The Tesseract in the Fourth Dimension: Mental Exercises

It is a four-dimensional cube with four faces that is called a tesseract. It was Einstein's seminal theory of General Relativity that presented the idea of a four-dimensional spacetime. When it comes to subatomic particles, there are more dimensions beyond the three we're famil

The three dimensions of length, breadth, and depth define our immediate surroundings. Just like the back of our hand, these three dimensions are natural and familiar.

Science, on the other hand, often has to move beyond the three dimensions that we're most acquainted with. Four-dimensional space-time was proposed by Einstein in his monumental theory of General Relativity. There are symmetries and dimensions beyond the three we're used to when it comes to subatomic particles. At the beginning of the Big Bang, astronomers postulate additional dimensions, which were then reduced to our present three spatial dimensions, height, breadth, and depth. '

 

Consequently, additional dimensions play a significant role in the development of robust scientific ideas. But because we are limited to three dimensions, we have a hard time imagining other dimensions. In our minds, there's no room to add another dimension.

 

In order to overcome our limitations, let's try some mental acrobatics, and see if we can imagine more dimensions. We will use a particular item, a tesseract or four-dimensional cube, to study a fourth spatial dimension. A tesseract is both familiar and unfamiliar, since it contains sides that are squares like a cube and lines that intersect at right angles like a cube; it is both familiar and foreign. Tesseract, on the other hand, is obscure due to its rarity as a geometric form and the fact that a tesseract needs four directions of space.

 

Increasing the Number of Lines

 

As previously stated, a tesseract is a four-dimensional cube with four faces. A tesseract, on the other hand, has four dimensions - w, x, y, and z - whereas a conventional cube has three. A tesseract is a four-dimensional shape made of of lines that cross at right angles.

 

In what ways may we assemble and depict a tesseract? Let's start with a basic, well-known item, in this example a line, and then simply add additional lines to that line to get a tesseract.

 

So begin by drawing a horizontal and vertical line in front of you. According to geometry, the line exists only in one dimension. With a finite line, which means it doesn't go on forever, we may have two end points on our line. A foot, a meter, or the length of a six-inch ruler are all good options for the length of the line segment.

 

To complete our tesseract, we must now successively add line segments.

 

The first step is to add two lines perpendicular to the original line at each of the end points. A counter top with the initial line going left and right would have these additional lines traveling away from us as well. These perpendicular lines form a U-shape, with the entrance facing away from us. Add a third line to the free ends of the previously connected ones, and join them together (i.e. close the opening). A square has been formed.

 

Figures like squares and rectangles have four corners and lines on each side to keep track of. The intersection of two lines is the place at which each corner is located. We've progressed from a single dimension to two (or 1D to 2D).

 

Keep up the good work. Add a line perpendicular to the square to each of the square's corners. The counter top will now have four more lines. It looks like a four-legged table that's been placed upside down on the counter top by the addition of these four lines. Add lines to link the four perpendicular lines' free end points. There will be a need of four. As a result, we now have a cube.

 

We now have eight corner points, twelve lines, six square surfaces, and one cube to keep track of. Three lines and three squares meet at each corner point. We've gone from two dimensions to three (or 2D to 3D).

 

You may want to look up photographs of squares and cubes on the internet to get a visual representation, as well as ensure that you can count the number of corners, lines, and squares.

 

Keep up the good work. However, be ready, as the fourth spatial dimension is about to be entered (which exists mathematically despite not existing in our visual field).

 

Add a line to each of the cube's eight corners. These lines must now be drawn diagonally outward from each of the eight corner points rather than perpendicularly (as we should, but our visual dimensions have been depleted). This offers us a cube-shaped spacecraft with eight antennas protruding in eight distinct directions, like a cube in orbit.

 

We now have eight free points, one at the unattached end of each of the newly added lines, in this structure. In order to form a cube, link the eight free end points with additional lines (twelve in all) that demarcate the eight free end points. There is a bigger cube that encircles the previous cube in the process.

