How to understand n-dimensional space and n-dimensional space-time

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user_ld3ui LV1 2022年1月15日 22:58 编辑

<section id="nice" style="padding-right: 10px; padding-left: 10px; overflow-wrap: break-word; font-family: Optima-Regular, Optima, PingFangSC-light, PingFangTC-light, "PingFang SC", Cambria, Cochin, Georgia, Times, "Times New Roman", serif; line-height: 1.6; letter-spacing: 0.034em; color: rgb(63, 63, 63); font-size: 16px;"><h1 style="font-weight: bold; color: black; font-size: 24px; text-align: center; background-image: url("http://img.macrozheng.com/mall/md/koala-1.png"); background-position: center top; background-repeat: no-repeat; background-size: 75px; line-height: 95px; margin-top: 38px;">How to understand n-dimensional space and n-dimensional space-time</h1> <h2 style="font-weight: bold; color: black; font-size: 22px; text-align: center; background-image: url("http://img.macrozheng.com/mall/md/koala-2.png"); background-position: center center; background-repeat: no-repeat; background-attachment: initial; background-origin: initial; background-clip: initial; background-size: 50px; margin-top: 1em;"><span style="display: inline-block; height: 38px; line-height: 42px; color: rgb(72, 179, 120); background-position: left center; background-repeat: no-repeat; background-attachment: initial; background-origin: initial; background-clip: initial; background-size: 63px; margin-top: 38px; font-size: 18px; margin-bottom: 10px;">Foreword</h2> Some friends must wonder why the author suddenly sent such an article that seems completely irrelevant to technology. In fact, this content is also extended by the author when studying space-time search. After reading some materials and deepening the understanding of n-dimensional space and n-dimensional space-time, I summarized it. If you are a friend who has no contact with this aspect, you will feel strange at the beginning. If you are a math major or major, you are a friend in this direction. If there are errors in the article, you are welcome to put it forward and discuss it together. <h2 style="font-weight: bold; color: black; font-size: 22px; text-align: center; background-image: url("http://img.macrozheng.com/mall/md/koala-2.png"); background-position: center center; background-repeat: no-repeat; background-attachment: initial; background-origin: initial; background-clip: initial; background-size: 50px; margin-top: 1em;"><span style="display: inline-block; height: 38px; line-height: 42px; color: rgb(72, 179, 120); background-position: left center; background-repeat: no-repeat; background-attachment: initial; background-origin: initial; background-clip: initial; background-size: 63px; margin-top: 38px; font-size: 18px; margin-bottom: 10px;">Space and time</h2> First of all, space and space-time are two concepts that are often confused. In fact, they are different. Einstein's general theory of relativity mentioned four-dimensional space, which is three-dimensional space plus one-dimensional time. This is not the concept of multidimensional space in mathematics. In fact, the time dimension is independent of the space dimension. One dimensional space can also have time, and two-dimensional space can also have time. Multidimensional space has time. However, the four-dimensional space mentioned in general relativity is actually a four-dimensional space-time composed of three-dimensional space and one-dimensional time. High dimensional geometry after Riemannian geometry has developed for many years. In superstring theory, the space of the universe is nine dimensional space plus one-dimensional time. In M theory, the universe is an eleven dimensional space-time of ten dimensional space and one-dimensional time. <h2 style="font-weight: bold; color: black; font-size: 22px; text-align: center; background-image: url("http://img.macrozheng.com/mall/md/koala-2.png"); background-position: center center; background-repeat: no-repeat; background-attachment: initial; background-origin: initial; background-clip: initial; background-size: 50px; margin-top: 1em;"><span style="display: inline-block; height: 38px; line-height: 42px; color: rgb(72, 179, 120); background-position: left center; background-repeat: no-repeat; background-attachment: initial; background-origin: initial; background-clip: initial; background-size: 63px; margin-top: 38px; font-size: 18px; margin-bottom: 10px;">How to describe the division of high-dimensional space</h2> In two-dimensional space, two vertically intersecting lines can form the x-axis and y-axis. In three-dimensional space, three mutually perpendicular directions constitute the x-axis, Y-axis and z-axis. The third line passes through the intersection(i.e. the origin ) in two-dimensional space and is perpendicular to two-dimensional space. Similarly, in the four-dimensional space, there