- What are the dimensions?
- Three-dimensional space
- The fourth dimension and time
- The coordinates of a hypercube
- Unfolding of a hypercube
- References
A hypercube is a cube of dimension n. The particular case of the four-dimensional hypercube is called a tesseract. A hypercube or n-cube consists of straight segments, all of equal length that are orthogonal at their vertices.
Human beings perceive three-dimensional space: width, height and depth, but it is not possible for us to visualize a hypercube with a dimension greater than 3.
Figure 1. A 0-cube is a point, if that point extends in a direction a distance a forms a 1-cube, if that 1-cube extends a distance a in the orthogonal direction we have a 2-cube (from sides x to a), if the 2-cube extends a distance a in the orthogonal direction we have a 3-cube. Source: F. Zapata.
At most we can make projections of it in three-dimensional space to represent it, in a similar way to how we project a cube onto a plane to represent it.
In dimension 0 the only figure is the point, so a 0-cube is a point. A 1-cube is a straight segment, which is formed by moving a point in one direction a distance a.
For its part, a 2-cube is a square. It is constructed by shifting the 1-cube (the segment of length a) in the y direction, which is orthogonal to the x direction, a distance a.
The 3-cube is the common cube. It is built from the square by moving it in the third direction (z), which is orthogonal to the x and y directions, a distance a.
Figure 2. A 4-cube (tesseract) is the extension of a 3-cube in the orthogonal direction to the three conventional spatial directions. Source: F. Zapata.
The 4-cube is the tesseract, which is built from a 3-cube moving it orthogonally, a distance a, towards a fourth dimension (or fourth direction), which we cannot perceive.
A tesseract has all its right angles, it has 16 vertices, and all of its edges (18 in all) have the same length a.
If the length of the edges of an n-cube or hypercube of dimension n is 1, then it is a unit hypercube, in which the longest diagonal measures √n.
Figure 3. An n-cube is obtained from an (n-1) -cube extending it orthogonally in the next dimension. Source: wikimedia commons.
What are the dimensions?
Dimensions are the degrees of freedom, or the possible directions in which an object can move.
In dimension 0 there is no possibility to translate and the only possible geometric object is the point.
A dimension in Euclidean space is represented by an oriented line or axis that defines that dimension, called the X-axis. The separation between two points A and B is the Euclidean distance:
d = √.
In two dimensions, space is represented by two lines oriented orthogonal to each other, called the X axis and the Y axis.
The position of any point in this two-dimensional space is given by its pair of Cartesian coordinates (x, y) and the distance between any two points A and B will be:
d = √
Because it is a space where Euclid's geometry is fulfilled.
Three-dimensional space
Three-dimensional space is the space in which we move. It has three directions: width, height, and depth.
In an empty room the perpendicular corners give these three directions and to each one we can associate an axis: X, Y, Z.
This space is also Euclidean and the distance between two points A and B is calculated as follows:
d = √
Human beings cannot perceive more than three spatial (or Euclidean) dimensions.
However, from a strictly mathematical point of view it is possible to define an n-dimensional Euclidean space.
In this space a point has coordinates: (x1, x2, x3,….., xn) and the distance between two points is:
d = √.
The fourth dimension and time
Indeed, in relativity theory, time is treated as one more dimension and a coordinate is associated with it.
But it must be clarified that this coordinate associated with time is an imaginary number. Therefore the separation of two points or events in space-time is not Euclidean, but rather follows the Lorentz metric.
A four-dimensional hypercube (the tesseract) does not live in space-time, it belongs to a four-dimensional Euclidean hyper-space.
Figure 4. 3D projection of a four-dimensional hypercube in simple rotation around a plane that divides the figure from front to left, back to right and from top to bottom. Source: Wikimedia Commons.
The coordinates of a hypercube
The coordinates of the vertices of an n-cube centered at the origin are obtained by doing all the possible permutations of the following expression:
(a / 2) (± 1, ± 1, ± 1,…., ± 1)
Where a is the length of the edge.
-The volume of an n-cube of edge a is: (a / 2) n (2 n) = a n.
-The longest diagonal is the distance between opposite vertices.
-The following are opposite vertices in a square: (-1, -1) and (+1, +1).
-And in a cube: (-1, -1, -1) and (+1, +1, +1).
-The longest diagonal of an n-cube measures:
d = √ = √ = 2√n
In this case the side was assumed to be a = 2. For an n-cube of side to any it will be:
d = a√n.
-A tesseract has each of its 16 vertices connected to four edges. The following figure shows how vertices are connected in a tesseract.
Figure 5. The 16 vertices of a four-dimensional hypercube and how they are connected are shown. Source: Wikimedia Commons.
Unfolding of a hypercube
A regular geometric figure, for example a polyhedron, can be unfolded into several figures of smaller dimensionality.
In the case of a 2-cube (a square) it can be divided into four segments, that is, four 1-cube.
Similarly a 3-cube can be unfolded into six 2-cubes.
Figure 6. An n-cube can be unfolded into several (n-1) -cubes. Source: Wikimedia Commons.
A 4-cube (tesseract) can be unfolded into eight 3-cubes.
The following animation shows the unfolding of a tesseract.
Figure 7. A 4-dimensional hypercube can be unfolded into eight three-dimensional cubes. Source: Wikimedia Commons.
Figure 8. Three-dimensional projection of a four-dimensional hypercube performing a double rotation around two orthogonal planes. Source: Wikimedia Commons.
References
- Scientific culture. Hypercube, visualizing the fourth dimension. Recovered from: culturacientifica.com
- Epsilons. Four-dimensional hypercube or tesseract. Recovered from: epsilones.com
- Perez R, Aguilera A. A method to obtain a tesseract from the development of a hypercube (4D). Recovered from: researchgate.net
- Wikibooks. Mathematics, Polyhedra, Hypercubes. Recovered from: es.wikibooks.org
- Wikipedia. Hypercube. Recovered from: en.wikipedia.com
- Wikipedia. Tesseract. Recovered from: en.wikipedia.com