Imagining the world in different numbers of dimensions can change how you perceive everything, including time, space, and depths. Think about how watching a 3D movie lets you experience an added depth you wouldn't normally be able to see, or how 3D printing allows you to experience and utilize additional properties of a 3D object.

It's easy to think about the difference between two dimensions and three dimensions because we can easily experience and see in real life. However, when we try to move beyond the dimensions of space that we can experience, it's important to understand what scientists and other researchers mean when they speak of different dimensions to better determine the differences between three dimensions and four dimensions because we can’t directly see or feel four-dimensional space.

## 3D vs. 4D

Our world is in three spatial dimensions, with a fourth dimension that is temporal (as in, the dimension of time); together, they form the fabric of space-time. Scientists and philosophers have wondered and performed research on what a fourth spatial dimension would be. Because these researchers can't directly observe a fourth dimension, it's all the more difficult to find evidence of it.

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Einstein’s Theory of Special and General Relativity rely on the concept of space-time to explain phenomenon like time dilation and length contraction. The theory of relativity described by Einstein was revolutionary in understanding dimensionality and the coordinate systems of our world.

To better understand what a fourth dimension would be like, you can take a closer look at what makes three dimensions three-dimensional, and – following these ideas – speculate on what a fourth dimension would be.

Length, width and height make up the three dimensions of our observable world. You observe these dimensions through the empirical data given to you by our senses like vision and hearing. You can determine the positions of points and directions of vectors in our three-dimensional space along a reference point.

You can imagine this world as a three-dimensional cube that has three spatial axes that account for width, height and length moving forward and backward, up and down, and left and right alongside time, a dimension you don't directly observe but perceive. In a purely mathematical sense, the number of dimensions are represented by the axes of data, in a cartesian system, there is typically an x-axis, y-axis, and z-axis.

When comparing 3D vs. 4D, given these observations of the three-dimensional spatial world, a four-dimensional analog of a cube would be a tesseract, an object that moves in these three dimensions that you perceive alongside a fourth dimension that you can't.

These objects are also called eight-cells, octachorons, tetracubes, or four-dimensional hypercubes, and, while they can't be directly observed, they can be formulated in an abstract sense.

It’s very hard to properly visualize or even picture 4D technology and objects, and perhaps the best method is to compare 2D images and 3D models. We can directly see the key differences between a 2D square and 3D cube, and the transition to 4D would be a similarly significant paradigm shift.

## 4D Shadow

Because three-dimensional beings cast a shadow onto a two-dimensional surface, this has lead researchers to speculate that four-dimensional objects would cast a three-dimensional shadow. For this reason, it's possible to observe this "shadow" in your three spatial dimensions even if you can't directly observe four dimensions. This would be a 4D shadow.

Mathematician Henry Segerman of Oklahoma State University has created and described his own 4-dimensional sculptures. He has used rings to create dodecacontachron-shaped objects which are made of 120 dodecahedra, a three-dimensional shape with 12 pentagon faces.

The same way a dimensional object casts a two-dimensional shadow, Segerman has argued his sculptures are three-dimensional shadows of the fourth dimension.

Though these examples of shadows don't give you direct ways of observing the fourth dimension, they're a good indicator of how to think about the fourth dimension. Mathematicians often bring up the analogy of an ant walking on a piece of paper in describing the limits of perception with respect to dimensions.

An ant walking on the surface of a paper can only perceive two dimensions, but this doesn't mean that the third dimension doesn't exist. It just means the ant can only directly see two dimensions and infer a third dimension through reasoning about these two dimensions. Similarly, humans can speculate on the nature of the fourth dimensions without directly perceiving it.

## Difference Between 3D and 4D Images

The four-dimensional cube tesseract is one example of how the three-dimensional world described by x, y and z can extend into a fourth one. Mathematicians, physicists and other scientists and researchers can represent vectors in the fourth dimension using a four-dimensional vector that includes another variables such as w.

