Have you ever been in Paris? I am still in love with Paris even though it has been more than a few years since my first and only visit to Paris. You can only image my excitement, when I learnt that behind one of the monuments I visited is math. This monument in question is Arche de la Défense (left).
We all know what a square and a cube are. A square is a two dimensional shape with four equal sides and a cube is a three dimensional shape with six square faces. We also know that a square and a cube are basically the same thing. They differ only by the dimension of the space where they are defined. A square is defined in a two dimensional space and a cube in a three dimensional space. Mathematicians can define “cubes” also in four, five or higher dimensional spaces but we can only think about them in an abstract way.
Can you imagine how a four dimensional cube looks like? Of course not! This is where the projections come in handy. The idea behind Arche de la Défense is a projection of a four dimensional cube to a three dimensional space. What does that mean? If I asked you to draw a cube on a piece of paper, you would probably draw something like this.
These two drawing are only two possible ways how to project a cube (which is defined in a three dimensional space) to a two dimensional space (a piece of paper). Projections usually come up when we want to draw a higher dimensional object in a lower dimensional space. This gives us an idea how does this higher dimensional object look like (remember we can only think of objects in four or higher dimensional spaces in an abstract way because we are confined to three dimensions). Using this idea we can project a four dimensional cube to a three dimensional space.
If we take a closer look at the right projection of a cube on the above picture, we can see that it is made up of a small square inside of a big square. If we do the same thing but in three dimensions instead of two, we have to draw a small cube inside of a big cube. Now this is one of the projections of a four dimensional cube to a three dimensional space. Below you can compare this projection with Arche de la Défense.
Arc de Triomphe
- Danish architect Johann Otto von Spreckelsen and Danish engineer Erik Reitzel designed it to be a 20th-century version of the Arc de Triomphe,
- it is a monument to humanity and humanitarian ideals,
- it complets the line of monuments that forms the Axe historique (among other monuments are Arc de Triomphe, Obelisque at Place de la Concorde and Le Louvre),
- it forms a secondary axe with the two highest buildings in Paris, the Tour Eiffel and the Tour Montparnasse.
Source: http://www.mathsinthecity.com/sites/4-dimensional-cube-paris