Trying to conceive of the geometry of a fourth dimension involves disregarding everything you thought you “knew” about reality, existence, and your ability to perceive the world around you. Typically when one hears about the fourth dimension, the mind goes directly to time, but in reality it doesn’t matter whether you call time the fourth dimension or the fortieth dimension. Each dimension must be perpendicular to all of the other dimensions, and it doesn’t matter what order you examine them in. In terms of this paper, the fourth dimension refers to a fourth spatial dimension. By looking at differences between dimensions, specifically the second and third dimension, it is possible to gain a greater understanding about our own dimension-dependent spatial perception, and to explore what this tells us about the fourth dimension.
Before delving into complex examples of perception differences between different dimensions, it is necessary to develop a foundational understanding of the spatial components of each dimension. Dimensions are built upon one another by moving perpendicularly in space from all points in the lower dimension. Speaking in these terms, the fourth dimension is all of the space that one can get to by traveling in a direction perpendicular to three-dimensional space. To start from the beginning, however, means starting with a single point in space. A point defines the zeroth dimension. Essentially a point has not height, width or length, and is infinitesimally small. 1-D is defined by a single line. 2-D is the x-y plane, defined by two perpendicular lines. It is useful to picture the different structural components of a cube when examining dimensions, and in the case of 2-D, the cube is just a square. Similarly, 3-D is the x-y-z plane, with three lines extending perpendicular to each other in space, creating a cube. The fourth dimension becomes tricky because it is difficult for our 3-D minds to visualize a spatial direction perpendicular to our three existing ones. A four-dimensional cube is known as a tesseract, and it is typically drawn as one cube inside a slightly larger cube with the vertices of both cubes connected. (1 [2]) This is, of course, a distorted image of a tesseract because it cannot be drawn in three dimensions, but it is a useful tool in trying to visualize or at least understand the geometry of the fourth dimension.
Another useful method of comprehending the fourth dimension is to note how objects are bounded in each dimension. In 2-D, a square is bounded by one-dimensional objects, lines. In 3-D the cube’s boundaries are composed of six squares. Similarly, in 4-D, a tesseract is surrounded by eight cubes, eight objects with volume. Additionally, looking at the shadows objects of differing dimensions project can be an interesting way to compare dimensions. Think of your own shadow: your 3-D body projects a 2-D image, and likewise, a 4-D object will project a 3-D object. (7 [3])
With a working knowledge of the basic concepts behind the construction of each dimension, and the geometry of each, it’s possible to begin to understand perception differences between the dimensions. Again, it’s useful to begin by examining lower dimensions so that we can readily visualize before moving on to explanations about the fourth dimension. Think of a square drawn on a sheet of paper. You can simultaneously see all of the points of square. If the square has dots drawn in the center of it, you can see these, even though they are on the inside of the square. For a being that can only perceive two dimensions, this would not be possible (more detailed explanations and examples of this below). Likewise, a being that can perceive four dimensions would be able to see all of the points in three-dimensional space “at once.” Our insides would be just as easily exposed as those of the square. Our perception abilities (and limitations) are a result of the arrangement of receptors in our retina: they are organized in a two-dimensional array (the third dimension is perceived through indirect information and visual cues such as shadowing). (7 [3]) A theoretical being who could visualize four dimensions would have a three-dimensional array of receptors.
