Exploration of non-Euclidean Geometries
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Exploration Activity
As two of these different geometries were on a computer program, anyone can use them. Below is a challenge so that personal exploration can be obtained.
Spherical Geometry
My exploration of spherical used a clear spherical model, and wet erase markers, but can easily be replicated with a basketball, soccer ball, etc. Use a protractor for angle measurements, and string as lines. Some challenges exist below, all of them are possible:
- Create a great circle
- Create a lesser circle
- Create a triangle
- Create a triangle with two right angles
- Create a triangle with three obtuse angles
- Create a Saccheri quadrilateral
Hyperbolic Geometry
For hyperbolic geometry, the program Non-Euclid was used. A link to the program is below, click on the button and go to the site. Some challenges exist below, all of them being possible.
- Create a diameter
- Create a circle intersection
- Create a triangle
- Create a circle in the center
- Create a circle on the edge
- See what happens to the angle measures when the triangle vertices come closer to the edge
How to use the program:
- Go to the bottom of the disc and turn of the animations.
- Click on the menu in the pink drop down menu, and choose clear all.
- Use the blue drop down menu to create line segments and lines.
- Then use the pink drop down menu to move objects, measure lengths, and measure angles.
- It is recommended to use the measure triangle option to measure the entire triangle.
- It is recommended to use the measure triangle option to measure the entire triangle.
Taxicab Geometry
For taxicab geometry, I used a program of Geogebra. The challenges for this program exist below, as well as the procedure that needs to be taken as there is no way to automatically calculate the measurements:
- Create two points, and create as many geodesics between the lines as possible (Note: all lines must go only horizontal or vertically)
- Create a triangle
- Create two triangles that are visually different with the same side lengths
- Create a unit circle
- Measure an angle
Steps:
- Open the link
- Click on geometry calculator
- Click on the side menu that has lines and a triangle and circle, located on the top right hand side.
- From this menu chose the coordinate grid option and turn the grid on, you can also turn on the axis by clicking the axis button.
- The menu on the top left hand side has all of the options for creating objects, explore using the tools and the challenge list above.
- Although lines are going to look like Euclidean lines, they will be measured by the number of units in the x-direction + the number of units in the y-direction.
- Angles are going to be harder to measure, create a taxicab unit circle around the angle being measured and use the tools to create intersection points on the unit circle where the circle and lines meet. Add the arc lengths on the circle to get the t-radian measurement of the angle.
How to measure lines, example.
Note: it is much easier if you scale your grid so that one space is one unit long, than you can count the spaces instead of doing math. |
How to measure angles, example.
Note to construct the unit circle, place the 4 points at one unit away from the vertex and then connect the points with line segments. |