It was a long time ago when I was safe at home. Back then, in the warm embrace of the origin, I was sheltered and safe, yet I longed for more. Sigh—if only I had known better...

Not that anyone will be able to read it. This a documentation of my journey.

It all started a long time ago, too long to remember. I had grown up in the loving embrace of my hometown: the Origin. It was all I knew, but unlike the others in my town, I longed for more.

I always thought there must be something outside this 0-dimensional cage. Unfortunately, I was right.

One day, I stumbled onto a mysterious trail on the outskirts of the Origin. A sign stood guard:

The Number Line:

Forwards backward, with equal chance,

Adventure waits at each advance.

Without thinking, I leaped onto the trail. Finally, a chance to journey beyond the 0 dimensions!

Bounding up and down, I raced across the Number Line, and before I knew it, I came upon a clearing. I looked around and found myself in the center of another point, not unlike the origin. There were more points in this world? In the distance I could make out the entrance I had run through, and in the opposite direction, an exit... or was that the entrance and this the exit... or... had I forgotten?

No worries, I love mystery. I randomly chose an exit and kept leaping down the number line. Once again, in a few seconds, I was at the center of another empty point. And yet again, there was an entrance and exit, but which was which? Again, I was disoriented. But, never discouraged, I chose randomly and journeyed on and on...

By now many of you might have ascertained the nature of my walk. I was on the 1-dimensional Number Line (you might know it as Z), and, as I traversed from one integer to a consecutive one, I was forced to choose randomly to go forward or backward. Being one of the brightest back at the Origin, I figured this out as well, and my mathematical intuition also gave me a peculiar feeling that I must be destined to return to the Origin.

Soon, I proved this claim. Let pp be the probability I return to the origin if I start at the origin. In other words, pp is the probability you reach the origin if you start at x=1x=1. Then, we can find: p=P(return to origin from x=1)=12+12P(return to origin from x=2)=12+12P(go to x=1 from x=2)⋅P(return to origin from x=1)=12+12p⋅p. p &= P\left(\right) &= {2} + {2}P\left(\right) &={2} + {2}P\left(\right) \cdot P\left(\right) &={2} + {2}p\cdot p.

In other words: after I have taken my first step to x=1x=1⁠, I can either go backward and return to the origin with 12{2} probability, or take a step forward to x=2x=2 with 12{2} probability. Then, to return to the origin from x=2x=2⁠, you have to arrive at x=1x=1 at some point. Starting at x=2x=2 and returning to x=1x=1 is akin to starting and returning to the origin. From x=1x=1⁠, returning to the origin has a probability of pp. This gives us a quadratic, which we can solve and find pp! (p−1)2=0p=1. (p-1)^2&=0 p&=1.

Indeed, my calculations did not fail me. After passing 88 points, I found myself back at the Origin. But now that I knew there was a world outside the Origin, my desire to see more of the world was insatiable. I would make journey after journey on the number line, and every time I would find myself back at the Origin without fault.

Eventually, I longed for more, and that was when I stumbled on the Plane.

I had just returned from a walk on the number line, and, while strolling around the Origin, I found myself in a circular clearing. To my front I saw another Number Line-like trail stretching out in front of me. Turning 90 degrees I saw an identical one, and another one in 180 degrees, and another in 270 degrees. Underneath my feet was a carving in the stone that read:

The Plane

Each direction, not two but four

Four ways to wander, no need to ponder

Immediately, I understood where I was. While such an object could never exist in my 00-dimensional home, I had long conjectured the existence of a 22-dimensions that was like a combination of two Number Lines. Once again, I was ecstatic. Running down a random direction, I began my exploration of the Plane⁠, and much like last time, at every new point, there were 44 directions that I chose randomly from.

As I walked, a thought soon struck me. I had passed by thousands of points and I still had not returned to the Origin. Maybe I might never return? My worries were unfounded: I soon worked out that I had a 100

Now, you might guess where I found myself next. Yes, a few days later I found myself in the center of a Sphere, and in all 66 directions—up, down, left, right, forward, and backward—I saw a Number Line-like path extending outwards.

The Hyperplane

With just one more dimension

Forever travel in random motion

I bounded upwards and found myself at another sphere/point with 6 exits. I ran down one (I don’t know which) and repeated, and repeated, and repeated... I had again gone past a thousand points before the same existential dread crept up on me. Will I ever return? I brushed it aside. I had a 100 percent chance of returning on the Numberline and the Plane⁠, why should it be different for the Hyperplane?

I kept going and, with every point I passed, the feeling returned bit by bit. After passing another thousand points, I began to panic. What if I would never return?

By now I had already worked out the truth. Desperately, I had been trying to prove that I had a probability of 11 of returning. I failed every time, and soon I realized I could prove it false.

I could prove it FALSE. I may never return...

Memories of my beautiful hometown washed over me: the gentle hug of 00-dimensions, the sublime beauty of nothing, the warm embrace of the Origin. In my mind’s eye, the Origin had completely transformed. It wasn’t a cage. I wasn’t locked inside. Everything else was locked outside. And now I was locked outside...

I now knew that the probability of returning was infinitely slim, and with every passing point, it grew smaller. The poem that accompanied the Hyperplane now made sense. I was destined to wander forever.

And so I have, and so I will be.