Dear experts, How to make a real tesseract ( 4d cube) with procedure if possible
Answer :
Constructing a 4d-Cube or Tesseract
Now we come to our main topic. We want to study projections of four- or higher-dimensional objects, in particular of the four-dimensional cube or Tesseract. To get an idea what we are talking about, let us first construct the Tesseract one dimension at a time:
We begin with a point that we might call a 0-cube as no dimensions are yet involved.
Then we choose our first dimension, say horizontal, and displace the 0-cube or point one unit in that direction, connecting the starting point and the endpoint. What we get is a line segment or 1-cube.
Now we find the first interesting shape: Choose a direction or dimension perpendicular to the first one and displace the 1-cube one unit in the new direction ("down" in our picture), recording the path of each vertex. What we end up with, is clearly a square or 2-cube. To keep track of our dimensions, we display as well the vector star displaying the directions or dimensions chosen.
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If we want to continue, we encounter the first difficulty: we need to choose a third direction, perpendicular to the first two, whence we have to move out of the paper plane . But our brain is so used to the three-dimensional world that it constructs the correct image from the plane projection! Moving the square in a southwest direction creates the image of the square moving towards us from the screen. Recording again the paths of each vertex, we obtain all the edges of an ordinary 3-cube:
Note that the visual clues are not quite sufficient for our eyes or brain to decide whether the grey square should be in front or in the back and accordingly the cube appears in two different ways: looking at the picture for some time, the grey square jumps in and out. We choose a fourth direction, that should be perpendicular to the three given ones, whence we now leave the physical world. It should help to visualise the following two-dimensional projection of the 4-cube or Tesseract : the dark grey cube moves in time along the fourth direction, chosen here to point northwest. Its future end position is shown in light grey.
Constructing a 4d-Cube or Tesseract
Now we come to our main topic. We want to study projections of four- or higher-dimensional objects, in particular of the four-dimensional cube or Tesseract. To get an idea what we are talking about, let us first construct the Tesseract one dimension at a time:
We begin with a point that we might call a 0-cube as no dimensions are yet involved.
Then we choose our first dimension, say horizontal, and displace the 0-cube or point one unit in that direction, connecting the starting point and the endpoint. What we get is a line segment or 1-cube.
Now we find the first interesting shape: Choose a direction or dimension perpendicular to the first one and displace the 1-cube one unit in the new direction ("down" in our picture), recording the path of each vertex. What we end up with, is clearly a square or 2-cube. To keep track of our dimensions, we display as well the vector star displaying the directions or dimensions chosen.
If we want to continue, we encounter the first difficulty: we need to choose a third direction, perpendicular to the first two, whence we have to move out of the paper plane . But our brain is so used to the three-dimensional world that it constructs the correct image from the plane projection! Moving the square in a southwest direction creates the image of the square moving towards us from the screen. Recording again the paths of each vertex, we obtain all the edges of an ordinary 3-cube:
Note that the visual clues are not quite sufficient for our eyes or brain to decide whether the grey square should be in front or in the back and accordingly the cube appears in two different ways: looking at the picture for some time, the grey square jumps in and out. We choose a fourth direction, that should be perpendicular to the three given ones, whence we now leave the physical world. It should help to visualise the following two-dimensional projection of the 4-cube or Tesseract : the dark grey cube moves in time along the fourth direction, chosen here to point northwest. Its future end position is shown in light grey.