Derivative
"Yes! We are about to learn the secrets of the universe, James!"
"First thing I wanna know is, how do you puke inside a warp!"
"We're not currently three-dimensional, so you can't! We have to go through a warp to adjust ourselves for paradimensional existence!"
"Why are we both shouting!"
"The warp magnifies energy, and sound is energy!"
"Where the hell are my ears then!"
And so it was that the three semi-fearless partial heroes went hurtling through the warp into an unknown dimension. Light that did not exist was flying past them at a speed that was much greater than the speed of light that did exist, creating a kind of shifting conical perception.
In other words, they were tripping on math.
Rachel was still asleep in James' arms, which were shifting back and forth between being arms and being Slinkies in a wavelike pattern. Dr. Poeingk was taking on the appearance of a sphere. They were all beginning to acquire a blurry double of their physical bodies; it was the effect of dimensionalization.
The sphere that was Dr. Poeingk seemed to say, "Turn here!" James maneuvered somehow to enter a large, black region in the warp-space. They felt themselves moving, although their surroundings didn't change for a few minutes. James felt a strange sensation, as if he was in every part of the universe at once. He decided that if he ever met Euclid, he would clock the man for not telling people that this experience feels like -
"Schfla!" He flopped onto the ground, if it can be called that. "My head hurts."
James pushed himself up and realized he had landed on Rachel. He decided that in itself wasn't such a big deal; then he saw that she was now awake.
"What the - James, where are we?" She walked up to the very large sphere and kicked it. It rolled down what looked like a hill, making small, murmured "Ow!" noises. Rachel watched it hit the bottom and spin around a little before settle into a small crevasse.
"Rachel, you just kicked Dr. Poeingk."
"Say wha!?"
The ball waddled very slowly back up the hill on tiny circular pods. Why had that boy brought her along? She wasn't interested in the secrets of the universe. Dr. Poeingk made it back to the top of the hill just in time to see Rachel on the verge of killing James.
"Ms. Jeuille, we have traveled through a dimension warp. The main concept behind the warp is the derivative. Each dimension has a certain formula to describe the relation among all the objects in that dimension. The formula for each dimension is related to the previous one through the integral-derivative pair." Dr. Poeingk looked up. The concept of 'up' was relative to the dimension, so it didn't change just because of differences in dimension. James looked up too. The sky was orange and blue, with a lucid black object sitting in the deep center of the colored areas. James didn't know what the lucid black object was, and Rachel didn't care at the moment, but Dr. Poeingk had seen it a million times already.
Rachel was looking at everything else around her. Everything in this dimension had a much cleaner, more polished appearance than her home world, and a much stranger one. Trees were waxed white and had the shape of tents with the tops cut into a V and little yellow needles sticking out. The ground was a rolling blue and red, as if the entire landscape had been stripped of all but its elevation and someone had dumped those two colors all over randomly. The colors and shapes of the objects that littered the realm both clashed and produced harmony at the same time.
"Ms. Jeuille, James, we're in dimension 4.72180945127. I trust you are both in good health?" Dr. Poeingk began waddling off, and James started to follow him. Rachel stood adamantly behind, watching them with her arms folded and her lips in a sneer. She overestimated the effect this would have on James, who continued walking without looking behind himself. When they disappeared over a hill, Rachel unfolded her arms and gave forth a triumphant "Hmmpf," feeling that she had made a choice as a liberated young woman.
She had chosen to be completely stranded in a bizarre dimension with no living beings for what seemed to be miles.
James and Dr. Poeingk finally noticed her absence when they stopped at a very large tree. James, as was his style, became extremely concerned. "Maybe we should go back for her?"
"My boy, we don't know how far this dimension extends. If our two parties moved in opposite directions, we'll never catch her. I am far too slow in my current form. The best thing to do is find some inhabitants of this dimension."
"PEOPLE LIVE HERE!?"
