You are searching about Define Completing The Square In Math, today we will share with you article about Define Completing The Square In Math was compiled and edited by our team from many sources on the internet. Hope this article on the topic Define Completing The Square In Math is useful to you.
Exercising the Mind: Tesseract in the Fourth Dimension
Height, width and depth – these delineate the world around us. These three dimensions are as natural and familiar, as well, just about anything, even the back of our hand.
However, science sometimes, in fact often, needs to go beyond these familiar three dimensions. Einstein, in his epic theory of General Relativity, postulated, with great success, a four-dimensional space-time structure. Physicists, for sub-atomic particles, operate in dimensions, and symmetries in dimensions, beyond our familiar three. When astronomers talk about events at the very, very beginning of the Big Bang, they hypothesize added dimensions, dimensions which collapsed down to our current slate of three spatial dimensions, the proverbial height, width and depth.
Thus, extra dimensions play a strong role in making sense of the world when building rigorous scientific theories. But bound as we are to our three dimensions, we have difficulty conceiving dimensions beyond our familiar three. As we build mental pictures, we just have no place to readily put an added dimension.
So let’s do a bit of mental gymnastics, and see if we can hurdle our mental constraints on picturing added dimensions. Our approach will be to examine a fourth spatial dimension, and do so through an examination of a specific object, a tesseract or four-dimensional cube. That object is both familiar and unfamiliar; a tesseract is familiar in that it is in the cube family, i.e. it has sides that are squares like a cube, and lines that join at right angles like a cube. A tesseract, however, is unfamiliar in that the tesseract is a geometric figure rarely mentioned, but more importantly in that a tesseract requires four spatial directions.
As just noted, a tesseract is a cube in four dimensions. So while a regular cube has three dimensions – generally labeled x, y and z in math terms – a tesseract has four – w, x, y and z. A tesseract is thus a figure composed of lines running at right angles in a four-dimensional space.
How can we construct and visualize a tesseract? Let’s start with a simple, familiar object, in this case a line, and then extend that line to a tesseract by simply adding more lines.
So start with a line, simply lying in front of you, with the line running left and right. The line, if you recall your geometry, exists in one dimension. We will use a finite line, i.e. one that does not run out forever, and thus our line will have two end points. As you build the mental picture, let the line segment be any convenient length, say a foot, or a meter, or the length of a small ruler, i.e. six inches.
Now let’s sequentially add line segments to construct our tesseract.
First, add a line at each end point of the original line, with the added two lines extending perpendicular to the original line. We can imagine the original line on a counter top, as noted running left and right, and we would put these added lines on the table also, running away from us. Adding these perpendicular lines gives a U-shaped figure, with the opening away from us. Now connect the free ends of the two added lines with another line (i.e. close the opening). We now have a square.
In terms of keeping track, our figure, our square, contains four corner points, four lines, and one square surface. Each corner point is the intersection of two lines. We have gone from one to two dimensions (or 1D to 2D).
Keep going. To each corner point of the square, add a line, extending perpendicular to the square. These four added lines will now extend up from the counter top. The addition of these four lines creates a figure like a four-legged table lying upside down on the counter top. Now connect the four free end points of the perpendicular lines with added lines. Four will be needed. That closes in the figure to give us a cube.
In terms of keeping track, we now have, with our cube, eight corner points, twelve lines, six square surfaces, and one cube. Each corner point is the intersection of three lines, and also of three squares. We have gone from two to three dimensions (or 2D to 3D).
Note at this point, you might search the web for images of squares and cubes, so you have a visual picture, and also check that you can count the number of corner points, lines and squares.
Keep going. But get ready, since we are now entering the fourth spatial dimension (which exists mathematically despite not existing in our visual field).
Okay, to each of the eight corner points of the cube, add a line. Now we can’t place these lines perpendicular (we should, but we have exhausted our visual dimensions), so draw theses lines running diagonally outward away from each of the eight corner points. This gives us a figure that could be analogous to a cube-shaped space satellite with eight antenna sticking out in eight different directions.
