# Connections and Dimensions

I wish I could say I came to this realization on my own. That would be fraudulent, a word that I had an unusual difficulty spelling, seriously who has trouble spelling Fraudulent? It sounds like it is spelled…good lord I’m hopeless sometimes.

Anyways back to the story. This is a really interesting piece of mathematics that has delighted me each time it is described. Hopefully I will not use incorrect terminology when I get to the last part, but mistakes are made as we build our minds! So sit back and enjoy this quite short ride.

Lets say we have a piece of paper, for this I’ll use MSPAINT and give us an awesome piece of fake paper!

Hopefully I haven’t lost you yet. We have two points, these are 0 sided objects that are not yet connected. They are hopelessly alone and wish to meet one another. It is a simple task to bring them together and make our first jump in something beautiful. Hopefully I’m not overstating how much I enjoy this.

They’ve joined! Friends forever! A line is a one dimensional object, it goes left to right but does not go up and down. It is connected by 2 0-dimensional objects. But they are connected now and are super happy with their lot in life. Where can we go from here? Well the next step up is a shape that is quite popular in Elementary school and a go-to slang name for boring people.

Welcome to the square! A square is a two dimensional object that is connected by 4 1 Dimensional Lines, which are themselves connected by 2 0-dimensional points (each). This next one is going to be a little hard for me seeing as I’m using paintbrush. But bear with me, we’ll see how it goes. What do you get if you step up the design? Where once there was a line connected by 2 0-dimensional points, it became a square consisting of 4 1-dimensional lines. What do we get if we connect 6 2-dimensional squares? Oh you can taste it I bet!

A Cube! We jumped from 0-dimensions, to 1-dimension, to 2-dimension, and now we’ve hit 3! In each case we took the amount of points and doubled it (2 for a line, 4 for a square, and 8 for a cube), each case we took the dimension N and added 1 from the previous amount (0+1=Line, 1+1=Square, 2+1=Cube), and we saw a beautiful evolution of the objects we made. We started with points, these lead into a line, the lines lead into a square, and the squares lead into a cube.

But where do you go next? It depends on who you ask. The math here is pretty simple, or at least it appears to be to me, a square is 4 lines connected at 90 degree angles, a cube is squares connected at 90 degree angles, and the next object know as a “4-Dimensional Square”, “Hypercube”, or my personal favorite “Tesseract”. Neil DeGrasse Tyson often says the first example, but the latter two, Hypercube and Tesseract, was spoken by Carl Sagan. I’m sure someone before him made it up, but he was the person I first heard it from. He mentioned it in the following great clip (an excerpt from Cosmos):

Back? Did he make your mind swoon? Truly a great loss. As Sagan mentioned the cube gives off the shadow of a square, and the Tesseract gives the shadow of a cube. We see the math, we see the patterns, and presumably we may be locked in our own flat land. I won’t go into it deeper, Sagan said it much more colorfully than I can. But obviously we come to the very popular gif (seeing as I can’t draw a Tesseract on paintbrush).

It’s a thing that can’t be truly be comprehended or observed in our universe. But it potentially provides an interesting consequence. What if, like the flat lander, we learned how to jump out of our dimension? To stand above our universe and move back and forth across the fourth dimension. What is this fourth dimension? Is it actually time? Could we indeed plop down anywhere in the timeline? Would we even come back in the correct universe when we did?

To close with Sagan’s words: “While we cannot imagine the world of 4th-Dimensions, we can certainly think about it perfectly well.” This is a philosophy I carry with me in all my mental and physical adventures. The universe is something truly out of scope of our imagination, so are things much smaller than that (as I’ve said before), but this does not mean we cannot think about it.

In fact the more unimaginable something becomes, the more delightful (I feel) it is to think about.