Louise Hay - one of the first masters of our time began to talk about ...
If you are a fan of the Avengers movies, the first thing that comes to your mind when you hear the word "Tesseract" is the transparent cube-shaped vessel of the Infinity Stone containing boundless power.
For fans of the Marvel Universe, the Tesseract is a glowing blue cube that makes people from not only Earth, but other planets also go crazy. This is why all the Avengers have banded together to protect the Earthlings from the extremely destructive forces of the Tesseract.
However, the following must be said: The Tesseract is an actual geometric concept, or rather, a form that exists in 4D. This isn't just a blue cube from the Avengers ... it's a real concept.
Tesseract is an object in 4 dimensions. But before we explain it in detail, let's start from the beginning.
What is dimension?
Everyone has heard the terms 2D and 3D, representing respectively two-dimensional or three-dimensional objects in space. But what are these dimensions?
Measurement is simply the direction you can go. For example, if you are drawing a line on a piece of paper, you can go either left / right (x-axis) or up / down (y-axis). Thus, we say that the paper is two-dimensional, since you can only walk in two directions.
There is a sense of depth in 3D.
Now, in real world besides the two directions mentioned above (left / right and up / down), you can also go to / from. Hence, a sense of depth is added in 3D space. Therefore, we say that real life 3-dimensional.
A point can represent 0 dimensions (since it does not move in any direction), a line represents 1 dimension (length), a square represents 2 dimensions (length and width), and a cube represents 3 dimensions (length, width, and height).
Take a 3D cube and replace each face (which is currently a square) with a cube. And so! The shape you get is the tesseract.
What is a tesseract?
Simply put, a tesseract is a cube in 4-dimensional space. You can also say that it is a 4D analog of a cube. It is a 4D shape where each face is a cube.
A 3D projection of a tesseract that rotates twice around two orthogonal planes. Image: Jason Hise
Here's a simple way to conceptualize dimensions: a square is two-dimensional; therefore, each of its corners has 2 lines extending from it at an angle of 90 degrees to each other. The cube is 3D, so each of its corners has 3 lines descending from it. The same way, the tesseract is a 4D shape, so each corner has 4 lines extending from it.
Why is it difficult to imagine a tesseract?
Since we, as humans, have evolved to visualize objects in three dimensions, anything that goes into extra dimensions such as 4D, 5D, 6D, etc., does not make much sense to us, because we cannot have them at all. introduce. Our brain cannot understand the 4th dimension in space. We just can't think about it.
However, just because we cannot visualize the concept of multidimensional spaces does not mean that it cannot exist.
September 19th, 2009
Tesseract (from ancient Greek τέσσερες ἀκτῖνες - four rays) is a four-dimensional hypercube - an analogue of a cube in four-dimensional space.
An image is a projection (perspective) of a four-dimensional cube onto three-dimensional space.
According to the Oxford Dictionary, the word "tesseract" was coined and used in 1888 by Charles Howard Hinton (1853-1907) in his book " New era thoughts". Later, some people called the same figure "tetracubus".
Geometry
An ordinary tesseract in Euclidean four-dimensional space is defined as the convex hull of points (± 1, ± 1, ± 1, ± 1). In other words, it can be represented as the following set:
The tesseract is bounded by eight hyperplanes, the intersection of which with the tesseract itself defines its three-dimensional faces (which are ordinary cubes). Each pair of non-parallel 3D faces intersect to form 2D faces (squares), and so on. Finally, a tesseract has 8 3D faces, 24 2D, 32 edges, and 16 vertices.
Popular Description
Let's try to imagine what the hypercube will look like without leaving three-dimensional space.