 

Finally, we've got our tesseract in hand. As with the cube and square, pictures of a tesseract will be useful.

 

Look at the picture. The cube-within-a-cube structure may be seen in the most frequent picture with a little focus. There are twelve trapezoid-shaped interior surfaces that link the inner cube to its exterior counterpart. Six trapezoid-shaped cubes are defined between the internal and exterior cubes by those interior surfaces. There are six sides to the trapezoid-shaped cubes, four of which are made up of trapezoids stretching from the larger exterior cube to the smaller inside cube. It should be noted that the trapezoids in a true tesseract are perfect squares, but they are transformed into trapezoids due to the restrictions of what we can depict in this example.

 

It's now possible to maintain track of 16 corner points and 32 lines, as well as 24 squares and 8 cubes. Four lines, six squares, and four cubes come together at each corner point to form the point's shape.. Despite the fact that the drawing is three-dimensional, we have now transitioned to four dimensions (so 3D to 4D).

 

From Lines to Beams.

 

A tesseract may be constructed in a rational manner by following this progression of adding lines. However, we drew the last eight lines, the crucial lines extending into the fourth dimension, as diagonals in our now existent three dimensions. In order to save time, we drew the four-dimensional lines as diagonals in three-dimensional space instead of painting them as lines in the four-dimensional space. We learned how to build a tesseract logically, but only had a limited understanding of the fourth dimension.

 

What can we learn from this logical progression of building a tesseract in order to increase our experience of our fourth dimension?

 

The only way to achieve this is to actually go through the actual procedures of building an actual tesseract out of solid stuff. As a structural component, we've decided on steel beams. Strong, weighty steel beams are ideal alternatives for providing an emotional and visual image. So, if one were to use steel beams to build a true tesseract, what would one do?

 

Take it for granted that we can visualize the construction of a three-dimensional cube out of steel beams, which is exactly what I am about to do. Four square beams would form the foundation of the cube, four columns would form the uprights, and four more beams would form the top square. There are a total of twelve beams, with three beams at each of the eight corner locations. The beams are all set at a 90-degree angle.

 

What should we do next? What will our next step be when we stand in front of the cube? We're going to do something no one has ever done before for our next move (and not yet possible, and maybe always impossible). We shall rotate out of the previous cube's three dimensions and enter a new set of three dimensions from our current vantage point in front. In other words, we're going to pivot from our beginning x-y-z set of dimensions to one that contains our fourth dimension via the looking glass. The x, z, and w measurements will be used to define a new space. This might be done by passing via a "Stargate"-like portal.

 

Think about that for a moment. Our beams, cranes, welding torches, and bodies all exist in three dimensions. We can't make them into anything else than three-dimensional images. Given that our beginning cube has three dimensions (x-y-z), if we want to expand it into a fourth dimension, we must relocate our apparatus and ourselves into an area that includes that fourth dimension. And given that everything in the structure (including ourselves) is fixed as three-dimensional objects, we need to leave a dimension behind if we want to add the w-dimension.

 

Our focus shifts to width as we leave behind the y-dimension of depth.

 

The new place we enter is almost identical to the one we left when we do this imaginary rotation and move to it. Our equipment works, the noises and sights are the same, and gravity is at work. We can hear what our coworkers in the building trades are saying. They look and feel just as they did before we left our old home.

 

However, there is one object that jumps out in a big way. It seems that eight of the twelve rays in our cube have been extinguished. Why? Keep in mind that we have to delete the "y" direction before picking up the "w" direction. As a result, we can no longer see the eight columns that were previously visible in the depth. Only height and width are available from the x, y, and z coordinates.

 

There are four beams, two of which are upright and two of which are horizontal, that form a symmetrical square in our new three-dimensional view. Our three-dimensional world doesn't include the other columns, yet they're still there.