The geometry of objects in the fourth dimension is more complex that include 4-polytopes, which are four-dimensional figures. These objects show the difference between 3D and 4D images.

Some professionals have used the "fourth dimension" to refer to adding more effects to forms of media that three dimensions can't accommodate. This includes "four-dimensional movies" that change the ambience of the theater through temperature, humidity, motion and anything else that can make the experience immersive as though it were a virtual reality simulation.

Similarly, ultrasound researchers that study three-dimensional ultrasound sometimes refer to the "fourth dimension" as ultrasound that carries a time-dependent aspect, as in, a live recording of it. These methods rely on using time as the fourth dimension. As such, they don't account for the fourth spatial dimension that tesseracts try to illustrate.

## 4D Shapes

Creating 4D shapes may seem complicated, but there are many ways of doing so. To take the tesseract as an example, you can express a three-dimensional cube along the w-axis such that it has a starting point and an ending point.

Imagining this expansion tells you that the tesseract is constrained by eight cubes: six from the faces of the original cube and two more from the starting and ending points of this expansion. Studying this expansion more closely reveals that the tesseract is constrained by 16 polytope vertices, eight from the starting position of the cube and eight from the ending position.

Tesseracts are also often portrayed with the variations in the fourth dimension imposed upon the cube itself. These projections show the surfaces intersecting one another, which makes things confusing in the three-dimensional world, but rely on your perspective in discerning the four dimensions from one another.

Mathematicians take into account the limits of perception in creating images of tesseracts. The same way you can view the three-dimensional wire frame of a cube to see the faces on the other side, the wire diagrams of a tesseract show the projections of the sides of the tesseract you can't directly observe without removing them completely from view.

This means rotating or moving the tesseract can reveal these hidden surfaces or parts of the tesseract the same way rotating a three-dimensional cube can show you all of its faces.

## 4-dimensional beings

What beings or life would look like in four dimensions has occupied scientists and other professionals for decades. Writer Robert Heinlein's 1940 short story "And He Built a Crooked House" involved creating a building in the shape of a tesseract. It involves an earthquake that shatters the four-dimensional house into an unfolded state of eight different cubes.

Writer Cliff Pickover imagined four-dimensional beings, hyperbeings, as "flesh-colored balloons constantly changing in size." These beings would appear to you as disconnected pieces of flesh the same way a two-dimensional world would only let you see cross-sections and remnants of a three-dimensional one.

The four-dimensional life form could see inside of you the same way a three-dimensional being can see a two-dimensional one from all angles and perspectives.

You could describe the positions of these hyperbeings using four-dimensional coordinates such as (1, 1, 1, 1). John D. Norton of University of Pittsburgh's department of history and philosophy of science explained that you can arrive at these conclusions on the nature of the fourth dimension by asking questions of what makes one-, two- and three-dimensional objects and phenomena the way they are and extrapolating into a fourth dimension.

A being that lived in the fourth dimension may have this sort of "stereovision," Norton described, to visualize four-dimensional images without being restrained by the three dimensions. Three-dimensional images that drift together and apart from one another in three dimensions show this limitation.

References

- Wolfram MathWorld: Octachoron
- Difference Between: Difference between 3d and 4d
- Wolfram Alpha: Hypercube
- New Scientist: Sculptures cast shadows from the fourth dimension
- Revolvy: 4-polytope
- Sprott's Gateway: The Fourth Dimension by Cliff Pickover
- HubPages: Tales of the Fourth Dimension: "And He Built a Crooked House" by Robert. A. Heinlein
- Einstein for Everyone: What is a four dimensional space like?
- Cut the Knot | Tesseract

About the Author

S. Hussain Ather is a Master's student in Science Communications the University of California, Santa Cruz. After studying physics and philosophy as an undergraduate at Indiana University-Bloomington, he worked as a scientist at the National Institutes of Health for two years. He primarily performs research in and write about neuroscience and philosophy, however, his interests span ethics, policy, and other areas relevant to science.