Kepler-Poinsot Solids
Kepler-Poinsot Solids |
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Kepler-Poinsot Solids
- The Kepler-Poinsot solids, or polyhedra, are four concave polyhedrons constructed of regular concave polygons. Along with the Platonic Solids, they are referred to as the "cosmic figures".
Contents
Basic Description
There are four Kepler-Poinsot solids: the great dodecahedron, the great icosahedron, the great stellated dodecahedron and the small stellated dodecahedron. The small stellated dodecahedron was first displayed by Paolo Uccello in 1430 and the great stellated dodecahedron was later published in 1568 by Wenzel Jamnitzer. These two polyhedra were then later rediscovered and described by Kepler in his 1619 work, Harmonice Mundi. Furthermore, Poinsot rediscovered Kepler's solids and went on to discover the great dodecahedron and great icosahedron in 1809.
They gained their current names in 1859 from Arthur Cayley. Further research by Cauchy in 1813 proved that these four polyhedrons exhaust all possibilities for regular star polyhedra ^{[1]}.
Additionally, although the ancient Greeks are credited with the Platonic Solids, stone carvings of shapes bearing heavy resemblance to the solids were dated to the period from 3000 to 1000 B.C to Indian civilizations. Interestingly enough, these civilizations also assigned certain elements to the shapes, just as the Greeks had done later on ^{[2]}.
A More Mathematical Explanation
Creation and Classification
The four Kepler-Poinsot solids are traditionally created via the s [...]Creation and Classification
The four Kepler-Poinsot solids are traditionally created via the stellation of a either a dodecahedron or icosahedron. This is done by extending the edges of each of faces of the polyhedron, or the faces themselves, until they meet again ^{[3]}.
The Schläfli symbol, given in the form of {P, Q}, denotes the number of edges on each face of the polygon (P) as well as the number of faces that meet at each vertex of polyhedron (Q). Below is a chart that provides the Schläfli symbols for each of the Kepler-Poinsot Solids:
Solid | Schläfli symbol | Faces | Edges | Vertices |
---|---|---|---|---|
Great Dodecahedron | {} | 12 | 30 | 12 |
Great Icosahedron | {} | 20 | 30 | 12 |
Great Stellated Dodecahedron | {} | 12 | 30 | 20 |
Small Stellated Dodecahedron | {} | 12 | 30 | 12 |
Upon closer observation of the Schläfli symbols on each of the solids, it should be noted that the Great Dodecahedron and the Small Stellated Dodecahedron do not fulfill Euler's formula of Vertices - Edges + Faces = 2, which led to Schläfli originally assuming that the polyhedrons with these numbers could not exist ^{[4]}..
Additionally, all four Kepler-Poinsot Solids have 30 edges.
Stellation
Edge Stellation
This is the simple approach of extending the edges of the polyhedron until they intersect at a later point. This is the only type of stellation polygons experience, but it does not apply to cubes, tetrahedrons and octahedrons, all of which do not intersect again after their original vertices. |
Face Stellation
There are two ways to approach face stellation. The first, the two-dimensional approach, involves selecting a face-plane and observing how the other planes intersect it. Through the information gathered at this one plane, we can deduce the possible faces for the stellated forms of these polyhedra.
The second method, the three-dimensional apprach, involves considering the stellations as being built upon layers of solid or bounded cells. These bounded cells surround the polyhedron that acts as the core of the new stellated polyhedron. These bounded cells can also be stuck together to form new polyhedra with faces that lie in the same plane as the original polyhedron.
Cubes and Tetrahedrons, however, do not produce any new stellated forms when attempting face-stellation because the extended faces will never intersect.
Creating the Kepler-Poinsot Polyhedra
The information below is, for the most part, taken from Kavitha d/o Krishnan's paper on Polyhedra: ^{[5]} with the right-hand side images from Tom Gettys' page on Kepler-Poinsot Solids ^{[6]}
Small Stellated Dodecahedron
To form the small stellated dodecahedron through edge stellation, one simply has to extend the edges of each pentagonal face until they intersect to form a pentagram. Then, repeat with all the edges until the edges all come together to create pentagonal pyramids above each face. Eventually, there will be total of 12 pentagonal pyramids on the polyhedron itself, making 12 vertices with five pentagrams meeting at each vertex and two pentagrams meeting at each edge. Thus, when the small stellated dodecahedron is considered as a self-intersecting polyhedron, it has:
When considering the small stellated dodecahedron as a non-intersecting polyhedron, it can also be called an elevated dodecahedron because the solid is now seen as a dodecahedron with a pyramid placed on each face. Thus, the solid consists of 60 isosceles triangular faces, and where the dodecahedron had 20 vertices, 12 more are added through the 12 vertices of the pyramids. Thus, you can calculate 32 vertices per small stellated dodecahedron.
Additionally, it is possible to create the small stellated dodecahedron through face stellation. By the two-dimensional approach to face stellation, we can visualize placing a dodecahedron on a flat surface with a flat top face-plane situated parallel to the base plane that is currently in contact with the surface. All of the planes barring the base do meet the top face-plane. The top five adjacent planes on the dodecahedron intersect in five lines as well, forming a pentagram. By applying this through all the faces of the dodecahedron, it is possible to visualize a small stellated dodecahedron.
