Difference between revisions of "Cross-polytope"
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| − | A cross-polytope is a higher-dimension analog for the octahedron. The graph of the nD cross-polytope can be formed by the following method: | + | A cross-polytope, also known as a orthoplex or hyper-octahedron is a higher-dimension analog for the octahedron. When no dimension is specified, it is assumed to be 4. The graph of the nD cross-polytope can be formed by the following method: |
1. Take the K 2n complete graph. | 1. Take the K 2n complete graph. | ||
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3. Remove all edges between two vertices in the same pair | 3. Remove all edges between two vertices in the same pair | ||
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| + | The area of an nD cross-polytope with side length s is given by: | ||
| + | <cmath>\frac{2^n}{n!}v^n~\text{where}~v=\frac{s}{\sqrt{2}}~\text{is the distance from the center of the cross-polytope to a vertex of it}</cmath> | ||
| + | This formula can be derived by splitting the cross-polytope into many hyper-tetrahedra. | ||
| + | ==16-cell== | ||
| + | A 16-cell is the 4th dimensional cross-polytope. It is made by combining two cubes. The net of a tesseract is composed of 8 cubes. It has the Schlaefli symbol <math>{3,3,4}</math>. One simple coordinate system for its vertices are <math>(\pm1, 0, 0, 0),(0, \pm1, 0, 0),(0, 0, \pm1, 0),(0, 0, 0, \pm1)</math>. The dual of the 16-cell is the tesseract. | ||
| + | ==Links== | ||
| + | * [[Hypercube]] | ||
| + | * [[Square]] | ||
| + | * [[Octahedron]] | ||
Revision as of 15:32, 20 August 2024
A cross-polytope, also known as a orthoplex or hyper-octahedron is a higher-dimension analog for the octahedron. When no dimension is specified, it is assumed to be 4. The graph of the nD cross-polytope can be formed by the following method:
1. Take the K 2n complete graph.
2. Find n pairs of vertices so no two pairs share a common vertex.
3. Remove all edges between two vertices in the same pair
The area of an nD cross-polytope with side length s is given by:
![\[\frac{2^n}{n!}v^n~\text{where}~v=\frac{s}{\sqrt{2}}~\text{is the distance from the center of the cross-polytope to a vertex of it}\]](http://latex.artofproblemsolving.com/6/9/e/69ee0b7bd69d3d407936f17a08ac99c1c9c4122e.png)
This formula can be derived by splitting the cross-polytope into many hyper-tetrahedra.
16-cell
A 16-cell is the 4th dimensional cross-polytope. It is made by combining two cubes. The net of a tesseract is composed of 8 cubes. It has the Schlaefli symbol 
. One simple coordinate system for its vertices are 
. The dual of the 16-cell is the tesseract.
