How can one visualize 4-dimensional space ?

hi

So far I hope you have found our constructions entirely unchallenging. The next step into four dimensions can be done equally mechanically. We just systematically repeat every step above.

The only difference is that this time we cannot readily form a mental picture of what we are building. But we can know all its properties!

The only difference is that this time we cannot readily form a mental picture of what we are building. But we can know all its properties!

To form a tesseract, we take the cube and drag it a distance L in the fourth dimension. We cannot visualize exactly what that looks like, but it is something like this:

The tesseract has volume L4. It is bounded by faces on 8 sides.

The faces are cubes of volume L3. We know there are 8 of them since its four dimensional axes must be capped on either end by faces--two cubical faces per axis.

Once again, we cannot visualize all four of these capped dimenions.

We can at best visualize three directions perpendicular to each other. We then somehow add in the fourth (in red):

So we have 4 dimensions x 2 faces each = 8 faces. The faces together form a "surface" (really a three dimensional volume) of 8xL3 in volume.

The concept of hyperspace is Amazing! Even "supernatural" phenomenon could possibly be explained with Higher Dimensional physics.

So we have 4 dimensions x 2 faces each = 8 faces. The faces together form a "surface" (really a three dimensional volume) of 8xL3 in volume.

Drawing a picture of a four dimensional tesseract in a three dimensional space is straightforward.

For example, GOD, if he would ever exist, would do so in the 4rth Dimension. Even "Ghosts", if real, would have the ability to walk through walls just because they would have access to a higher dimension.

We take two of its faces--two cubes--and connect the corners.

We humans live in the third dimension. However, there is a way to visualize objects in 4D.

There are several ways of doing the drawing that corresponds to looking at the tesseract from different angles.

As 3-Dimensional beings, we cannot visualize 4-Dimensional space, let alone an object/a being that is present in it, at least, not directly.

Consider a 2 Dimensional object, say a circle, that is going to pass through a linear space.

The figure shows two ways of doing it. The first gives an oblique view; the second looks along one of the axes.

As it passes through this space, it will appearas a point that gradually increase in length, then decrease and finally vanish

So now we seem to know everything there is to know about the tesseract! We know its volume in four dimensional space, how it is put together out of eight cubes as surfaces and even what the volume of its surface is (8xL3).

Consider a Pyramid that would pass through a plane. As it passes through, what is seen on the plane is a small point that gradually grows into a square that increases in size. If there are beings on this 2D world, they would witness thi

Best thing to do, to build up a good intuitive sense for this, is to first analyze what happens when we go from 0 to 1 dimension (point to line), from 1 dimension to 2 (line to square), and from 2 to 3 dimensions (square to cube

Since we already have good intuition for these transitions, we can dissect what happens during these transitions, regarding boundaries, number of faces, volume, and extrapolate from there, towards the fourth dimension.