 

As we watch the four beams, we discover something odd. We can only glimpse the surface of things; we have no idea how deep they go. There are four facades floating in the air that look to be made of paper. Why? Keep in mind that in order to get the w-dimension, the y-dimension had to be given up. We have so lost visual contact with the eight other beams, as well as with the depth of the four beams that we can see in part.

 

Four horizontal (or perpendicular) beams are erected from the corners of the square formed by the four visible beams, since we are now comfortable in our new three-dimensional environment. Because the x-z plane is perpendicular to the square formed by the four visible beams, our new beams must extend into the w-dimension in order to be perpendicular. As a last step, we construct four parallel beams between the just-installed ones. This creates a cube in the x, z, and w dimensions.

 

We have finished our job and are satisfied with our dimensional pivot, so we pivot back to the original x, y, and z coordinate systems. Our "looking-glass" pivots us into a separate 3D environment, this time in the x-z-w coordinate system, but with a different "y" coordinate. We go around the back of the previous cube. Whereas before we left the y-dimension behind at y = 0, we now leave it at the rear of the cube at y = 1 instead.

 

We've now had a taste of what it's like to live in 4D. Because the range of y-values is so wide, a distinct 3D x-z-w space exists for each one. You may go on and on like this: y=1.11, 1.111, etc. Many three-dimensional areas there. To understand how many 2D squares are contained inside a 3D cube, consider the comparison.) However, in our x-z-w space at y=1 we attach four extra beams, perpendicular to the back of our original cube, and link the free ends of those beams with four beams. As a result, we return to the x-y-z coordinate system.

 

There are four connecting beams at the ends of the eight perpendicular beams that are missing. Only eight of these enclosing beams have been constructed, when we require twelve of them. To pivot, we go to the cube's left side (where x=0 is). The y-z-w coordinate system replaces the x-y coordinate system in this pivot. Four beams protrude out from the cubes, but only two beams join the open ends. We completed the installation of the last two connecting beams. On the right side of the initial cube (x=1), we pivot to the y-z-w space (where x=1), and create the two remaining beams.

 

So far, our efforts have resulted in eight beams extending from the original cube, and twelve connecting beams at the free ends of those eight additional beams. Our tesseract is now complete. As a result, we followed these steps:

 

In our x-y-z area, we built twelve beams.

Additional x-z-w space beams were added to the front of the structure.

Perpendicular and surrounding beams were added in an entirely new x-z-w space.

On each of my two trips to the y-z-w spaces, I added two more enclosing beams.

We used a fictitious gateway to traverse between the various 3D environments.

 

3D in a 2D area, thanks to the magic of the traveling cube.

 

Let's not stop now. To construct our tesseract, we've traversed a variety of 3D environments. We want to appreciate it from a distance. What does it seem to be like?

 

Our tesseract now exists in 4D, whilst we only exist in 3D. This is a stark contrast. As a result, we are only able to perceive a 3D piece of the tesseract rather than the whole thing at once. But even if we can only see a piece of the three-dimensional scene, we may switch between various views by swiveling the tesseract in front of us. When it comes to imagining a moving 4D object, we're likely to have little (if any) experience with it. For now, let's try seeing an item of a higher dimension traveling across a place of a lower dimension to improve our skills. We'll take a step back and see a cube in 3D from the perspective of a being who sees things in 2D, i.e. from a place on a flat plane.

 

We now want some background information about a two-dimensional creature. Such an entity could only look in front and to the side, not up or down, and it would be confined to a two-dimensional plane. To see this, visualize a three-foot-high flat plane. It was possible for our 2D self to move about and view the sides of anything that penetrated the plane, but we were unable to see or move above or below it. This may seem to be restrictive, yet this 2D creature would know nothing other. We don't feel constrained by not being able to see in 4D, just as we don't feel constrained by knowing just in 3D.