will also be a straight line passing through the intersection of the three lines in the three-dimensional space(the origin of the three-dimensional coordinate axis ) and perpendicular to the first three lines. The line perpendicular to the three-dimensional space in the four-dimensional space cannot be expressed or drawn in the three-dimensional space. This line is located in four-dimensional space inside the coordinate origin. So how does the four-dimensional space relate to the three-dimensional space? After all, three-dimensional space is the most familiar spatial structure of human beings. We know that three-dimensional space has x-axis, Y-axis and z-axis, so their three axes can divide the whole space into six faces, up and down, left and right, front and back. How can four-dimensional space be divided? It has two directions, inside and outside, more than three-dimensional space. The top inside is a different space from the top outside. Although it is above in three-dimensional space. Similarly, if we continue to extend these theories to high-dimensional space, there must be a line perpendicular to n-1 lines, and N-1 lines also intersect each other vertically. The above is to describe multi-dimensional space from the perspective of space division. <h2 style="font-weight: bold; color: black; font-size: 22px; text-align: center; background-image: url("http://img.macrozheng.com/mall/md/koala-2.png"); background-position: center center; background-repeat: no-repeat; background-attachment: initial; background-origin: initial; background-clip: initial; background-size: 50px; margin-top: 1em;"><span style="display: inline-block; height: 38px; line-height: 42px; color: rgb(72, 179, 120); background-position: left center; background-repeat: no-repeat; background-attachment: initial; background-origin: initial; background-clip: initial; background-size: 63px; margin-top: 38px; font-size: 18px; margin-bottom: 10px;">The form of things in high-dimensional space</h2> In the high-dimensional space, things are very abstract and may not be drawn with graphics, but we can understand the high-dimensional space through the low-dimensional space we can understand, so we need to study the display form of things in the high-dimensional space in the low-dimensional space. In two-dimensional space, an equilateral triangle has three vertices. And assume that the side lengths are equal to 1. If there is a fourth point in space, the distance from this point to the three vertices can be equal to 1. Then this point must not exist in two-dimensional space, but must exist in three-dimensional space(the mathematical proof here is omitted, which is too difficult for interested students to prove ) . If these four points are connected in three-dimensional space, a three-dimensional regular tetrahedron can be formed. Similarly, if a fifth point can be 1 away from the three-dimensional regular tetrahedron, then the point must also exist in the four-dimensional space and form a four-dimensional super tetrahedron together with the three-dimensional regular tetrahedron. The super tetrahedron is beyond the dimension of our life, so we can't draw its shape in three-dimensional space. But we can observe it in three-dimensional space by projection. Let's first review how three-dimensional regular tetrahedra are generated. Because it is an equilateral triangle, the distance from the vertical center of the equilateral triangle to the three vertices must be equal. Then we take this heart out and pull it into three-dimensional space until the distance from the other three vertices is 1. In this way, a three-dimensional regular tetrahedron is generated. The inner three obtuse triangles divided by the vertical center follow the vertical center, and when pulled out, they will become three faces outside the regular tetrahedron. Similarly, in the of a three-dimensional regular tetrahedron, take out its perpendicular center. The distance between the vertical center and the four vertices is equal. This vertical center divides the regular tetrahedron into four flat tetrahedrons. Then if you pull the vertical center to the fifth vertex in four-dimensional space, it will become a super tetrahedron. The four flat tetrahedrons divided internally will also evolve into the four outer surfaces of the super tetrahedron. The four-dimensional super tetrahedron is a super body composed of 5 vertices, 10 edges, 10 triangular surfaces and 5 tetrahedrons. It cannot be described in three-dimensional space. Cube is our common three-dimensional object. What does the cube look like in four-dimensional space? The above figure is a cube in four-dimensional space, called hypercube. The figure above shows that each side of a four-dimensional cube is equal in length, and you can also see how the cubes are connected to each other. The simplest step to construct a hypercube