I began the day by carefully cleaning my guitar. It was my first stage performance. The guitar would shine along with me on the stage. As I wiped, I just looked at the curved shape at the guitar’s sides. I liked the shape a lot, for no particular reason. A friend of mine, a mathematician, had once mentioned the curved shape of the guitar’s sides is a hyperbola
I was not interested in knowing different geometric shapes. However, the word hyperbola has remained in my mind since then. I just placed the guitar on the table and went in to get my bag. Thud! I heard a huge sound and was on the ground in the next movement. I slowly opened my eyes; there was no pain in my body, which meant I was not hurt.
The new visual world!
I looked up, finding myself rooted to the spot in shock. I then felt something was wrong with my eyes, or maybe my brain was just imagining things. The flat walls of my room look curved! Why doesn’t anything around me look flat anymore? I slowly get up and turn around to see if everything is fine with my guitar.
I stood shocked again, the table I had just placed my guitar on looked so far away! When did my room expand so much that I had to walk a long distance to get to my table? I glanced around; my room had lost its cuboid shape. I remember that my room was cuboid until a few minutes ago, but now it is no longer cuboid; it’s like a deflated cuboid!
Image credits: math.stackexchange.com
I was now scared; I wanted to escape my room. I just ran out of the room through the door near me.
This isn’t my house, I shouted! Everything looks so weird. Everything has expanded? No, but still, everything looks so far away.
The space around me appeared to stretch out infinitely, my house now unrecognizable. The door looks so far away, and the walls seem to meet at the door that looks like a point!
Did the door shrink, or is it because I am far away?
I start running straight to the door. But why do I feel like I am running on a curved path? I look back to see even the staircase looks no more parallel but more curved. The width of the stairs no longer seems to be constant. The width seems almost zero at the top end.
I ran towards the front door to escape the house and find help. Is something wrong with me or my house? I thought it would take a long time to reach the door. But no, I just took the same time as I used to take always. And now the door, too, seems to have not shrunk!
What is happening?
I step out of my house. All the roads, buildings, everything is curved and disoriented. I spotted a person walking away from me; I tried to walk to him soon. Just a second, I tripped but balanced myself before I fell. But I have lost the person who was just here a second ago. How could he disappear so fast? It looks like he doubled his pace of walking at each step.
Image credits: coffeeshopphysics.com
There was a vast map hung at the colony where I live; I had unknowingly approached it. I just walked to it. The map was no more on a rectangular outline but a circular outline. Lost in confusion, I dialed the helpline no, below the map. The helpline was introduced to help out the people new to the colony who are lost with none around and maps need to be helping them.
Calling the helpline number felt strange, considering I had lived in the same colony since birth. The helpline person was quick to arrive. They asked me how they could help me.
I could only stammer out: why is everything curved? Why is the map circular, with even its routes curved with no straight roads?
But the person looking at me weirdly told me those were the straight paths. The shortest and parallel routes are the curves, he said.
I was dumbfounded. Aren’t straight lines supposed to be the shortest paths between two points? And how can parallel lines be curved lines? Curved lines are the ones which would eventually converge into a loop.”
He looked like my mathematics friend who would start lecturing at random comments, “The parallel lines do seem to converge or intersect but eventually will never do. And these are the parallel lines. You can try checking yourself by measuring the distance between the lines at random places. They are constant.” He said.
I tried to prove to him that the curved lines meant the distance between lines varied, but I proved myself wrong. They were indeed equidistant lines.
“Hold on,” I told myself, trying to calm my racing heart. “Ok, and why are the buildings so disoriented? As if someone pulled it on the four edges?” I asked.
Image credits: Pinterest
“That is how the cuboid looks,” He said calmly. I stood speechless. I lost myself.
I murmured to myself, “How can a cuboid be like this in a flat Euclidean space?”
“Flat Euclidean space!” the person gasped in wonder and started laughing. “Where do you hail from? This isn’t a flat Euclidean space where we live. It is in curved hyperbolic space.” He said in between laughter.
Image description: Left: Euclidean / Flat space; Right: Non-Euclidean / Hyperbolic space
Image credits: Research gate
I just fainted. When I regained consciousness, I found myself in a hospital, still with curved walls around. It was my luck that I hadn’t fainted for the second time when I looked at a world map hanging over the wall. The map was to show the locations of the hospitals on various continents. Still, my brain never registered the locations as I was shocked to see the map itself.
Image credits: phys.org
While I was still coping with my new reality, I overheard a child asking for a scoop of ice cream, but their parent was trying to trick them with an empty cone. Is that a cone?
Image description: Crochet model of pseudosphere – the hyperbolic equivalent of a cone.
Image credits: theiff.org
The angles in triangles are less than 180 degrees, which contradicts the reality I have always known.
Image credits: mat.uab.cat
Where am I? Am I in some other world? The thoughts wandered around. The world is curved; parallel lines are curved; maps are circular; the distance seems to increase exponentially! Imagine a giant white ball moving away; it looked like the one below.
But all I remember was it should have looked like the below:
Image credits: virdir.ncsa.uiuc.edu
I would not even be able to play with a ball in this world. But why has the world changed like this? And why is it only me who feels new to this world?
My consciousness began to slip away once more. Slowly murmuring to myself. We live in Euclidean space, not a hyperbolic space.
Back to the reality
Once I regained consciousness, I found myself back in my room. I woke up with a start. Everything around me now looked normal, with flat walls, a cuboid room, door not looking far away. I must have tripped and fallen unconscious; it was a bad dream! I exhaled slowly and relaxed.
I now laughed at myself! Why did I even have such a dream? Our world remains a Euclidean space in short distances, even though the Earth is a sphere. There is no reality of my dream.
As I sat on the sofa, I sat on the remote of the TV, turning it on accidentally. As I took the remote to switch it off, the voice came out of the TV.
“Hyperbola has real-life applications; You know how satellites orbit the Earth? They actually follow a path similar to the hyperbola, the same shape you might see in the curves of your guitar or the arch of a bridge. Radio systems’ signals employ hyperbolic functions, microscopes, telescopes, and televisions all are made up of hyperbolas.
Scientists have proposed that the universe might not be flat Euclidean, it might be hyperbolic. Mathematicians have found that considering the hyperbolic universe can explain the universe’s expansion without considering any dark matter.”
The program on TV continued, but my brain was too tired to take in anything else. Even the flat space I now live in keeps reminding me of the curved world I had experienced. It makes me keep questioning reality. Was it a dream, or was I teleported into a different hyperbolic world? Is the existence of a hyperbolic world a possible reality? I had no answer to the question posed by my mind.