 

Imagine a cube hovering above the 2D plane in front of you as we practice seeing items of a higher dimension move through an environment of a lower dimension (the one floating three feet above the ground that is home to our 2D being). Our 2D self is aligned (parallel) to the bottom face of the cube, which is about a foot on each side. The sides of the cube are empty, and the cube itself is nothing more than a framework.

 

The cube should now begin to gently descend. The cube's bottom face will eventually come into contact with the aircraft.

 

The smooth descent of the cube is all that is required for this point of intersection of the cube face with the plane to appear to us. However, what does the 2D creature perceive? A two-dimensional creature can only perceive items on the plane, hence he can't notice the cube coming from the side. The 2D creature can only perceive the cube when it touches the square.

 

Because of this, while you're in a two-dimensional square, the bottom face of a cube suddenly appears in front of you. Suddenly, the bottom face bursts out of the thin air, appearing as an item.

 

The cube is now being moved in a clockwise direction. It seems to us that the cube's top and bottom faces have crossed across the plane of the 2D creature, and the cube's front face has moved beyond it. As far as we are concerned, the cube is merely continuing its downward trajectory.

 

A 2D person's bottom face vanishes just as suddenly. Due to its limited field of view, the 2D creature cannot see the full cube. As a result, the bottom face of the cube may be followed in 3D, but the bottom face of the cube cannot be followed in 2D since the surface of the plane is opaque.

 

What does the 2D entity perceive now that the bottom face has been removed? The 2D creature is only able to perceive the portion of the cube that is directly on top of the plane. How does the cube's bottom face meet the plane after it's passed? Those four lines that link the bottom face of the cube to the top face. And I'm not talking about all the way down here. The 2D creature is only able to view the four points that make up the two-dimensional square. So, exactly like the bottom face, these four points would float out in thin air, shimmering weirdly like the bottom face.

 

By now, you've undoubtedly had an idea of what's going to happen next. On its way down, the cube's descending following top face comes into contact with the plane of 2D geometry. As with the bottom face, the 2D entity now watches the top face appear and then vanish of its own own.

 

The 3D cube's enigmatic appearance to a 2D entity begs the question: why? Because of this, this example explains why. Seeing just 2D slices, the 2D creature can't see ahead to see what's coming. Objects that can be moved into the environment of a 2D creature have an excellent hiding spot in the third dimension.

 

Let's imagine for a moment what a 2D creature would observe if the cube traveled across the 2D plane at a 45-degree angle. Otherworldly cubes may be tilted and rotated such that one of its eight corners points towards the 2D plane.

 

It would be impossible for the cube to be aligned with the surface of the 2D square in this situation. Initially, the 2D creature would see the cube as a single point, followed by three, then six, then three, and finally one, as the tilted cube travelled across its 2D space.

 

A 4D Tesseract in a 3D Space:

 

This idea of higher-dimensional things travelling through lower-dimensional space, let's look at the tesseract.. The tesseract would travel across our 3D slice while we stood on the edge of a huge open space, such as a park. With the tesseract in line with our space, we'll get the ball rolling. Our 3D is aligned with the tesseract's x, y and z dimensions (x,y and z).

 

As a parallel to the two-dimensional scenario, we could expect to see a section of the tesseract spring out out of nowhere, dangling in mid-air over the park. That's exactly right. The front section of the tesseract would not be visible until it reached w=0, since we cannot look into the fourth dimension. If the tesseract travelled from w=1 to w=0, where our x-y-z space is located, we would not be able to see it.

 

Would we see anything? We were able to construct a perfect cube in any three-dimensional space thanks to the tesseract. The front cube of the tesseract would appear in a flash if the tesseract was properly aligned. When we move a cube along a two-dimensional plane, the square suddenly appears in the middle of the cube.

 

It would be gone in a split second. Now what? Eight unconnected spots would now be visible in midair. Essentially, they are beams that link together the two front cubes, and they're also beams that comprise the three-dimensional cubes themselves. Mathematically, these additional cubes may be found in the w-y-z-z, wx-z, and wx-y-z spaces if we are in the x-y-z-z space.