is to connect the 8 vertices of two cubes with the vertices of another hypercube. The above figure shows that the hypercube is essentially obtained by combining two cubes and connecting the corresponding vertices. The figure above is arranged according to the length of each vertex starting from the bottom vertex along the edge. If we want to use hypercube as in <a href="https://zh.wikipedia.org/wiki/%E5%B9%B6%E8%A1%8C%E8%AE%A1%E7%AE%97" style="overflow-wrap: break-word; font-weight: bold; color: rgb(72, 179, 120); border-bottom: 1px solid rgb(72, 179, 120);">parallel computing</a> Connecting different processors in <a href="https://zh.wikipedia.org/wiki/%E7%BD%91%E7%BB%9C%E6%8B%93%E6%89%91" style="overflow-wrap: break-word; font-weight: bold; color: rgb(72, 179, 120); border-bottom: 1px solid rgb(72, 179, 120);">network topology</a> These images can be very useful if they are basic. In a hypercube, there are at most 4 different paths between any two vertices, and there are many paths that are identical. Hypercube or a bipartite graph(https://zh.wikipedia.org/wiki/%E4%BA%8C%E5%88%86%E5%9B%BE ) , just like squares and cubes. The following two figures are perspective projections The figure above shows that the positive octahedron rotates around a plane cut through the figure from front left to rear right and <a href="https://zh.wikipedia.org/w/index.php?title=%E5%8D%95%E6%97%8B%E8%BD%AC&action=edit&redlink=1" style="overflow-wrap: break-word; font-weight: bold; color: rgb(72, 179, 120); border-bottom: 1px solid rgb(72, 179, 120);">from top to bottom</a> <a href="https://zh.wikipedia.org/wiki/%E9%80%8F%E8%A7%86%E6%8A%95%E5%BD%B1" style="overflow-wrap: break-word; font-weight: bold; color: rgb(72, 179, 120); border-bottom: 1px solid rgb(72, 179, 120);">Perspective projection</a> 。 The figure above shows a positive octet surrounding two in four-dimensional space <a href="https://zh.wikipedia.org/wiki/%E6%AD%A3%E4%BA%A4" style="overflow-wrap: break-word; font-weight: bold; color: rgb(72, 179, 120); border-bottom: 1px solid rgb(72, 179, 120);">orthogonal</a> <a href="https://zh.wikipedia.org/w/index.php?title=%E5%8F%8C%E6%97%8B%E8%BD%AC&action=edit&redlink=1" style="overflow-wrap: break-word; font-weight: bold; color: rgb(72, 179, 120); border-bottom: 1px solid rgb(72, 179, 120);">Double rotation</a> Perspective projection at. In addition, four-dimensional space and the above space belong to high-dimensional model. High dimensional models are also divided into two concepts: Mathematics and physics. Mathematically, multidimensional has many models. Theoretically, the dimension can be very high. There are many models. However, few satisfy the commutative invariant property. Therefore, some people think that the four-dimensional space is the physical upper limit. However, some people think there will be higher dimensional physics. Thinking is good for intelligence, because it is only constrained by mathematical conditions. In physics, multidimensional has many models. Theoretically, the dimension cannot be very high. In order to explain the finite boundless nature of the universe as a whole, we must introduce multidimensional, generally four-dimensional space-time(a pair of relative composition properties ) , and there are also some other finite countable dimensions. There are few models that may be established in physics. It is very difficult to think because it is constrained by physical phenomena. <h2 style="font-weight: bold; color: black; font-size: 22px; text-align: center; background-image: url("http://img.macrozheng.com/mall/md/koala-2.png"); background-position: center center; background-repeat: no-repeat; background-attachment: initial; background-origin: initial; background-clip: initial; background-size: 50px; margin-top: 1em;"><span style="display: inline-block; height: 38px; line-height: 42px; color: rgb(72, 179, 120); background-position: left center; background-repeat: no-repeat; background-attachment: initial; background-origin: initial; background-clip: initial; background-size: 63px; margin-top: 38px; font-size: 18px; margin-bottom: 10px;">Does perspective and wall piercing really not exist?