Tag Archives: Polyhedra

Polyhedra: Truncations

Truncation is the act of cutting the corners from the faces of a shape around its vertices. When truncating a regular polyhedron, each vertex will create a new face. This new face will have the same amount of edges as the number of faces meeting at the vertex. Therefore, when truncating a tetrahedron which has three triangle meeting at each of its vertices, a new triangle is formed at the vertex.

Now, what happens when a regular polyhedron is truncated to the point where there are no edges from the original shape? In this case, the only edges of the original shape would be of those that were cut form the corners. The number of faces on the new shape would therefore be dependent on the number of vertices on the old shape. A tetrahedron, for example, has four vertices. Therefore, the new shape would have four faces. These faces, as mentioned previously, would be triangles. However, notice that the tetrahedron has four triangular faces already. Therefore, a fully truncated tetrahedron produces a new tetrahedron!

tetra1tetra2tetra3tetra4

Next we will explore what happens when we truncate a cube. As stated before, the number of faces will be dependent on the number of vertices. With a cube, there are eight vertices. After we truncate the cube to the point where there are no more faces of the regular shape, we should expect to see a new polyhedron with eight faces. These new faces would have the shape of a triangle. Our new polyhedron will be an octahedron. We see a relation between these two polyhedra. The octahedron has eight faces and six vertices, and the cube has six faces and eight vertices. Both of these polyhedra when truncated completely will create one another.

cube1cube2cube3cube4

The final two regular polyhedra are the dodecahedron and the icosahedron. It follows the same formula as before with relation to the vertices and faces. Once you truncate the dodecahedron completely it ends up as the icosahedron. Similar to the cube and octahedron, the icosahedron truncates fully to the dodecahedron.

This is the truncation of the dodecahedron to the icosahedron:

dode1dode2dode3dode4dode5

 

If you need a visual on these shapes truncating, the site http://www.vandeveen.nl/Wiskunde/Diversen/Archimedean.html can be very useful.\

With all of our regular polyhedra, we see that when you truncate them completely you will end up with another regular polyhedra. The tetrahedra truncates to another tetrahedra. A cube to an octahedron and vice versa. Finally, the dodecahedron will truncate completely into the icosahedron and, similar to the cube and octahedron, the icosahedron truncates to the dodecahedron.

 

Polyhedra: Conclusion

In this lab, we set out to explore the five regular polyhedra including their qualities as well as their exclusive nature. Through analyzing these shapes, it is clear that regular polyhedra are only created through three basic 2D shapes: the equilateral triangle, the square, and the pentagon. It is also clear that the vertices on a regular polyhedra require at least three of these shapes to obtain their 3D nature. Another requirement for their 3D nature is that each vertex can only be surrounded by interior angle equaling less than 360 degrees in sum. This is because 3D shapes must have a height and this comes from bending the shapes at their connected edges.

These findings helped to prove that there are only five regular polyhedra. It was found that there are only five regular polyhedra because the limits on the the total amount of degrees that can surround a vertex in the third dimension as well as the minimum amount of shapes that must share the vertices. Because equilateral triangles have small interior angles of 60 degrees, the triangle is capable of forming three regular polyhedra with 3, 4, and 5 triangles around a vertex. The 90 degree angles of the square allow for the cube to form, and the 108 degree angles allow for the dodecahedron to form. It is clear that no other regular polyhedra exist because the interior angles of regular shapes only grow larger the more edges they have. Therefore, no sum of three interior angles is less than 360 after the pentagon and never will be. Another experiment that helped to further show the exclusivity of regular polyhedra was the truncating of their vertices which resulted in the shapes to form into one-another. It is clear that the truncating of a shape creates a new shape with the number of faces equal to the number of vertices from the previous shape. These new faces have the same number of edges as the number of old faces shared at the cut vertex.

In further studying regular polyhedra, it would be interesting to analyze the incomplete truncation of these solids. When truncating, polygons are seen at the corners that eventually become the faces of the new shape when truncation is complete. When truncation is incomplete, each corner is inhabited by these miniature polygons while the old faces shrink in size and gain an extra edge at each of their vertices. However, are these shapes still symmetrical throughout their entire transformation? If so, how many of these solids can be created. For example, a cube with slight truncation would have triangles on its vertices and the squares would have four new edges, making them octagons. How many of these shapes exist? Is there a finite amount? These are the questions that would be explored in further experimentation.

 

The Polyhedra

Definition: A polyhedra is a solid figure with many plane faces, typically more than six.

I’ve always learned better with some sort of visual help. In this chapter, it has to do with the polyhedra and its properties. In geometry there are usually a lot of visual aids. The book that I have doesn’t actually have the questions written out (accidentally ordered the teacher’s guide). It seems to be focusing on the number of vertices, edges, and faces of polyhedra and the extension of some simple formulas in plane geometry to polyhedra.

Another important aspect of this chapter that I like is that it does not require a computer. The coding has definitely been difficult for me in this class and it would be nice to get a little break from that.

Polyhedra! -Jordan

This lab is all about polyhedrons and investigating their properties. This is the only chapter that does not use the computer to view data. This instead requires you to use models to help you visualize and and think about the observations that are made.

Some of the questions that are explored include finding the diameter of each of the regular solids if the edge length is one and, “what is the diameter of each of the regular solids if the faces each have area 1?.”

A reason why this topic is of interest to me because it does involve geometry which is a subject that I enjoy. Also the fact that this lab does not use the computer means to me that it will be somewhat hands on which interests me a well.