What is ‘four-dimensional space’?

If I have to explain it in such a way that one can imagine it directly, understand: I cannot!

I try it slowly and sometimes a bit simplified, the physicists among the users may forgive me!

Everyone has a different idea of it.Some people say very quickly: time.

But the question was about the “room”.Or was it a mistake?

So let’s just like in the Feuerzangenbowle: There mer put us again janz stupid-

So a cube is three-dimensional, everyone knows that!There is also a four-dimensional body, a cube in four-dimensional space. To understand this we have to go backwards and look at a 2D cube.

Put a lamp on a cube and look at the shadow.

A two-dimensional cube is nothing more than a quadrilateral:loud right angles and four corners.

And one step lower; a square from the side, so to speak?

Right, a line.

And a line?Exactly one point.

  • Conversely, one could say that if we pull a point in one direction apart, we get a line .
  • If we draw a line, we get a square, 4 corners
  • let’s draw a square, we get a cube of 8 corners
  • if we draw a cube, we get a 4 D body.

16 corners

Nochmal: If you expand a square (2D, four corners) to a cube (3D, 8 corners), you can see that the number of corners has doubled.

Because when the square is “pulled apart”, the 4 old corners stop and 4 new ones form on the opposite side.

The same thing happens now with the extension of the 3D cube to the four-dimensional hypercube: the old corners remain standing, and in the new dimension there is a new opposite of each old corner, so the hypercube has 16 corners.

Ok, let’s just ask ourselves:

How many edges do these bodies have?

  • The shadow, the square has as already said 4,
  • Let’s come tum cube.

If you pull apart the edges of the square, a new square is created as usual on the opposite side, thus making 8 edges first. From each of the 4 corners of the square, however, a new edge is created. So makes a total of 12 edges on the 3D cube. Nal recount…. Agrees!

  • Now we go to the edges of the hypercube and pull the apart; 12 edges of the old plus 12 of the opposite 3D cube makes 24, and each of the 8 corners of the 3D cube becomes an edge in the hyper cube, yielding a proud 32edges.
  • And how many areas?, that would be the next question.

    Count ,,,,

    Let us now come to the areas: we have counted:

    • The square has one,
    • the 3D cube has 6

    When the square is pulled apart from the cube, the original surface stops and a new one forms on the opposite side.But that wasn’t quite it, in between 4 have been created, these come from the 4 edges of the square. So we found the known 6 areas.

    This is also the case with the 4D object: when the 3D cube is expanded, the original cube stops for the first time and a new one forms opposite (this counterpart is already in the fourth dimension).So we get 12 surfaces of old and new cubes. And what happens in between? Each of the 12 edges of the original 3D cube expands to a surface as it pulls apart. This means we have 12+12=24 surfaces on the 4D hypercube.

    So our 4-D object has

    16 corners,

    24 side surfaces and

    32 edges.

    ….. and now please think a little!

    • A one-dimensional straight line is delimited by two zero-dimensional points, in German “the sausage has two ends”.
    • A two-dimensional square is delimited by 4 one-dimensional straight lines.
    • And a three-dimensional cube is enclosed by 6 two-dimensional side surfaces.
    • Thus, the narrowing objects always have one dimension less than the object itself.

    This means that the4D hypercube is fenced by three-dimensional objects.

    Capt?We are not so cute in the 3-D universe! By the way, there are 8, and who is in the others?

    Oh, I love Dali, he tried it:

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