Getting into more specific examples about perception differences between the dimensions and their inhabitants illuminates some fascinating spatial anomalies in our 3-D world that do not seem physically possible based on our understanding of space, but are completely valid within the context of the fourth dimension. First, however, as usual we begin with discussion of 2-D space and perception. A two-dimensional being experiences restrictions on movement and perception compared to beings that can perceive three dimensions. For example, in terms of movement, a 2-D being can only move along two axes in space. Let’s say this is forward and backwards, as well as up and down. This being has no concept of moving left or right. Think of a 2-D drawing on a piece of paper: the figures are “trapped” within the plane of the page. In this same scenario, perception of even a 2-D object, a square, is limited, and depends on its orientation in relation to the 2-D space the being occupies. If the square lies within the 2-D plane outlined, then the 2-D being looking at the square will only see a line, the boundary of the square facing it. This 2-D being can perceive a square by imagining many lines stacked together, similar to how we see in two dimensions but can perceive three. (2 [4]) However, if the square is perpendicular to the 2-D plane, then our being will see two dots, the points where the lines of the square intercept the plane. Likewise, a cube can appear as a line (where two sides of the cube meet), a square (one side of the cube), or a single point (one of the vertices of the cube) within 2-D space, depending on its orientation within the 2-D space. Three-dimensional objects can only appear as cross-sections of themselves within two dimensions, and their shape will depend on their orientation in relation to the plane of the two-dimensional space. It is possible to use the 2-D to 3-D transition model as an analogy for what happens when we, as 3-D perceivers, encounter four-dimensional objects. Coming across a tesseract, for example, we would see it simply as a cube. (3 [5])
Existing in three dimensions gives us a third option for movement that doesn’t exist for 2-D beings. Because of this option, it is possible for 3-D inhabitants to move around objects in 2-D. One interesting phenomenon is object rotation. Imagine a painting hanging on the wall. You can take this painting and flip it along one of its edges so that the picture is now facing the wall. Now imagine the painting has no depth to it. Flipping such an object in two dimensions results in the creation of the mirror image of the original object being rotated. For a being in 2-D, this seems impossible because this kind of rotation cannot happen in only two dimensions. The same thing is theoretically possible for cubes in the third dimension. If a being in the fourth dimension wished to rotate a box in the third dimension, the axis of rotation would be a plane (one step above the line used in 2-D), and the box would return as a mirror-image of itself. If you were to write a letter on each side of the box, for example, the box would be returned to you with the letters flipped and the sides they were on reversed. (4 [6]) Trying to imagine this happening is mind-boggling. For our 3-D oriented brains, such a transformation is not spatially comprehendible.
Another interesting event is the locked-safe example. Say our 2-D being has its 2-D treasure stored away in a 2-D safe. As members of the third dimension, we can see the treasure inside the safe because we can see all 2-D points simultaneously. Additionally, we can move in and out of the 2-D plane as we choose, and move around it by moving in our third spatial option. (5 [7]) Because of this option, it is possible to take the treasure out of the safe without “opening” it. Theoretically, the same thing is possible for a 4-D being. What we perceive as “solid” is no longer so. Any stability we perceive in our environment can be circumnavigated within the fourth dimension.
The implications of the fourth dimension are profound. Once again we find that our reality, our view of the environment around us is not complete. Not being able to see four-dimensional events because of our receptor arrangement in the retina is very similar to our inability to view ultraviolet colors because we are lack a fourth color-sensitive cone. Four-dimensional activity is potentially more disturbing than just missing out on colors: the laws of motion and direction within a 4-D space are drastically different from what we are used to. Creating the mirror-image of a box, altering its seemingly stable physical appearance, simply by rotating it is eerie and fascinating. Other differences occur between the dimensions, such as the change in porosity. The higher in the dimensions you go, the more porous things become because of the additional directions for things to slip through. (6 [8]) A fourth dimension is only the tip of the iceberg as far as dimensions are concerned: “The mathematics used in superstring theory requires at least 10 dimensions. That is, for the equations that describe superstring theory to begin to work out—for the equations to connect general relativity to quantum mechanics, to explain the nature of particles, to unify forces, and so on—they need to make use of additional dimensions. These dimensions, string theorists believe, are wrapped up in…curled-up space…” (8 [9]) The possibilities are unimaginable for what kinds of actions are possible, and what laws hold true in these dimensions.