As you visualize this construction, we now have eight free points, one at the unattached end of each of the added lines. With a bit more visualization, we see that the eight free end points demarcate a cube, so connect the eight free endpoints with added lines (twelve in total) so as to create that cube. That added cube sits as a larger cube that encompasses the cube from the step before.
We now have our tesseract. Again, as with the cube and square, it will be helpful to search for images of a tesseract.
Study the image. In the most common image, with a bit of concentration, you can see the cube-within-cube structure. You can also see the series of twelve trapezoid-shaped internal surfaces connecting the inner cube to the outer cube. Those internal surfaces define six trapezoid-shaped cubes between those internal and external cubes. The trapezoid-shaped cubes consist of a side from the larger external cube, a side from the smaller internal cube, and four sides from internal web of trapezoids extending between the larger and smaller cube. Note, in an actual tesseract, the trapezoids are perfect squares, but become trapezoids given the limitations of what we can draw.
In terms of keeping track, we now have 16 corner points, 32 lines, 24 squares, 8 cubes, and of course one tesseract. Each corner point is the intersection of four lines, six squares, and four cubes. Though the drawing is in three dimensions, we have gone from three dimensions to four (so 3D to 4D).
From Lines to Beams
This construction sequence, of progressing adding lines, shows – logically – how a tesseract can be built and what components it contains. But we drew the last set of lines, the set of eight, the critical lines extending into the fourth dimension, as diagonals in our existing three dimensions. Though logically sufficient, we took a short cut (drawing the 4D lines as diagonals in 3D) on the very step of interest, the step involving the fourth dimension. We thus gained a logical procedure for constructing a tesseract, but probably only a partial intuitive grasp of the fourth dimension.
So from this logical sequence of constructing a tesseract how can we strengthen a visceral sense of our fourth spatial dimension?
Let’s do that by actually progressing through the real-life steps needed to construct a physical tesseract with solid material. We will pick steel beams as our structural element. If anything can provide a visceral and visual picture, then strong, hefty steel beams qualify as leading candidates. So what would one actually do in constructing a real tesseract with steel beams?
Let’s assume, correctly I will presume, that we can picture the creation of a three-dimensional cube of steel beams. We would need twelve beams, four in a square at the base, four upright as columns, and four more for a square at the top, to create the cube. We have twelve beams, and eight corner points with three beams each. All the beams are at right angles.
How would we now proceed? Standing in front of the cube, what would our next move be? For our next move, we will execute a move never humanly done before (and not yet possible, and maybe always impossible). From our position in front, we will pivot out of the three dimensions of this original cube and emerge into another set of three dimensions. This will be a “through-the-looking-glass” pivot and move us out of our starting x-y-z set of dimensions to one containing our forth dimension. We will pivot to a space defined by the x-z-w set of dimensions. We can imagine going through a “Stargate” type portal to accomplish this.
Let’s think about this. Our beams, our cranes to lift the beams, our welding torches, our physical bodies, exist in three dimensions. We can’t transform them into four-dimensional objects. Thus, if we are to extend our cube into a fourth dimension (the w-direction, given our starting cube began in the x-y-z set of three dimensions), we must move our machinery and ourselves into a three-dimensional space that contains that fourth dimension. And given, as just noted, that all the construction items (including ourselves) are decidedly fixed as three-dimensional objects, if we are to add the w-dimension, we need to leave a dimension behind.
We thus leave behind the y-dimension, depth, to pick up the w-dimension.
When we execute this hypothetical pivot, and arrive in our new space, we enter a space just about as normal as the one we left. Gravity works, our equipment works, sounds and sights are the same. Our fellow construction works talk and we hear. Our surrounding environment has the same three-dimensional look and feel as the one we left.
One item, though, stands out as seriously different. We look at our cube, and eight of the beams (of the twelve) in the cube have vanished. Why? Remember, in picking up the “w” direction we needed to remove the “y” direction. This removes the depth we previously could see, which was where the eight columns resided. We only have height and width, from the x-y-z space.
Thus, in our new set of three dimensions, we would see just the four beams, two upright and two horizontal, that make up the square at the front of the initial cube. The other columns of course still exist, but they are outside our three dimensions.
