Difference between revisions of "Hypercube"

m
Line 1: Line 1:
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.
  
 
==Links==
 
==Links==
Line 8: Line 8:
 
==Tesseract==
 
==Tesseract==
 
A tesseract is the 4th dimensional hypercube. It is made by combining two cubes.
 
A tesseract is the 4th 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>. Its vertices are <math>{\pm1, \pm1, \pm1, \pm1}</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 also it's 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.
  
 
To see an <math>\mathfrak{e}</math>xample of a 4D cube, click here: [https://latex.artofproblemsolving.com/3/d/5/3d5fc91ddaa1838f367ade6a2baa0649edd32317.png]
 
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]]

Revision as of 16:19, 20 August 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.

Links

Tesseract

A tesseract is the 4th dimensional hypercube. It is made by combining two cubes. The net of a tesseract is composed of 8 cubes. It has the Schlaefli symbol ${4,3,3}$. One simple coordinate system for its vertices are $(\pm1, \pm1, \pm1, \pm1)$. The alternated tesseract is a 4D cross-polytope, which coincidentally, is also it's 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.

To see an $\mathfrak{e}$xample of a 4D cube, click here: [1]