In one-dimensional "space" - on a line - select a segment AB of length L. On a two-dimensional plane at a distance L from AB, draw a segment DC parallel to it and connect their ends. The result is a square ABCD. Repeating this operation with the plane, we get a three-dimensional cube ABCDHEFG. And by shifting the cube in the fourth dimension (perpendicular to the first three) by a distance L, we get a hypercube ABCDEFGHIJKLMNOP.
http://upload.wikimedia.org/wikipedia/ru/1/13/Construction_tesseract.PNG
The one-dimensional segment AB is the side of the two-dimensional square ABCD, the square is the side of the cube ABCDHEFG, which, in turn, will be the side of the four-dimensional hypercube. A straight line segment has two boundary points, a square has four vertices, and a cube has eight. Thus, in a four-dimensional hypercube, there will be 16 vertices: 8 vertices of the original cube and 8 shifted in the fourth dimension. It has 32 edges - 12 each give the initial and final positions of the original cube, and 8 more edges will "draw" its eight vertices, which have moved into the fourth dimension. The same reasoning can be done for the faces of the hypercube. In two-dimensional space, it is one (the square itself), the cube has 6 of them (two faces from the moved square and four more will describe its sides). A four-dimensional hypercube has 24 square faces - 12 squares of the original cube in two positions and 12 squares from its twelve edges.
In a similar way, we can continue the reasoning for hypercubes of a larger number of dimensions, but it is much more interesting to see how a four-dimensional hypercube will look like for us, inhabitants of three-dimensional space. Let's use the familiar analogy method for this.
Unfolding the tesseract
Take a wire cube ABCDHEFG and look at it with one eye from the side of the face. We will see and can draw two squares on the plane (its near and far faces), connected by four lines - side edges. Similarly, a four-dimensional hypercube in three-dimensional space will look like two cubic "boxes" inserted into each other and connected by eight edges. In this case, the "boxes" themselves - three-dimensional faces - will be projected onto "our" space, and the lines connecting them will stretch in the fourth dimension. You can also try to imagine a cube not in projection, but in a spatial image.
Just as a three-dimensional cube is formed by a square shifted by the length of a face, a cube shifted into the fourth dimension will form a hypercube. It is limited by eight cubes, which in perspective will look like a rather complex figure. The part of it, which remained in "our" space, is drawn with solid lines, and that which has gone into hyperspace, with dotted lines. The very same four-dimensional hypercube consists of an infinite number of cubes, just as a three-dimensional cube can be "cut" into an infinite number of flat squares.
Having cut six faces of a three-dimensional cube, you can expand it into flat figure- sweep. It will have a square on each side of the original face plus one more - the face opposite to it. A three-dimensional unfolding of a four-dimensional hypercube will consist of the original cube, six cubes "growing" from it, plus one more - the final "hyperface".
Tesseract properties are continuation of properties geometric shapes smaller dimension into four-dimensional space.
Projection
Into two-dimensional space
This structure is difficult for the imagination, but it is possible to project a tesseract into 2D or 3D spaces. In addition, projection to plane makes it easy to understand the location of the vertices of the hypercube. In this way, images can be obtained that no longer reflect spatial relationships within the tesseract, but which illustrate the structure of vertex connections, as in the following examples:
Into three-dimensional space
The projection of a tesseract onto a three-dimensional space is represented by two nested three-dimensional cubes, the corresponding vertices of which are connected by segments. The inner and outer cubes have different sizes in three-dimensional space, but in four-dimensional space they are equal cubes. To understand the equality of all the cubes of the tesseract, a rotating tesseract model was created.
The six truncated pyramids at the edges of the tesseract are images of equal six cubes.
Stereo pair
A stereopair of a tesseract is depicted as two projections onto three-dimensional space. This tesseract image was designed to represent depth as a fourth dimension. A stereopair is viewed so that each eye sees only one of these images, a stereoscopic picture appears that reproduces the depth of the tesseract.
Unfolding the tesseract
The surface of a tesseract can be expanded into eight cubes (similar to how the surface of a cube can be expanded into six squares). There are 261 different tesseract unfolding. The unfolding of the tesseract can be calculated by drawing connected corners on the graph.
Tesseract in art
In Edwine A.'s New Abbott Plains, the hypercube is the storyteller.
In one episode of The Adventures of Jimmy Neutron: Genius Boy Jimmy invents a four-dimensional hypercube identical to the foldbox from Heinlein's 1963 novel Road of Glory.
Robert E. Heinlein has mentioned hypercubes in at least three science fiction stories. In The House of Four Dimensions (The House That Teale Built) (1940), he described a house built as an unfolding of a tesseract.
Heinlein's novel Road of Glory describes an oversized dish that was larger on the inside than on the outside.
Henry Kuttner's story "Mimsy Were the Borogoves" describes an educational toy for children from the distant future, similar in structure to a tesseract.
In the novel by Alex Garland (1999), the term "tesseract" is used for a three-dimensional unfolding of a four-dimensional hypercube, not the hypercube itself. This is a metaphor designed to show that the cognizing system should be broader than the cognizable one.
Cube 2: Hypercube focuses on eight strangers trapped in a hypercube, or network of interconnected cubes.
The TV series Andromeda uses tesseract generators as a conspiracy device. They are primarily designed to manipulate space and time.
Painting "Crucifixion" (Corpus Hypercubus) by Salvador Dali (1954)
The Nextwave comic book depicts a vehicle that includes 5 tesseract zones.
On the Voivod Nothingface album, one of the songs is called “In my hypercube”.
In the novel by Anthony Pierce "Route Cuba" one of the orbiting moons of the International Development Association is called a tesseract, which was compressed into 3 dimensions.
In the series "School" Black hole“” In the third season there is a series “Tesseract”. Lucas pushes a secret button and the school begins to take shape like a mathematical tesseract.
The term "tesseract" and the term "tesserate" derived from it is found in Madeleine L'Engle's story "The Fold of Time"
In one-dimensional "space" - on a line - select a segment AB of length L. On a two-dimensional plane at a distance L from AB, draw a segment DC parallel to it and connect their ends. The result is a square CDBA. Repeating this operation with the plane, we get a three-dimensional cube CDBAGHFE. And shifting the cube in the fourth dimension (perpendicular to the first three) by a distance L, we get the hypercube CDBAGHFEKLJIOPNM.
The one-dimensional segment AB is the side of the two-dimensional square CDBA, the square is the side of the cube CDBAGHFE, which, in turn, will be the side of the four-dimensional hypercube. A straight line segment has two boundary points, a square has four vertices, and a cube has eight. Thus, in a four-dimensional hypercube, there will be 16 vertices: 8 vertices of the original cube and 8 shifted in the fourth dimension. It has 32 edges - 12 each give the initial and final positions of the original cube, and 8 more edges will "draw" its eight vertices, which have moved into the fourth dimension. The same reasoning can be done for the faces of the hypercube. In two-dimensional space, it is one (the square itself), the cube has 6 of them (two faces from the moved square and four more will describe its sides). A four-dimensional hypercube has 24 square faces - 12 squares of the original cube in two positions and 12 squares from its twelve edges.
As the sides of a square are 4 one-dimensional segments, and the sides (faces) of a cube are 6 two-dimensional squares, so for a "four-dimensional cube" (tesseract), the sides are 8 three-dimensional cubes. The spaces of opposite pairs of tesseract cubes (that is, the three-dimensional spaces to which these cubes belong) are parallel. In the figure, these are cubes: CDBAGHFE and KLJIOPNM, CDBAKLJI and GHFEOPNM, EFBAMNJI and GHDCOPLK, CKIAGOME and DLJBHPNF.
In a similar way, we can continue the reasoning for hypercubes of a larger number of dimensions, but it is much more interesting to see how a four-dimensional hypercube will look like for us, inhabitants of three-dimensional space. Let's use the familiar analogy method for this.
Take a wire cube ABCDHEFG and look at it with one eye from the side of the face. We will see and can draw two squares on the plane (its near and far faces), connected by four lines - side edges. Similarly, a four-dimensional hypercube in three-dimensional space will look like two cubic "boxes" inserted into each other and connected by eight edges. In this case, the "boxes" themselves - three-dimensional faces - will be projected onto "our" space, and the lines connecting them will stretch in the direction of the fourth axis. You can also try to imagine a cube not in projection, but in a spatial image.
Just as a three-dimensional cube is formed by a square shifted by the length of a face, a cube shifted into the fourth dimension will form a hypercube. It is limited by eight cubes, which in perspective will look like a rather complex figure. The very same four-dimensional hypercube consists of an infinite number of cubes, just as a three-dimensional cube can be "cut" into an infinite number of flat squares.
Having cut six faces of a three-dimensional cube, you can expand it into a flat shape - a sweep. It will have a square on each side of the original face plus one more - the face opposite to it. And the three-dimensional unfolding of the four-dimensional hypercube will consist of the original cube, six cubes "growing" from it, plus one more - the final "hyperface".
Tesseract properties are the continuation of the properties of geometric figures of lower dimensions into four-dimensional space.
Tesseract sweeps
Just as the surface of a cube can be expanded into a polygon of six squares, the surface of a tesseract can be expanded into a three-dimensional body, consisting of eight cubes.
There are 261 unfolded tesseract. Unfolds of a hypercube can be found by listing "doubled trees", where "doubled tree" ( paired tree) is a tree with an even number of vertices, which are split into pairs so that no pair consists of two adjacent vertices. There is a one-to-one correspondence between the "twin trees" with 8 vertices and the unfolding of the tesseract. In total, there are 23 trees with 8 vertices, splitting the vertices of which into pairs of non-adjacent vertices results in 261 "doubled trees" with 8 vertices.
The cruciform scan of the tesseract is an element of Salvador Dali's painting "Corpus Hypercubus" (1954).
Projection
Into two-dimensional space
This structure is difficult for the imagination, but it is possible to project a tesseract into 2D or 3D spaces. In addition, projection onto a plane makes it easy to understand the location of the vertices of the hypercube. In this way, images can be obtained that no longer reflect spatial relationships within the tesseract, but which illustrate the structure of vertex connections, as in the following examples:
Into three-dimensional space
One of the projections of the tesseract onto three-dimensional space is represented by two nested three-dimensional cubes, the corresponding vertices of which are connected by segments. The inner and outer cubes have different sizes in three-dimensional space, but in four-dimensional space they are equal cubes. To understand the equality of all the cubes of the tesseract, a rotating tesseract model was created.
- The six truncated pyramids at the edges of the tesseract are images of equal six cubes. However, these cubes are to a tesseract - like squares (faces) to a cube. But in fact, the tesseract can be divided into an infinite number of cubes, like a cube - into an infinite number of squares, or a square - into an infinite number of segments.
Another interesting projection of the tesseract onto three-dimensional space is a rhombic dodecahedron with its four diagonals drawn, connecting pairs of opposite vertices at large rhombus angles. In this case, 14 of the 16 vertices of the tesseract are projected into 14 vertices of the rhombododecahedron, and the projections of the remaining 2 coincide in its center. In such a projection onto a three-dimensional space, the equality and parallelism of all one-dimensional, two-dimensional and three-dimensional sides are preserved.
Stereo pair
A stereopair of a tesseract is depicted as two projections onto a plane of one of the variants of a three-dimensional representation of a tesseract. A stereopair is considered so that each eye sees only one of these images, a stereoscopic effect arises, which makes it possible to better perceive the projection of the tesseract into three-dimensional space.
Tesseract in culture
- In one episode of The Adventures of Jimmy Neutron, "genius boy" Jimmy invents a four-dimensional hypercube identical to the foldbox from The Road to Glory (1963) by Robert Heinlein.
- Heinlein's novel Road of Glory describes an oversized box that was larger on the inside than on the outside.
- In the story "... And he built himself a crooked house" (in another version of the translation "The House That Teal Built") by Heinlein, an eight-apartment building is described in the form of an expanded tesseract.
- Henry Kuttner's story "All tenals of the Borogovs" describes an educational toy for children from the distant future, similar in structure to a tesseract.
- In Alex Garland's 1999 novel The Tesseract, the term “tesseract” is used to refer to a three-dimensional unfolding of a four-dimensional hypercube, not the hypercube itself. This is a metaphor designed to show that the cognizing system should be broader than the cognizable one.