We also notice something unusual about the four beams we can see. We only see their front surface; we can not see any depth to them. The four beams appear as four facades floating in space, as thin as paper. Why? Remember the depth dimension, the y-dimension, was relinquished to acquire the w-dimension. We thus have lost not only visual contact with the eight other beams, but also with the depth of the four beams we can partially see.
Now at ease in our new set of three dimensions, we undertake construction of four beams extending horizontally (aka perpendicularly) from the corners of the square created by the four visible beams. The square created by the four visible beams lies in the x-z plane, so to be perpendicular our new beams will extend out into the w-dimension. We then complete our work by building four beams between the free ends of the perpendicular beams just installed. This gives us a cube in the x-z-w space.
We complete (and admire) our work, and as we are now reasonably comfortable with our dimensional pivot, we pivot back to the original x-y-z space. We walk around to the back of the original cube, and pivot in the same “through-the-looking-glass” fashion into a different 3D space, this time still the x-z-w space, but a different “y” co-ordinate. While previously we left behind the y-dimension in the front of the cube (at y = 0), we now leave behind the y-dimension at the back of the cube (at y = 1).
We have now experienced a key nuance of 4D space. The space is sufficiently broad that a different 3D x-z-w space exists at every different y-value. Every value, i.e. y=.1 and.11 and.111 and so on. That’s lots of 3D spaces. (Think by analogy how many 2D squares exist stacked inside a 3D cube.) In any event, in our x-z-w space at y=1, we attach four more beams, perpendicular to the back of our initial cube, and also connect the free ends of those beams with four beams. We pivot back to our original x-y-z space.
We do have something missing, specifically four connecting beams at the end of the eight perpendicular beams. We built only eight of those enclosing beams, and we need twelve. We go to the left side of our cube (which is x=0), and pivot. For this pivot, we leave behind the x-dimension, and enter a y-z-w space. We do see four beams extending out from original cubes, but only two beams connecting the free ends. We put the other two connecting beams in place. We pivot to the y-z-w space on the right side of the original cube (where x=1), and construct the two remaining beams at the free ends.
So our efforts have built eight beams extending from the original cube, and twelve connecting beams at the free ends of those eight new beams. This completed our tesseract. In doing so, we executed the following steps:
- Constructed twelve beams in our home x-y-z space
- Added four perpendicular beams and four enclosing beams in the x-z-w space, in the front
- Added four more perpendicular and four more enclosing beams in a different x-z-w space
- Added two more enclosing beams in each of two visits to two different y-z-w spaces
We moved between the different 3D spaces using a hypothetical (more like fictional) portal.
A Traveling Cube: 3D in a 2D space
Let’s keep going. We have traveled in and out of different 3D spaces to build our tesseract. We want to step back and admire it. What does it look like?
Now, our tesseract exists in 4D, while we exist in only 3D. So we can not see the whole tesseract all at once; we can only see a 3D portion of the tesseract. But while we can only see a 3D portion, we can see different 3D portions by moving the tesseract across in front of us. Now, the art of visualizing a moving 4D object is likely something we have limited (more likely no) experience at. So to get skilled at how to do that, how to see a higher dimensional object moving in a lower dimensional space, let’s practice, but using a simpler situation. We will step down and imagine a 3D object, a cube, as would be seen by a being in 2D, i.e. a being in a flat plane.
Now we need a bit of background about a 2D being. Such a being could only see in front and to the side, not up or down, and further such a being would be stuck to a 2D plane. To picture this, imagine a thin flat plane three feet off the ground. Our 2D being could move along and around this plane, and see the sides of anything that pierced the plane, but could not look or move above or below the plane. This may see restricting, but that would be all this 2D being would know. (Just like all we know is 3D, and just like we don’t feel restricted not being able to see in 4D.)
Now, to build our skill at higher dimensional objects moving through a lower dimensional space, picture a cube, floating in front of you, above the 2D plane (the one floating three feet above the ground that is home to our 2D being). The cube is say a foot on each side a side, and the bottom face of the cube is aligned (parallel) to the plane of our 2D being. The cube consists just of the frame pieces, i.e. the sides are empty and the cube is just the framework.
Now start the cube moving slowly downward. Eventually, the bottom face of the cube hits the plane.