 

The rear cube would emerge in a second flash. Remember, a tesseract contains eight cubes, just as a cube has six squares on the perimeter. The x-y-z space contains two of such cubes. (There are two cubes in each of the other three-dimensional spaces, i.e. w-y-z, w-x-z, and w-x-y.)

 

A significant comparison for the 4D tesseract going through a 3D space may be drawn from our picture of the 3D cube passing through 2D space. 2D object (square) may be constructed by traveling through 1D space (line). The comparison is appropriate. There are only slices of the whole thing visible in lower-dimensional space when an object of higher dimensions travels across it. As we've seen, a lower-dimensional entity is unable to notice the approaching item.

 

The tesseract point is now tilted forward. We would no longer be able to see the tesseract's extended pieces if we tilted the cube point forward. Since the tesseract's slanted rays penetrate our three-dimensional space but never align with it, we could only perceive points. One point, four points, twelve points, four points, one point, and so on. We wouldn't be able to perceive a cube, square, or line if the tesseract was completely tilted.

 

The tilted tesseract has end points with various w coordinate values if our x-y-z space is at coordinate w=0 mathematically.

 

We could get squares to line up if we slanted the tesseract just a little bit. When the tesseract is slightly tilted and aligned correctly, two squares will emerge, with their sides parallel. There were no more startling appearances of the figures, which hung in the air like ghosts.

 

The Panoramic Lens

 

Let us now briefly discuss another method of conceptualizing a 4D object. Space movies often commence with a long, wide view of an intergalactic ship. We can't see the whole ship at once since the camera is so near and the ship spans so far.

 

If we go with that, we may imagine ourselves on a tiny space shuttle, shooting down from a few feet over the hull of a massive space cruiser. Since the ship's different appendages and shapes would very probably obstruct our ability to see much beyond, our x-y-z range of vision would be restricted to a few dozen yards in any direction.

 

It's possible to see the whole ship at once, moving up and down as well as around and in and out, as long as we don't go too close to the hull. Time, like the w-spatial direction, operates as a fourth dimension here. In the fourth dimension (here, time), we can view the whole ship from our spaceship. However, even if we cover the whole ship, we may never be able to piece together the many little pieces we witness into a coherent whole since the ship's outer shapes may be so intricate and vast that we may never be able to put them all together.

 

The tesseract may be compared to this analogy. At first, we can't make sense of what is going on. Disjointed enough 3D bits we can see prohibit us from creating an image of the item as a whole.

 

Significance

 

Is there a point to this foray into 4D visualization? Perhaps this mental activity is intriguing, or challenging, or elicits curiosity, or acts as a distraction or time-filler for certain people.

 

Is there a broader purpose to seeing in many dimensions?

 

Physicists, astronomers, and mathematicians are likely to say "yes" to this question. A mental image isn't necessary for such persons while working with solely mathematical statements with additional dimensions. Mathematical formality is useful, but mental images help illuminate the underlying dynamics. Images in the mind help us make intuitive and logical leaps, as well as reveal patterns and answers we would not have seen on our own.

 

Is this a worthwhile endeavor for someone who isn't interested in serious research? Yes, I'd say so, without a doubt. Popular science publications referring to more dimensions would be encountered by anybody seeking some kind of conceptual completeness in exploring the world. A mental tool kit is needed to integrate these allusions together and into other conceptions, such as those from theology (where may God be?) or metaphysics (what is existence and what does it mean to exist). As with any tool, the ability to handle 4D visualization is beneficial even if it is not at the top of the list. When you use a tool, your general ability to utilize all tools, or in this example, your general ability to manipulate a wide variety of ideas (whether in 4D or not) grows.

 

Think of this study of 4D and tesseracts as something you may use to improve your mental agility in general, rather than solely for its own sake.


Larry Gaza

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