Category 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!

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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.

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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:

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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.

 

Polyhedra: Proof Of The Five

While the perfect structure of the regular polyhedra are fascinating, there are only five that are currently known: the tetrahedron consisting of four triangular faces; the octahedron with eight triangular faces; the icosahedron with twenty triangular faces; the cube with six square faces, and the dodecahedron with twelve pentagonal faces. These five unique polyhedra have been recognized throughout the ages dating all the way back to the ancient Greeks. The Greeks believed that these five shapes were the only in existence and none others exist naturally. Through analyzing these five polyhedra, we have come to the conclusion that the Greeks were right in their theory.

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Why is this, however? First, it helps to understand the differences between 2D and 3D. Think of the vertices of the faces on a 2D polygon. When connecting polygons at their vertices, each vertex has a 360 degree radius for which shapes can be placed. Once all of the degrees are occupied, the vertex can no longer hold any other shapes. The angles that center around a vertex are the interior angles of the polygons attached. Regular polygons have equal interior angles, and regular polygons make up regular polyhedra.

Now, what makes 2D different from 3D? In order for an object to shift from the second dimension to the third dimension, it must have a height. An object that has no height is simply flat, a 2D polygon. Therefore, in order to be a 3D object, the object must also have faces. As a result, a regular polygon cannot enter the third dimension on its own. There must be more polygons. In the case of the regular polyhedra, there must be more regular and equal polygons. These polygons are attached at their vertices and line up along their edges. Also notice that two regular polygons are not enough to create an enclosed object because the two polygons would simply fold onto one-another. A third polygon is needed, and as a result, a vertex must connect a minimum of three polygons. Notice that this is the case for all of the five regular polyhedra: the tetrahedron has three triangles connected at each vertex while the octahedron has four and the icosahedron has five. The cube has three squared at each vertex and the dodecagon has three pentagons at each vertex.

Remember that a vertice has 360 degrees to work with in two dimensions. Now, remember that a 3D object must have a height. In order for a polyhedra to obtain this height, its regular polygon faces must be at angles to one-another, or they must bend at their edges. When the faces bend, the vertices no longer have a 360 degree radius. In other words, the vertices can only connect a total number of polygons whose interior angles equal a total less that 360 degrees!

Notice that each for each of the five regular polyhedra, this is the case. The equilateral triangle has interior angles of 60 degrees. In a tetrahedron, each vertex needs only to hold 180 degrees. In an octagedron, each vertex needs only hold 240 degrees. In a icosahedron, each vertex needs only to hold 300 degrees of interior angles. However, notice that there exists no regular polyhedron where each vertex connects six triangles. This is because six triangles would require a vertex to hold 360 degrees, something possible in 2D but not in 3D. The same applies to the cube which has vertices connecting three squares with interior angles of 90. The sum of these three interior angles is 270 which is possible, but four squares would contain a sum of interior angles too high for a 3D vertex. The dodecagon has three pentagons at every vertex. Pentagons have interior angles of 108 degrees. Three pentagons would therefore take up 324 degrees on a vertex which is enough for the vertex to handle. Four, once again, is too much.

The next regular polygon, the hexagon, does not form a regular polyhedron. This is because the hexagon has interior angles of 120 degrees. Because at least three regular polygons are needed at each vertex in order to form a regular polyhedron, a vertex would need to be able to connect three hexagons. However, 120 multiplied by three is 360 which just barely exceeds the maximum angular capabilities of a vertex. Three are able to share a vertex in 2D but not 3D as shown in the figures below.

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How do we know that no regular polygons besides triangles, squares, and pentagons can form regular polyhedra, through? Notice that, as sides are added to regular polygons, their interior angles are always increasing. Therefore, the interior angles of the hexagon are smaller that that of every other regular polygon with a larger number of sides. Therefore, their angles will never be small enough to form a regular polyhedra.

Polyhedra; The Five Shapes

As stated in the previous post, there are five regular polyhedra. The tetrahedron, icosahedron, cube, dodecahedron, and octahedron. The Greek believed these to be the only regular polyhedra. After analyzing these polyhedra, multiple characteristics can be found that might help describe their uniqueness.

It helps to look at the basic 2D shapes that make up each of these polyhedra: the regular triangle for the tetrahedron, octahedron, and icosahedron; the square for the cube; and the pentagon for the dodecahedron. The regular triangle has 60 degree interior angles at all three of its vertices, The cube has 90 degree interior angles at all four of its vertices, and the pentagon has 108 degree interior angles at all five of its vertices.

It also helps to analyze how these shapes move from their two dimensional forms to their three dimensional forms. When looking at a cube, each of its vertices have three squares connected them.

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When singling out a single vertex on a cube, we can analyze this in 2D by folding the 3 squares flat.

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When thinking of this in another 2D scenario with 2 squares, folding this back into 3D would not be possible be because the two squares would simply fold flat onto each other. This helps show that, in order to make a regular polyhedron, three or more shapes must connect at a single vertex. This is seen from all of the five regular polyhedra listed. The tetrahedron, cube, and dodecahedron all have three of their respective shapes at each of their vertices while the icosohedron and octahedron have more of their respective shapes at their vertices.

So it is clear that the vertices of the regular polyhedra as well as the angles of the shapes surrounding the vertices contribute to the creation of the regular polyhedra. In further research, we will attempt to prove these to be the only regular polyhedra based on these characteristics they share.

Polyhedra

A polyhedron is a geometrical 3D shape whose faces are all polygons. A regular polygon is a 2D shape that has equal length sides as well as equal interior angles. A regular polyhedron, therefore, is a polyhedron with sides that are all equal regular polygons that meet at their vertices. Each vertex on a regular polyhedron connects an equal number of regular polygons. These figures have been studied since ancient times by the Greeks, and remnants of architecture has been uncovered in Scotland as well as Egypt.

There are five currently known regular polyhedra. Three have been formed with regular triangles: the tetrahedron with 4 faces, the octahedron with 8 faces, and the icosahedron with 20 faces. The commonly used cube is also a polyhedron with equilateral squares for faces, and the dodecahedron  is represented as a 12-sided shape with pentagonal faces. The five known regular polyhedra can be seen here.

The Greeks believed these to be the only regular polyhedra possible. In this lab, we will attempt to prove (or disprove) this theory of the ancient Greeks. We also plan to analyze the truncation of these regular polyhedra. Truncating is when the vertices of a 3D shape are cut off, creating a shape in their place.  Truncating a regular polyhedron at all vertices can create entirely new shapes and this interesting phenomenon will be explored further. As we delve further into the lab, more questions might arise that we will analyze and perhaps attempt to answer.