Difference between revisions of "Hypercube"
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| − | As used in geometry, a '''hypercube''' is an extrapolation of the cube or square to n dimensions. For example, a 4th dimensional hypercube is called a [[tesseract]]. Therefore, an n-dimensional hypercube is also known as an n-cube. It is best drawn and represented in non-Euclidean geometry. | + | As used in geometry, a '''hypercube''' is an extrapolation of the cube or square to n dimensions. When n is not specified, it's generally assumed to be 4. For example, a 4th dimensional hypercube is called a [[tesseract]]. Therefore, an n-dimensional hypercube is also known as an n-cube. It is best drawn and represented in non-Euclidean geometry. |
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==Tesseract== | ==Tesseract== | ||
| − | A tesseract is the | + | A tesseract is the <math>4</math>th dimensional hypercube. It is made by combining two cubes. |
| − | The net of a tesseract is composed of 8 cubes. It has the Schlaefli symbol <math>{4,3,3}</math>. | + | The net of a tesseract is composed of 8 cubes. It has the Schlaefli symbol <math>{4,3,3}</math>. One simple coordinate system for its vertices are <math>(\pm1, \pm1, \pm1, \pm1)</math>. The alternated tesseract is a 4D [[cross-polytope]], which coincidentally is dual. |
| + | ==Extra Notes== | ||
| + | The alternated hypercube is known as a demicube. The dual of the hypercube is known as the cross-polytope. For dimensions n≥3, the only n-dimensional regular honeycomb is made of the hypercube. | ||
| + | ==Links== | ||
| + | * [[cube (geometry) | cube]] | ||
| + | * [[square (geometry) | square]] | ||
| + | * [[dimension]] | ||
| + | * [[cross-polytope]] | ||
| − | To see an | + | To see an <math>\mathfrak{e}</math>xample of a 4D cube, click here: [https://latex.artofproblemsolving.com/3/d/5/3d5fc91ddaa1838f367ade6a2baa0649edd32317.png] |
[[Category: Geometry]] | [[Category: Geometry]] | ||
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Latest revision as of 17:47, 6 December 2024
As used in geometry, a hypercube is an extrapolation of the cube or square to n dimensions. When n is not specified, it's generally assumed to be 4. For example, a 4th dimensional hypercube is called a tesseract. Therefore, an n-dimensional hypercube is also known as an n-cube. It is best drawn and represented in non-Euclidean geometry.
Tesseract
A tesseract is the 
th dimensional hypercube. 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 alternated tesseract is a 4D cross-polytope, which coincidentally is dual.
Extra Notes
The alternated hypercube is known as a demicube. The dual of the hypercube is known as the cross-polytope. For dimensions n≥3, the only n-dimensional regular honeycomb is made of the hypercube.
Links
To see an 
xample of a 4D cube, click here: [1]
![[1]](https://latex.artofproblemsolving.com/3/d/5/3d5fc91ddaa1838f367ade6a2baa0649edd32317.png){kind=link}