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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.

Hypercube animation

 

This film happen in a two dimensional space; maybe the best way to represent a hypercube is by building the object in a three dimensional setting. Shadows and surfaces suggest the existence of the extra-dimension. 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):

 


Dimensions-math.org has created several films available for free online.




A NECESSARY REMARK

Almost every day math instructors draw points, lines and circles on the board. Be advised that, indeed, none of these mathematical objects can be drawn. What we instructors draw are dots, bars and discs. Check this figure out:

The space, in general, is a set of points. Whenever we refer to a specific point, for instance, the center of the disc, is a non-dimensional place, a position in the space equidistant from all others points on the curve, circumference. Imagine now lines from one corner of the square to the opposite corner, diagonals. We only draw them in order to facilitate our understanding of the figure. Once drawn, they have width, thickness. They are no longer lines.

In this regard of drawings geometric figures, there is something problematic. For me as instructor a poor drawing is a mathematical offense of first degree: the fact that teachers and professors alike keep drawing freehand such imperfect circles, lines, squares and triangles in such a way that the aim of the explanation is obscured. We should try compasses, rulers, protractors, multimedia projections, etc. Do not disturb my circles: someone, by the name of Archimedes, died in defense of these magnificent figures: those shapes of constant curvature, whose area is the greatest for a given perimeter and also the ones that encode the mistery of a number: