This new hypercube is now in the second dimension because it is able to differ in two measurements: width and length.
We can call this the 2-HC, and similarly to your 1-HC line, if you expand the square infinitely, you would create a two-dimensional space. The only direction left to expand our hypercube in is height. To do so, you must generate your 2-HC and then raise it into the third dimension to create a cube.
The width, length and height of this new shape can be measured, and all of its angles equal 90 degrees. Again, this can also be expanded infinitely to make a whole three-dimensional space, and this is obviously the dimension in which we humans live.
Try to imagine a fourth direction. Honestly have a go at it for a few seconds. We have no more option for expanding our hypercube in our three-dimensional world, but it can be done in the fourth dimension of spacetime.
We call this a tesseract. There are several ways of trying to illustrate this expansion, but they would be very difficult to explain in this article. Instead, I will take a practical approach, but it is interesting to ponder this fourth expansion, and there are some good motion graphics on the web if you give it a search. Now, picture yourself looking down on a piece of paper and imagine that this piece of paper is home to a world existing only in two-dimensional spacetime.
How does this helps us visualize 4-D objects? I searched that we can at least see their 3-D cross-sections. A Tesseract hypercube would be a good example. Can we conclude that a 3-D cube is a shadow of a 4-D tesseract? But, how can a shadow be 3-D? Was the screen used for casting shadow also 3-D ; or else, what way is it different from basic physics of shadows we learnt? This makes me think about higher dimensions discussed in string theory.
What other areas of Mathematics will be helpful? First things first: your brain is simply not made to visualize anything higher than three spacial dimensions geometrically. All we can do is using tricks an analogies, and of course, the vast power of abstract mathematics. The latter one might help us understand the generalized properties of such objects, but does not help visualizing it for us 3D-beings in a familiar way.
I will keep things simple and will talk more about the philosophy and visualization than about the math behind it. I will avoid using formulas as long as possible. If you look at an image of a cube, the image is absolutely 2D, but you intuitively understand that the depicted object is some 3D-model. Also you intuitively understand its shape and positioning in space. How is your brain achieving this? There is obviously information lost during the projection into 2D.
Well, you or your brain have different techniques to reconstruct the 3D-nature of the object from this simplified representation. For example: you know things are smaller if they are farther away. You know that a cube is composed of faces of the same size. However, any perspectively correct image shows some face the back face, if visible as smaller as the front face. Also it is no longer in the form of a square, but you still recognize it as such.
We use the same analogies for projecting 4D-objects into 3D space and then further into 2D-space to make an image file out of it. Look at your favorite "picture" of a 4D-cube. You recognize that it is composed of several cubes.
For example, you see a small cube inside, and a large cube outside. Actually a tesseract technical term for 4D-cube is composed of eight identical cubes.
But good luck finding these cubes in this picture. They look as much as a cube as a square looks like a square in the 2D-depiction of a 3D-cube. The small cube "inside" is the "back-cube" of the tesseract. It is the farthest away from you, hence the smallest. The "outer" cube is the front cube for the analogue reason. It might be hard to see, but there are "cubes" connecting the inner and outer cube. These cubes are distorted, and cannot be recognized as cubes. This is the problem of projections.
A shape might not be kept. Even more interesting: look at this picture of a rotating tesseract. Science The Human Brain Why are people's brains different sizes? Science The Human Brain Is the human brain still evolving? Science The Human Brain Can brain damage lead to extraordinary art? Science The Human Brain Could a brain scan tell you if you're going to become a criminal?
Science The Human Brain Are you really only using 10 percent of your brain? Science The Human Brain Do men and women have different brains? Science The Human Brain How does the brain create an uninterrupted view of the world? Entertainment Brain Games Brain Games.
How does gravity work? Sources Cole, K. July March 7, May 12, Cite This! Try Our Sudoku Puzzles! For the object below, the curves on the sphere cast shadows, mapping them to a straight line grid on the plane. With stereographic projection, each side of the 3D object maps to a different point on the plane so that we can view all sides of the original object. Stereographic projection of a grid pattern onto the plane. Just as shadows of 3D objects are images formed on a 2D surface, our retina has only a 2D surface area to detect light entering the eye, so we actually see a 2D projection of our 3D world.
Our minds are computationally able to reconstruct the 3D world around us by using previous experience and information from the 2D images such as light, shade, and parallax. The faces of the cube radially projected onto the sphere. Placing a point light source at the north pole of the bloated cube, we can obtain the projection onto a 2D plane as shown below.
View the 3D model here. Applied to one dimension higher, we can theoretically blow a 4-dimensional shape up into a ball, and then place a light at the top of the object, and project the image down into 3 dimensions.
Right : computer render of the same, including the 2-dimensional square faces. Here the 4-dimensional edges of the hypercube become distorted cubes instead of strips. Just as the edges of the top object in the figure can be connected together by folding the squares through the 3rd dimension to form a cube, the edges of the bottom object can be connected through the 4th dimension.
0コメント