The three-dimensional approach to face stellation involves acknowledging that the first stellation of the dodecahedron is consticted of 12 pentagonal pyramids, which are bounded cells with each pyramid stock to a corresponding face on the dodecahedron much in the manner the elevated dodecahedron is considered.
In the image above, the red and blue star refers to the small stellated dodecahedron.
Great Dodecahedron
The great dodecahedron is formed with 12 self-intersecting pentagonal faces and created through face-stellation of the dodecahedron.
With a two-dimensional approach, it is possible to create the great dodecahedron through the face-stellation of a dodecahedron. By extending the five planes containing the faces adjacent to the top face until they intersect and create five lines of intersection, the great dodecahedron is created. The face itself is still a pentagon, as seen as the yellow pentagon in the image above, the third shape from the left.
By approaching the creation of a great dodecahedron three-dimensionally, this polyhedron can also be created through the face-stellation of a small stellated dodecahedron. The bounded cells, then, would be 30 wedges with isosceles triangles as faces. Each wedge is then stuck between the pyramids of a small stellated dodecahedron to form the great dodecahedron.
((Figure out if we can use image 5.6 to help with explanation?))
Based on the information known about the small stellated dodecahedron, which is constructed with 12 pentagonal pyramids, by needing to add wedges that cover two adjacent triangular faces from different pyramids, the number of wedges needed is (12 * 5) / 2 = 30.
Also, since this polyhedron can be obtained through the stellation of the small stellated dodecahedron, it can also be called the stellated small stellated dodecahedron.
Great Stellated Dodecahedron
This solid can be created through the edge stellation of the great dodecahedron. By extending the edges of each pentagonal face within the great dodecahedron, pentagrams are formed to construct the great stellated dodecahedron.
Thus, because the great dodecahedron was formed of 12 intersecting pengtagons, the great stellated dodecahedron is created of 12 intersecting pentagrams, much like the small stellated dodecahedron. However, instead of 5 pentagrams meeting at each vertex, only three meet. Thus, the great stellated dodecahedron has 20 vertices and 30 edges.
This polyhedron can also be obtained through the face stellation of the dodecahedron or great dodecahedron. In this manner, by taking the two-dimensional approach, we must consider how the faces of the great dodecahedron and dodecahedron both have pentagonal faces. Thus, through stellation, the resulting face that the great stellated dodecahedron must have a pentagram as its face shape.
As for the three-dimensional approach, the face stellation of the great dodecahedrom must occur. The bounded cells that must be applied to the base polyhedron, then, must be 20 spikes, each of which are asymmetric triangular dipyramids. The spikes are placed upon where the yellow pyramid wedges were the bounding cells placed in order to create the great dodecahedron. More specifically, the spikes fit into the depressions between each of the wedges. Because each pentagonal face of a great dodecahedron has five depressions and each depression is shared between three face, (12 * 5) / 3 = 20 depressions are present, meaning 20 spikes are necessary to fill them.
Great Icosahedron
The great icosahedron is constructed of 20 self-intersecting equailateral triangular faces with five such triangles meeting at each vertex. Through this information, we calculate that the great icosahedron has 12 vertices and 30 edges in total. The only difference between the great icosahedron and the icosahedron (?) would be that the equilateral faces of the great icosahedran are larger than the faces of the icosahedron, thus why it is called the great icosahedron. It should be noted that in both this and the great dodecahedron, 'great' refers to how the resulting stellated polyhedron has larger faces than the polyhedron it was developed from. ^{[7]}
Duals
Much like the Platonic Solids, the duals of Kepler-Poinsot solids are themselves Kepler-Poinsot solids ^{[8]}:
Keppler-Poinsot Solid | Dual |
---|---|
Great Dodecahedron | Small Stellated Dodecahedron |
Great Icosahedron | Great Stellated Dodecahedron |
Great Stellated Dodecahedron | Great Icosahedron |
Small Stellated Dodecahedron | Great Dodecahedron |
Calculating Attributes
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References
- ↑ Weisstein, Eric W. "Kepler-Poinsot Solid." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Kepler-PoinsotSolid.html
- ↑ http://blazelabs.com/f-p-solids.asp
- ↑ http://www.waset.org/journals/waset/v38/v38-7.pdf
- ↑ Weisstein, Eric W. "Kepler-Poinsot Solid." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Kepler-PoinsotSolid.html
- ↑ http://www.math.nus.edu.sg/~urops/Projects/Polyhedra.pdf
- ↑ http://home.comcast.net/~tpgettys/kepler.html
- ↑ http://www.math.nus.edu.sg/~urops/Projects/Polyhedra.pdf
- ↑ Weisstein, Eric W. "Uniform Polyhedron." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/UniformPolyhedron.html
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[[Description::The Kepler-Poinsot solids, or polyhedra, are four concave polyhedrons constructed of regular concave polygons. Along with the Platonic Solids, they are referred to as the "cosmic figures".|]]