Expanding to the fourth dimension

Is it possible for a person in the third dimension to contemplate the fourth. You start with a single point. Stretch it into a line. Pull it into a plane. Lift it into three dimensional space. Then you take your space and unfold it to the next logical place.

Think of a graft. If you wish to express an equitation that has no variables 1+2=3 then you would make a point on paper and give it that value.
If the equitation has one variable x+1=2 then it would be expressed on a line. This way you would be showing not only where it is but also where it is not.
Two variables x+y=4 would be expressed on common graph paper.
Three dimensions would require a room. The two dimensional 'line ' would express itself wrapping through space.

I know that four dimensional space exists because of the equitation x+y+z+a=1.
Problems present everyday with many more variables than this. Therefore I know it exists. The difficulty is that I have no cognitive remembrance of being in these dimensions.

If I wish to express three dimensions on a sheet of paper I could chart two of the dimensions and write the third variable next to integers on the graph. A person who has an awareness of three dimensional space could then imagine the graph expanded upwards. This would work with simple equations but would be lost quickly in the more complex ones such as a sphere. Likewise, I could express four dimensions by placing values on integer points of a three dimensional graph.

In the movie The Time Machine we are given an example of travel through the dimension of time. Strapped to a seat one travels forward or backwards through time. We are allowed to observe how one spot would experience such movement. It is however only expressing one dimension. The seat does not move during the travel. If the seat could move freely in the first three dimensions while traveling in time then we would see it in four dimensions. If it were possible to interact with the environment then we would be in four dimensions.

In reality we are already there.