To us this intersection of the cube face with the plane occurs just from the smooth traveling of the cube downward. But what does the 2D being see? Remember, the 2D being can only see objects in the plane, and thus does not, can not, see the cube approaching from above. Only when the cube touches the square can the 2D being see the cube.
So to the 2D being in the square, the cube, or rather the bottom face of the cube, bursts into view with no warning. The bottom face explodes out of nowhere, as an object floating out in the middle of the thin air.
Now continue moving the cube. To us, being in 3D, we see the front face of the cube move past the plane of the 2D being, and the four lines connecting the bottom and top faces of the cube now cross the plane. To us this is simply the cube continuing its smooth downward move.
But for the 2D being, the bottom face disappears as mysteriously as it appeared. Remember, the 2D being can not see outside its 2D plane, so can not see the entirety of the cube. Thus, while we in 3D can follow the travel of the bottom face of the cube, the 2D being can not see below the surface of the plane and thus can not follow the bottom face leaving.
With the bottom face gone, what does the 2D being now see? The 2D being can only see the part of the cube intersecting the plane. And once the bottom face of the cube passes, what part of the cube is intersecting the plane? The four lines connecting the bottom face and the top face of the cube. And not the whole length of these lines. The 2D being can just see the four individual points intersecting the 2D square. And just like the bottom face floated out in thin air, these four points would be just out their, shimmering strangely.
You probably can now picture what will happen next. As the cube continues downward, the trailing top face of the cube intersects the 2D plane. The 2D being now sees the top face emerge spontaneously into view, then disappear, in a manner just like the bottom face.
In summary then, why does the 2D being see the 3D cube in such a mysterious fashion? This example illustrates the reason. The 2D being only sees 2D slices, and further, can not look ahead to see what’s approaching. The third dimension creates a great hiding place for objects that can pushed into the 2D being’s world.
Without going through the details, let’s briefly consider what the 2D being would see if the cube went through the 2D plane point forward. In other world’s, tilt and turn the cube so one of the eight corner points faces downward to the 2D plane.
In such a case, the cube would never have a face, or even a line, aligned with the surface of the 2D square. The 2D being would observe the cube as a series of strangely moving points, starting with one point, then three, then six, then three, then one, as the slanted cube moved through its 2D plane.
A Traveling Tesseract: 4D in a 3D space
With this concept of higher dimensional objects moving through lower dimensional space, let’s tackle the tesseract. As we stand expectantly, beside say a large open park, what would we see as the tesseract moved through our 3D slice? We will start with the tesseract in alignment with our space. This means the height, depth, and width (x, y and z) of the tesseract is aligned with our 3D.
A bit of thought might suggest, by analogy with the 2D illustration, that we would see a part of the tesseract suddenly appear, from nowhere, just hanging in mid-air, over the open park. And that would be correct. We can not see into the fourth dimension, so as the tesseract moved towards our 3D space (say from w=1 to where our x-y-z space resides at w=0), we would not observe it until precisely when the front part of the tesseract reached w=0.
What would we see? When we built the tesseract, we could build a complete cube within any given 3D space. So, given the proper alignment of the tesseract, we would see the front cube of the tesseract suddenly emerge, in a flash. This mimics, in 3D, the sudden appearance of the square as we moved the cube through the plane for our 2D being.
Then in a flash the front cube would be gone. What next? We would now see eight points hanging disconnected in the air. These would be the beams connecting the front and back cubes, and would be beams comprising cubes in the 3D space that includes the fourth dimension. Mathematically, if we exist in the x-y-z space, these other cubes exist in the w-y-z, w-x-z and w-x-y spaces.
In another flash, the back cube would appear. Remember, just like a cube has six squares on the outside, a tesseract has eight cubes. Two of those cubes lie in the x-y-z space. (And two lie in each of the other 3D spaces, i.e. w-y-z, w-x-z and w-x-y, so two cubes times four spaces gives eight cubes.)
If we use our visualization of the 3D cube passing through a 2D space, that provides a strong analogy for the 4D tesseract passing through a 3D space. We could even construct a 2D object (square) passing through a 1D space, a line. The analogy would fit. In concept, then, when a higher dimensional object passes through a lower dimensional space, only slices of the object can be observed in the lower dimensional space. And, as just seen, a being in the lower dimensional space can not see the object approaching.