Sources
1. Jones, Garrett. “Introduction to the fourth dimension.” Avail 30 April 2007. http://tetraspace.alkaline.org/introduction.htm [2]
Describes components of each dimension, introduces the tesseract
2. Jones, Garret. “From the second dimension to the third dimension.” Avail 30 April 2007. http://tetraspace.alkaline.org/page1.htm [4]
Describes events and differences in perception between the second and third dimensions
3. Jones, Garrett. “From the third dimension to the fourth dimension.” Avail 30 April 2007. http://tetraspace.alkaline.org/page2.htm [5]
Describes events and differences in perception between the third and fourth dimensions
4. Jones, Garrett. “Rotation.” Avail 30 April 2007. http://tetraspace.alkaline.org/page3.htm [6]
Describes rotational events in both second and third dimensions
5. Jones, Garrett. “Flatness and Levitation.” Avail 30 April 2007. http://tetraspace.alkaline.org/page4.htm [7]
Describes the “locked safe” event
6. Jones, Garrett. “Bodies of Water.” Avail 30 April 2007. http://tetraspace.alkaline.org/page8.htm [8]
Describes water in multiple dimensions, and the composition of matter in multiple dimensions
7. “Fourth dimension.” Avail 30 April 2007. http://en.wikipedia.org/wiki/Fourth_dimension [3]
General introduction to the fourth dimension, as well as a lot of interesting tidbits, such as the implication of different dimensions on shadow projection
8. Groleau, Rick. “Imagining Other Dimensions.” Avail 30 April 2007. http://www.pbs.org/wgbh/nova/elegant/dimensions.html [9]
Article discussing multiple dimensions and their applications to string theory
Comments
holding a tesseract in your hand [10]
so if we were to hold a miniature model of a real 4-dimensional object in our hand such as a tesseract, does that mean that we would only feel (and more importantly see) only a solid cube that we are holding in our hands even if it is a real bonafide 4 dimensional object?
Not on your palm , maybe IN it. [11]
First things first , you can't hold the Tesseract on your palm. He has more dimensions that ur palm does. He may choose to stand on a position that co-ordinates to ur palm , but not on it physically. Thats because , although we only perceive 3d (yes we can only see his 3d form) he exists in a 4d space , his fourth dimension just not visible. Its like an apple thats half way through a plane. While it has its other dimension , 2d objects in the plane perceive it as only 1d with possible cues as to its 2nd dimension , just like we can see in only 2d with cues to the 3rd dimension.
Suppose this apple moved up and down in through the plane , the 2d creatures will perceive a series of 1d projections aks lines (corresponding to where the apple's form intersected with the plane). Something similar would happen if we were to interact with the Tesseract. We might see a series 3d forms - thats only theory of course , bur you get the essence.
Did 2nd dimensional eyesight [12]
Did 2nd dimensional eyesight evolve into 3rd dimensional eyesight and acquiring knowledge of the 4th dimension is limited to an evolutionary process? How do physics, biology and philosophy and their various theories (string theory, quantum physics, evolution, etc.) play into this 4th dimension discussion?
quite possible. if everything [13]
quite possible. if everything evolved from single cell ameoba, it would take trillions of years for human to evolve into a fourth dimensional being. But is it possible that fourth dimensional beings are all around us? on earth?
There's still depth! [14]
We evolved from unicellulars - true that. But even they have a 3d factor , depth - say in the order of 10 power -20 or even -1000. Thats still depth. So we have always been evolving into better 3d beings not progressing up the "d-scale". If however the process of evolution is convoluted by a human effort to reach for the fourth d , then perhaps we can all one day be 4d boogey-man creatures , just like the deer becoming giraffe story.
yo [16]
What if we arrage our receptors in the retina in a 3-d pattern; can we see 4-d?
yo [17]
No, because they would still be pointing into the third dimension, not the 4th. Consider the lower-dimensional analogue: Can a square, in a flat universe, arrange it's receptors in 2d, in order to see in 3d? Even if you spread out the receptors into a circle, they would still be pointing into the plane of the square's universe (you really want the circular group of receptors to be seeing "up", not forward). In fact, just like in real life, the receptors at the back of the circle would receive no light at all, since they would be in the shadow of those in front. In order for the square to see into the 3d world, his receptors would have to be 'tilted', (or better, tiltable) out of the plane of his flat universe and into the 3rd dimension. Same with us, if we wanted to see in the 4th dimension, we would need a way for our eyes to rotate in the 4th dimension. As far as I know, this is an ability we humans do not possess...