- Cube 2: Hypercube focuses on eight strangers trapped in a hypercube, or a network of interconnected 3D projections of a single hypercube.
- In the Marvel Cinematic Universe film series, the Tesseract is a key plot element, a space artifact shaped like a hypercube.
- The plot of the film "The Avengers" focuses on the use of the "Tesseract" cube as an inexhaustible source of cosmic energy, to open a portal to another "dimension" in order to implement a plan to conquer the world (in exchange for the Tesseract - the Chitauri will provide Loki with an army to seize the Earth). However, this material has almost nothing to do with the general theory of four dimensions.
- In the comic "Deadpool Destroys the Marvel Universe" main character with the help of the supervillain Arcade, uses the tesseract to catch Kitty Pryde: her abilities were unable to help her get out of the cube.
- TV series "
An image is a projection () of a four-dimensional cube onto three-dimensional space.
The generalization of a cube to cases with more than 3 dimensions is called hypercube or (en: measure polytopes). Formally, a hypercube is defined as four equal segments.
This article mainly describes 4-dimensional hypercube called tesseract.
Popular Description
Let's try to imagine what the hypercube will look like without leaving our 3D.
In one-dimensional "space" - on a line - select AB of length L. On a two-dimensional space at a distance L from AB, draw a segment DC parallel to it and connect their ends. The result is a square ABCD. Repeating this operation with the plane, we get a three-dimensional cube ABCDHEFG. And by shifting the cube in the fourth dimension (perpendicular to the first three!) By a distance L, we get a hypercube.
The one-dimensional segment AB serves as the face of the two-dimensional square ABCD, the square as the side of the cube ABCDHEFG, which, in turn, will be the side of the four-dimensional hypercube. A straight line segment has two boundary points, a square has four vertices, and a cube has eight. Thus, in a four-dimensional hypercube, there will be 16 vertices: 8 vertices of the original cube and 8 shifted in the fourth dimension. It has 32 edges - 12 each give the initial and final positions of the original cube, and 8 more edges will "draw" its eight vertices, which have moved into the fourth dimension. The same reasoning can be done for the faces of the hypercube. In two-dimensional space, it is one (the square itself), the cube has 6 of them (two faces from the moved square and four more will describe its sides). A four-dimensional hypercube has 24 square faces - 12 squares of the original cube in two positions and 12 squares from its twelve edges.
In a similar way, we can continue the reasoning for hypercubes of a larger number of dimensions, but it is much more interesting to see how for us, inhabitants of three-dimensional space, it will look like four-dimensional hypercube... Let's use the familiar analogy method for this.
Take a wire cube ABCDHEFG and look at it with one eye from the side of the face. We will see and can draw two squares on the plane (its near and far faces), connected by four lines - side edges. Similarly, a four-dimensional hypercube in three-dimensional space will look like two cubic "boxes" inserted into each other and connected by eight edges. In this case, the "boxes" themselves - three-dimensional faces - will be projected onto "our" space, and the lines connecting them will stretch in the fourth dimension. You can also try to imagine a cube not in projection, but in a spatial image.
Just as a three-dimensional cube is formed by a square shifted by the length of a face, a cube shifted into the fourth dimension will form a hypercube. It is limited by eight cubes, which in perspective will look like a rather complex figure. The part of it, which remained in "our" space, is drawn with solid lines, and that which has gone into hyperspace, with dotted lines. The very same four-dimensional hypercube consists of an infinite number of cubes, just as a three-dimensional cube can be "cut" into an infinite number of flat squares.
Having cut eight faces of a three-dimensional cube, you can expand it into a flat shape - a sweep. It will have a square on each side of the original face plus one more - the face opposite to it. A three-dimensional unfolding of a four-dimensional hypercube will consist of the original cube, six cubes "growing" from it, plus one more - the final "hyperface".
Tesseract properties are the continuation of the properties of geometric figures of lower dimensions into 4-dimensional space, presented in the following table.
