must be satisfied. The next issue that I will address for these three geometries is the definition of an angle on all three surfaces. The definition that I will give applies to all three surfaces. There are at least three different perspectives from which we can define "angle". These include: a dynamic notion of angle-angle as movement, angles as measure, and angles as a geometric shape. A dynamic notion of angle involves an action which may include a rotation, a turning point, or a change in direction between two lines.

Angles as measure may be thought of as the length of a circular arc or the ratio between areas of circular sectors. When thinking of an angle as a geometric shape an angle may be seen as the delineation of space by two intersecting lines. I have provided my homework assignment on my definition of an angle so that one can see the reasoning of my definition for all three surfaces. However, my homework assignment does not ask to define an angle on a hyperbolic plane. This is because a region on a hyperbolic plane can be looked at locally to have the same results as a Euclidean Plane. Since we are on the topics of angles I need to mention the Vertical Angle Theorem.

In my homework I used two different proofs to prove the Vertical Angle Theorem on a Euclidean plane and a sphere. The first idea I used was looking at the Vertical Angle Theorem using angle as measure. The second idea I used was looking at the Vertical Angle Theorem using angle as rotation. I have provided my homework so that one can see my reasoning behind both of these proofs. I found that they worked on a Euclidean plane and a sphere. Although I did not have to say if my proofs worked on a hyperbolic plane, I can say that they would because we can look at a hyperbolic plane locally.

From Chapter 6 in our textbook Experiencing Geometry by Henderson and Tai mina, we formulated a summary of the properties of geodesics on the plane, spheres, and hyperbolic planes. I feel this is a good homework assignment to mention in this paper. For the first part of the problem we were to explain why for every geodesic on the plane, sphere, and hyperbolic plane there is a reflection of the whole space through the geodesic. For the second part of the problem I showed that every geodesic on the plane, sphere, and hyperbolic plane can be extended indefinitely (in the sense that the bug can walk straight ahead indefinitely along any geodesic).

The third part asked to show that for every pair of distinct points on the plane, sphere, and hyperbolic plane there is a (not necessarily unique) geodesic containing them. In the fourth part of the problem I showed the every pair of distinct points on the plane or hyperbolic plane determines a unique geodesic segment joining them On the sphere there are always at least two such segments. Finally, in the fifth part of the problem I showed that on the plane or on a hyperbolic plane, two geodesics either coincide or are disjoint or they intersect in one point. On a sphere, two geodesics either coincide or intersect exactly twice.

I have provided the homework assignment as an artifact so that one can read through my explanations. Instead of rewriting them for this part of the essay I decided to just include my previous work. I also discovered why the Isosceles Triangle Theorem (ITT) held on all three of the surfaces. I have provided the homework assignment as an artifact for this because I would need to explain the entire proof. So, this gives the reader the opportunity to see why the Isosceles Triangle Theorem holds on all three surfaces. For this homework assignment I also proved the Corollary for the ITT and the converse of the ITT.

One can see these by looking at my homework assignment Through this class we have also looked at many triangle congruences. The first one that we looked at was Side-Angle-Side. First we proved this congruency on a plane. Then we took the idea to a sphere. However, when we took the idea to a sphere we noticed that we had to restrict our triangle. So, I was given an assignment that defined different types of small triangles.

I was to explain if the definition allowed Side-Angle-Side congruency to hold. I have included this homework assignment for one to see how I used reasoning to support my conclusions. The next congruency that we looked at in class was Angle-Side-Angle congruency. After I proved this congruency for a plane, next I proved it for sphere.

I had to use the same process as I used for Side-Angle-Side congruency. I had to find a class of small triangles where the Angle-Side-Angle congruency would hold. I defined my small triangle to one that has three distinct vertices, all angles with a measure of less than 90 degrees, of the possible geodesic segment between two distinct vertices the shortest path is always chosen, and each side of the triangle is strictly less than half of a great circle. This definition of a small triangle allowed me to use the planar proof and apply it to the sphere. The proof I used for the plane also worked on a hyperbolic plane because the ideas we used to prove the planar proof also work on a hyperbolic plane. I have included my homework as an artifact so that one can see my conclusions.

For this portfolio I have also shown through problems 9.1, 9.2, and 9.3 why Side-Side-Side congruency, Angle-Side-Side congruency, and Side-Angle-Angle congruency work or do not work for these three surfaces. One can see these from looking at the section entitled "Exploration" in this portfolio. The final main idea that we studied in this Modern Geometry class was parallel transport. Through chapter eight of our textbook we investigated this idea.

The first section of this chapter focuses on Euclid's Exterior Angle Theorem. I proved this theorem and showed when it works on a plane, a sphere and a hyperbolic plane. The main ideas of this proof were vertical angles, Side-Angle-Side congruency, and the definition of a midpoint. Since we can perform all three of these ideas on all three of the surfaces discussed the proof applied to all three surfaces.

When looking at parallel transports we also looked at the symmetries of parallel transported lines. We looked at all three surfaces. We were asked to consider two lines, r and r', that are parallel transports of each other along a third line, l. Then to consider the geometric figure that is formed by the three lines and look for the symmetries of that geometric figure. Then we were asked what we could say about the lines r and r'.

I have provided my notes that include an outline to this proof for all three surfaces so that one can see the conclusions that we made as a class. We found that on a Euclidean plane parallel transported lines do not intersect and are equidistant. For a hyperbolic plane we found that parallel transported lines diverge in both directions. Finally for a sphere we found that parallel transported lines always intersect. Using all the above material, we can see that there are many different similarities and differences when looking at a Euclidean Geometry, Spherical Geometry, and Hyperbolic Geometry. Using my artifacts will help one understand many of my conclusions about these three surfaces.

This essay was an excellent opportunity to reflect on my growing understanding of these three surfaces. I hope you, the reader, can benefit from my conclusions and gain a better understanding of the similarities and differences of these three surfaces.