Imagining fourth dimension

We live in three dimensional space. (In fact, it could be only limitation of our mind, which makes us to believe that space is 3 dimensional). It's hard to imagine how the objects in 4th and higher dimensional space look like. A point is zero dimensional entity, a line has one dimension, a square has two dimensions (length and breadth). A cube is three dimensional and the analogous object in fourth dimensional is called Tesseract or Hypercube.

When I first time came across "Tesseract" (Hypercube) I could not just
understand how it is made. Strictly speaking, nobody can imagine the objects in 4th dimension space at all. But still we can draw them. How ? Can we draw a 3D cube on a 2D paper ? yes, in the same way we can draw 4D Tesseract on 3D or even 2D space. What we draw is actually projection of 4D object on 3D space.

Courtesy: http://www.instructables.com/id/Make-a-%22real%22-tesseract---4D-cube./
Projection of Tesseract on 3D space

 Courtesy:  http://upload.wikimedia.org/wikipedia/commons/7/7c/Hypercube_cubes.png
Projection of Tesseract  on 2D space (drawing on paper)

It's important to understand maths behind hypercube. It goes like this - A point is zero dimensional entiry. It has no edges (boundaries). A line is one dimensional entity whose boundaries are 2 points. Square is two dimensional entity which has lines as its boundaries. And it has 4 lines. Similarly cube is made up of squares as its boundaries. And it has totally 6 squares. So a general rule can be stated as - the entities in lower dimensional space form boundaries of entities in higher dimensional space. And number of boundaries of entity increases by 2 when you move above by one dimension. By this rule, we can easily conclude that the Tesseract is made up of cubes as its boundaries and it has 8 boundaries.

We can draw tesseract in the same way we draw a cube. For drawing a cube, we draw two squares and join their respective corners. So for drawing a tesseract we first draw two cubes on a paper and then join their respective corners points by 8 lines. Now, can you find out 8 cubes in above figures ? What we have drawn is a projection of tesseract, but the animals in 4 dimensional world (if exist any) would not see it like this. It's completely beyond our perceptions because our cognition is designed only for perceiving 3D objects.