Now let’s tilt the tesseract point forward. Like tilting the cube point forward, we would no longer see any extended pieces of the tesseract. We would only see points since the now slanted beams of the tesseract pierce our 3D space, but never align with it. We would observe one point, four points, twelve points, then four, then one. We would not see a cube, or square or line, with a complete tilt of the tesseract.
Mathematically, if our x-y-z space is at coordinate w=0, then for the tilted tesseract, all the beams have end points with different w coordinate values.
If we partially tilted the tesseract, we could get squares to align. In particular, with a partially tilted tesseract, at the proper alignment, we would get two squares appearing, with the squares parallel. As before, the figures would appear and vanish suddenly, and hang in thin air.
The Moving Camera
Now quickly, let’s touch on a further way to conceive a 4D object. We have likely all seen the opening shots of space movies, where the camera pans down the length of an enormous galactic interstellar cruiser. The camera stands so close, and the ship spans so large, that we can not see the whole ship at once, only small partial views.
If we go with that, if we picture a huge space cruiser, several thousand yards long, we could visualize ourselves in a small shuttle, holding the camera, hovering a few feet above the hull as we pan down. Our x-y-z field of vision would be limited to a few dozen yards in any direction, since the various appendages and contours of the ship would almost certainly obscure our ability to see much more.
We could view the entire ship over time, traveling up and down and around and in and out, staying within restriction (for the analogy) that we must stray no more than a few feet from the hull. So time here acts like a fourth dimension, just like the w-spatial direction. With our space ship, as we travel in the fourth dimension (time here) we can eventually see the whole ship. However, we might not ever build a complete visual picture, even after traveling over the entire ship, since the exterior contours of the ship could be so complex and extensive as to prevent our piecing together the numerous small snippets we observe into a cohesive whole.
This analogy mimics the tesseract. As we watch it go by, we can’t quite piece together a holistic image. The individual 3D snippets we can observe are disjointed enough that we are prevented from building a picture the entire object.
Does this voyage into 4D visualization serve some function? Certainly, to some, this mental exercise is interesting, or provides a challenge, or triggers a bit of curiously, or a serves as diversion or time-filler.
But does visualization in added dimensions have a larger purpose?
The answer is likely yes to (some) physicists and astronomers and mathematician performing serious, rigorous work. Those individuals undoubtedly can work with purely mathematical expressions of added dimensions without needing a mental picture. But mental pictures add to math formalism to reveal the dynamics of the situation. Mental pictures trigger intuitive and logical leaps, and uncover symmetries and solutions that may not otherwise emerge.
For a person not involved in rigorous exploration, does this have large purpose? I would say yes, definitely. Any person seeking some level of conceptually completeness in studying the world would be confronted with popular science articles referencing added dimensions. To integrate such references together, and into other concepts, such as from theology (where might God be?) or metaphysics (what is existence and what does it mean to exist), requires a tool kit of mental constructs. While an ability to handle 4D visualization may not be on the top of the list, having that ability, like having just about any tool, provides a benefit. And using the tool, just like using any tool, increases the generic ability to use the entire range of tools, or in this case the generic ability to manipulate an entire array of concepts, 4D or not.
So consider this study of 4D and of tesseracts to not just be a possible item of specific interest, but also an item of general mental agility to add to you intellectual tool box.
Video about Define Completing The Square In Math
You can see more content about Define Completing The Square In Math on our youtube channel: Click Here
Question about Define Completing The Square In Math
If you have any questions about Define Completing The Square In Math, please let us know, all your questions or suggestions will help us improve in the following articles!
The article Define Completing The Square In Math was compiled by me and my team from many sources. If you find the article Define Completing The Square In Math helpful to you, please support the team Like or Share!
Rate Articles Define Completing The Square In Math
Rate: 4-5 stars
Search keywords Define Completing The Square In Math
Define Completing The Square In Math
way Define Completing The Square In Math
tutorial Define Completing The Square In Math
Define Completing The Square In Math free
#Exercising #Mind #Tesseract #Fourth #Dimension