## Dimensions

The line has magnitude in one way, the plane in two ways, and the
solid in three ways, and beyond these there is no other magnitude
because the three are all.
Aristotle

For centuries there was no discussion: we all live in a three dimensional world.

History of knowledge changed on June 10 th, 1854. Bernhard Riemann gave his lecture On the Hypotheses which lie at the Bases of Geometry.Mathematics, science by extension, entered the n-dimensional world.

Then, in the early nineteen hundreds, the Special Theory of Relativity introduced the Minkowski spacetime: in this model popularized by Einstein's famous theory, time appears as the fourth dimension. For further information on Relativity and Minkowski spacetime, follow this link.

Such approximation to the multidimensional space goes beyond the scope of the page.

ONE & TWO

Let's explore all three dimensions and how are they related. In the Greek tradition of the Pythagoreans the Number One (they did not know the Zero) was considered the dimension generator; this way two points generate a line, the first dimension; there points a triangle, or the two dimensional plane; then, the fourth point make possible the construction of the three dimensional space, the tetrahedron:

Euclid's Elements, Part I, Definitions, define a straight line as “a line which lies evenly with the points on itself” in a clear reference to the line being generated by points; following the same pattern of thought, define a plane surface as a “surface which lies evenly with the straight lines on itself”.

THREE

Solids: The solid is generated by the connection of two planes –think of a square that is connected to another square or a square that is dragged perpendicularly to its original position--. Every face of the solid, a cube, for instance, is a two dimensional surface, a plane.

FOUR?

Is it possible to build a cube in such a way that every face of the cube becomes a cube? Or every corner of the cube is connected to another cube? If this is conceivable, then a hypecube is generated: a four dimensional object.

There is a monument in Paris that simulate the idea of the hypercube: One cube inside the other; sides connected, as shadows:

La Grande Arche de la Défense, Paris, by Johan Otto von Spreckelsen, Paul Andreu and and Peter Rice.

Perhaps a return to the second dimension allows us to understand the difficulties of how different dimensional spaces interact. This time a nice trip to Flatland, a novel populated by two dimensional beings —written by the British author Edwin Abbott, published in 1884: Flatland: A Romance of Many Dimensions.

I call our world Flatland, not because we call it so, but to make its nature clearer to you, my happy readers, who are privileged to live in Space. Imagine a vast sheet of paper on which straight Lines, Triangles, Squares, Pentagons, Hexagons, and other figures, instead of remaining fixed in their places, move freely about, on or in the surface, but without the power of rising above or sinking below it, very much like shadows — only hard and with luminous edges — and you will then have a pretty correct notion of my country and countrymen. Alas, a few years ago, I should have said "my universe": but now my mind has been opened to higher views of things.
Edwin Abbott, Flatland

Flatland: The Movie-- Official Trailer

Carl Sagan 4th Dimension Explanation-- (Flatland):