</h2> The world in the eyes of ants is almost two-dimensional. In its eyes, there is only length and width, not height. Any three-dimensional object is a "face" for it, and it will climb. Or in a two-dimensional space, people living in the riverside map on the Qingming Festival only have chaotic points, lines and surfaces in their eyes, and the people in the painting can't have a complete understanding of the world in the whole painting. But living in three-dimensional space, we can see the whole world at a glance. Similarly, we in three-dimensional space can't see the objects in three-dimensional space at a glance. For example, if we want to see the high-rise building in front of us, plus the roof and bottom, we can't see it at a glance. We need to circle around it. But these creatures in the four-dimensional space can actually see what a tall building looks like at a glance. Therefore, we can get a perhaps incorrect conclusion that low-dimensional space is just the skin of high-dimensional space, because low-dimensional space degenerates into "skin" due to the collapse of a dimension in high-dimensional space. Recall the two-dimensional equilateral triangle, three-dimensional regular tetrahedron, four-dimensional super tetrahedron mentioned before. Isn't the low dimension the skin of the high dimension? In the high dimension, look down on the low dimension, at a glance. Another example is the "perspective" in the sketch. Through some imaging principles, we can see the blocked parts of the object. Of course not real. If it is true to see, then this "perspective" is through the dimension. Besides, the Monkey King Circle in the journey to the West protects the Tang monk. In the two-dimensional space, this circle can protect the Tang monk, but in the three-dimensional space, you only need to gently jump out of this circle to get rid of the shackles of the monkey king. If you want to protect a person in three-dimensional space, you need to lock him up in a closed space. But if this person is a person in four-dimensional space, he can easily jump out of the four-dimensional space. This is why people in three-dimensional space can't understand "wall piercing", but people in four-dimensional space can do it easily. <h2 style="font-weight: bold; color: black; font-size: 22px; text-align: center; background-image: url("http://img.macrozheng.com/mall/md/koala-2.png"); background-position: center center; background-repeat: no-repeat; background-attachment: initial; background-origin: initial; background-clip: initial; background-size: 50px; margin-top: 1em;"><span style="display: inline-block; height: 38px; line-height: 42px; color: rgb(72, 179, 120); background-position: left center; background-repeat: no-repeat; background-attachment: initial; background-origin: initial; background-clip: initial; background-size: 63px; margin-top: 38px; font-size: 18px; margin-bottom: 10px;">Does transformers really not exist?</h2> In the three-dimensional space of our life, there are few creatures that can constantly change their forms. In the three-dimensional world, there are not many that can deform like transformers, especially from the inside to the outside. So in the world of high-dimensional space, is there such a thing as transformers? The answer is that there may not be many in the same dimension, but there are many across spatial dimensions. For example, a cube or polyhedron in three-dimensional space, how do two-dimensional things understand them? Take an example of hyperbola: Two inverted cones, placed top to top. Use a plane to cut them. The curve left by a three-dimensional object on this plane is called a conic. When the tangent direction of the surface is different, different conic curves can be formed, including circle, parabola, hyperbola and ellipse. In the two-dimensional world, we can only recognize these different conics. But in the three-dimensional world, we can understand that these are two cones. The above figure also clearly shows that objects in high-dimensional space have different sections on the plane and different shapes. Then we expand to four-dimensional space. If an object in four-dimensional space is cut continuously by three-dimensional space, the three-dimensional body left in three-dimensional space will change constantly? So we can't understand transformers because we are in low-dimensional space. Transformers will occur when high-dimensional objects are cut by low-dimensional space. <h2 style="font-weight: bold; color: black; font-size: 22px; text-align: center; background-image: url("http://img.macrozheng.com/mall/md/koala-2.png"); background-position: center center; background-repeat: no-repeat; background-attachment: initial; background-origin: initial; background-clip: initial; background-size: 50px; margin-top: 1em;"><span style="display: inline-block; height: 38px; line-height: 42px; color: rgb(72, 179, 120); background-position: left center; background-repeat: no-repeat; background-attachment: initial; background-origin: initial; background-clip: initial; background-size: 63px; margin-top: 38px; font-size: 18px; margin-bottom: 10px;">Can't time really be reversed?