- A tesseract is a four-dimensional hypercube with 24 faces, 32 edges, and 16 vertices.
- It can be created by thickening up a cube in a fourth dimension.
- Although it is impossible to construct a tesseract physically, we can visualize it in our 3D world.
The idea of the fourth spatial dimension has tantalized people since its conception. In physics, the three dimensions represent space (x,y,z), while the fourth dimension represents time (t). However, in an abstract mathematical concept, there could be an infinite number of spatial dimensions.
Let’s try to understand the fourth dimension. In geometry, the four-dimensional analogue of a cube is called a tesseract. It is easy to extrapolate by looking at lower dimensions –
- Zero-dimensional cube is a point, a vertex.
- One-dimensional cube is a line segment with 2 vertices (one at each end). It can be created by thickening up a point in one dimension.
- Two-dimensional cube is a square with 4 vertices. It can be created by thickening up the line segment in the second dimension.
- Three-dimensional cube is a cube with 8 vertices, created by thickening up the square in a third dimension.
Similarly, a four-dimensional cube (also known as hypercube or tesseract) has 16 vertices. It can be created by thickening up a cube in a fourth dimension. But since we live in a 3D world, it’s impossible to construct a 4D object.
Overall, you can say that a tesseract is to the cube as the cube is to the square. It has 24 faces, 32 edges, and 16 vertices.
An object shifting in dimensions, from Point to Tesseract | Credit: Wikimedia Commons
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Tesseract Is Very Hard To Visualize
It is extremely difficult, if not impossible, to visualize tesseract or any other four-dimensional object. This is because our imaginations are not strong enough to project our consciousness into an artificial world that is very different from our own.
Our brains are wired to convert two-dimensional input into a three-dimensional view. More specifically, our eyes send a pair of two-dimensional images to the brain, from which the brain constructs a 2D+depth model of the visual field.
This is what our brains are best suited for thinking about. Three-dimensional space is easy to visualize because we literally see it all the time. However, we have no direct experience of higher dimensions, and thus people have no clear prototype to use as a springboard for visualizing it.
Physicists and mathematicians trained in higher-dimensional spaces, on the other hand, are more capable than the rest of us to visualize it in their brains.
Let’s Try To Visualize A Tesseract
Like a cube can be projected into two-dimensional space, it is possible to project tesseracts into three-dimensional space.
Figure 2 | Image credit: Wikimedia Commons
The surface of a 3D cube contains 6 square faces; similarly, the hypersurface of the tesseract contains 8 cubical cells.
One can unfold a tesseract into 8 cubes into three-dimensional space (figure 2). This is similar to unfolding a cube into 6 squares into two-dimensional space. The unfolding of a geometric object [with flat sides] is known as a net. The tesseract features 261 different nets.
There are two types of four-dimensional rotations:
1) Simple Rotations: A three-dimensional projection of the Tesseract (figure 3) performing a simple rotation about a plane bisecting figure from top to bottom and front-left to back-right.
Figure 3 | Alternative projection of tesseract | Wikimedia Commons
2) Double Rotation: A three-dimensional projection of the tesseract (figure 4) showing a double rotation about 2 orthogonal planes.
Figure 4 | Alternative projection of tesseract | Wikimedia Commons
The tesseract can also be shown from a hidden volume elimination perspective. In figure 5, for instance, the red face is closest to the fourth dimension and has four cubical cells meeting around it.
Figure 5 | Tesseract from hidden volume elimination perspective
It Was Discovered In 1888
The word “tesseract” was coined by a British mathematician and science fiction writer Charles Howard Hinton. He used the word for the first time in 1888 in his book ‘A New Era of Thought‘. He also invented several new words to describe elements in the fourth dimension.
Since then, the word “tesseract” has been used in different kinds of art, architecture, and science fiction stories (such as Avengers and Agents of S.H.I.E.L.D), where it has nothing to do with the four-dimensional hypercube.
Human spatial representations are not constrained by ur 3D world
A team of researchers at the University of Illinois, United States, conducted a study to find out whether humans can develop an intuitive understanding of 4D space. They utilized virtual reality (VR) to get accurate results.
The evidence shows that people without special practice can learn to make spatial judgments on the length of, and angle between, line segments embedded in 4D space viewed in VR. Their judgment incorporated data from both the 3D projection and the fourth dimension. The underlying representations were based on visual imagery (installed of algebraic in nature), though primitive and short-lived.
The total number of possible dimensions in the universe
While general relativity paints a picture of a four-dimensional universe, superstring theory says it has 10 dimensions, and an extended version called M-Theory says that it has 11 dimensions. In bosonic string theory, spacetime is 26-dimensional. These theories just represent mathematical equations. They are so complicated that no one knows their exact form.
Experiment to study theoretical materials in 4D space
An international team of researchers has been able to develop a two-dimensional experimental system that enables them to analyze the physical properties of “materials” that were theorized to exist only in 4D space.
More specifically, they have demonstrated that four-dimensional quantum Hall effects can be emulated using photons flowing through a two-dimensional waveguide array.
How is this research helpful in our 3D world? Let’s say quasicrystals (widely used to coat some non-stick pans) have been shown to have hidden dimensions. This experiment could help us understand the physics of that hidden dimension. The physics could then be used as a design principle for novel photonic equipment.