</h2> When I was a child, I often thought about such a time. Can't time really be reversed? Can't a broken mirror really be reunited? To understand this matter, we must first talk about the four-dimensional space-time where we are now. In Einstein's general theory of relativity, he talked about four-dimensional space-time, three-dimensional space plus one-dimensional time. A person's life is like a timeline, from birth to old age. People can't go back to the past and childhood in four-dimensional space-time. How do we define time and space? <h2 style="font-weight: bold; color: black; font-size: 22px; text-align: center; background-image: url("http://img.macrozheng.com/mall/md/koala-2.png"); background-position: center center; background-repeat: no-repeat; background-attachment: initial; background-origin: initial; background-clip: initial; background-size: 50px; margin-top: 1em;"><span style="display: inline-block; height: 38px; line-height: 42px; color: rgb(72, 179, 120); background-position: left center; background-repeat: no-repeat; background-attachment: initial; background-origin: initial; background-clip: initial; background-size: 63px; margin-top: 38px; font-size: 18px; margin-bottom: 10px;">How to describe high-dimensional space-time</h2> At the beginning of the article, we divide the high-dimensional space by the way of coordinate axis and space division, and extend it to the n-dimensional space. Here, let's change our perspective and extend it from low dimensional space-time to n-dimensional space-time in the way of probability theory. First look at one-dimensional space. Two points can form a line. When an infinite number of lines cover a layer, it becomes a two-dimensional space. So many lines fill all the possibilities. So one dimension has only length, no width and height(depth ) . In two-dimensional space, there are faces. When countless surfaces are paved with a space, it will become a three-dimensional space. So many faces occupy all the possibilities of the space. Two dimensional space also has length and width, but no height(depth ) . We are all familiar with three-dimensional space, so we won't repeat it. Everything in three-dimensional space has length, width and height. In four-dimensional space-time, there is one-dimensional time more than three-dimensional space. We still use the previous probability theory to define four-dimensional space-time, so many one-dimensional time is from the generation of objects to the final destruction. For people, it is life. The time of this life is full of all activities that can be done in one's life, representing all possibilities. There is a concept of parallel universe. In such a long life, people will make many choices, which will change the development of their future life. Every choice has the possibility of choice. If there are n choices, there are n results. If each result goes down, you may get a different life. The same time axis may correspond to n possibilities at the same time. There are n main lines in the game, but each game character can only choose one of them. Of course, each choice is not necessarily one of two. You can also select more than one. Multiple choices will lead to different results. For example, I chose to go abroad for postgraduate entrance examination, find a foreign girlfriend and buy a house abroad. The choice accumulation of multiple dimensions will have an impact on the future. It is also possible to study hard as a child, enter a famous school and live a winner's life when you grow up. In quantum theory, ultra-small particles make up the whole world. Various possibilities, as waves, weaken to a certain point. We continue to make choices in life, but we don't weaken these waves until the choices are finished, which determines that this point is the final result. So many main lines of the game can also form a face. This surface is a two-dimensional surface. Of course, this surface is very special. The lines inside are time lines. This constitutes a five-dimensional space-time. These timelines are full <hr style="margin-top: 10px; margin-bottom: 10px; border-width: initial; border-style: none; text-align: center; background-image: linear-gradient(to right, rgba(93, 186, 133, 0), rgba(93, 186, 133, 0.75), rgba(93, 186, 133, 0));"> Spatial search series articles: <hr style="margin-top: 10px; margin-bottom: 10px; border-width: initial; border-style: none; text-align: center; background-image: linear-gradient(to right, rgba(93, 186, 133, 0), rgba(93, 186, 133, 0.75), rgba(93, 186, 133, 0));"> Reference： <a href="http://v.youku.com/v_show/id_XMTc3ODM4MTE2OA==.html?spm=a2h0j.8191423.module_basic_relation.5~5!2~5~5!6~5!2~1~3~A" style="overflow-wrap: break-word; font-weight: bold; color: rgb(72, 179, 120); border-bottom: 1px solid rgb(72, 179, 120);">Deciphering the mysterious four-dimensional space</a> <a href="http://v.youku.com/v_show/id_XNTYzNzQ4OTY0.html?spm=a2h0k.8191407.0.0&from=s1.8-1-1.2" style="overflow-wrap: break-word; font-weight: bold; color: rgb(72, 179, 120); border-bottom: 1px solid rgb(72, 179, 120);">Evolution from one-dimensional space to ten-dimensional space</a></section>

How to understand n-dimensional space and n-dimensional space-time

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