"Not people, James. You'll need to get used to the term 'sentient beings.' I've met some exceptionally intriguing beings in the 6th and 7th dimensions. They taught me about these partial dimensions. The integer dimensions look a lot like the 3D plane, but partial dimensions are completely different. The correct term for them is 'fractal dimensions,' but I thought you might not understand fractals."
"I don't understand anything you say."
"Good. Let's find some beings."
Rachel had decided to just go in one direction until she found something. She wandered over hills, across plains, and down rivers of shimmering gray. Nothing presented itself for several hours, but there was no sun to set, so she kept walking. Along the way, she finally took time to notice that she hadn't escaped dimensionalization unscathed.
She was completely green. Her skin and clothes were totally verdant. She was infinitely saddened by the fact that despite her being all one color, there wasn't a single thing in this forsaken realm that could camouflage her. Onward she continued, with no thought in her mind save walking, until she reached a place where a small pig-like creature was chopping down one of the strange trees.
"Hello! Hello! Hello! Hello! Hello!"
"Gahh!" Rachel was taken by shock that the creature was talking to her. It never entered her mind that the creature was like herself; no one for miles around to talk to. The poor little being had forgotten how to properly use its mouth, and it was saying everything multiple times to make sure it was said right.
"How are you? How are you? How are you?"
"I'm...um...fine....and you?"
"Good! Good! Good!"
"Where are we?"
"I don't know! I don't know!"
"Can you tell me how to get to the third dimension?"
"No! No! No! No! No!"
Rachel had made an error in judgement that is common to first-time dimensional travelers; she had assumed that creatures in other dimensions knew everything about warping.
"What do you mean, you can't!?"
"I don't know! I don't know!"
"Argh!" Rachel walked past the tiny pig and simply kept going, with no thought in her head except to get to a real civilization. Ironically, James and Dr. Poeingk had found one.
"Greetings, travelers! Welcome to Julia. Our population is in excess of 4000, and we have many excellent restaurants and museums you can stop to check out. Which dimension are you from?"
"We are from the 3rd dimension." As Dr. Poeingk explained this, the being who had greeted them broke a broad grin of amusement.
"We don't get very many from that realm, you know. How did you come across this place?"
"I calculated a derivative-warp based on the fractal dimension 4.72180945127. The number just popped into my head."
"It is most excellent that it did! Please, enjoy yourselves in our humble realm of Julia!"
James and the doctor walked into the city limits and scanned it. It was a place of culture, almost like home to them. The beings of this realm had discovered how to utilize their surroundings, just like men of Earth long ago. Though James looked at all the wonders of the city, Dr. Poeingk was in search of something very specific.
"Ah! There's one!"
Fifty yards in front of them, a little to the left side, stood a device from the greatest science fiction of Earth. It was hued with a shining, lucid black. James looked up at the sky.
"Yes, my boy, make that connection! Come with me, we're going to find a way to get in that ship and go!"
"What about Rachel?"
Dr. Poeingk stopped and turned to face James with a slight sigh. "James, I think it's time I told you about this town. It is named Julia."
"Yes - I know its name, but what does that - "
"The next town is also named Julia. In fact, that's the name of all the towns here."
"What are you saying?"
"I'm saying that Rachel will appear right next to you in about 3 seconds."
Pop! She did, with an irritating noise like a cat clinging to dear life as it slides down a glass skyscraper. "Where the hell am I!?"
"Rachel, you're...green!" James laughed uncontrollably. Dr. Poeingk motioned towards the ship. Seeing no response, he waddled towards it as fast as he could.
The ship was shaped like a nose that had transmuted partially into a carrot and grown into the ground. Rachel strode purposefully toward the ship, although her only purpose was to appear angry. Upon reaching the ship, she had no way of doing that anymore, so she turned around and walked back towards James.
"What were you trying to accomplish by kidnapping me and taking me here? I don't want to be here! I want to get back to my own world, right now!"
"Well, sweetie, I think we're gonna do that right now. Look at Dr. Poeingk and that ship, then come with me."
James began walking towards the ship. When Dr. Poeingk saw this, he got in. James followed shortly. They strapped themselves in and prepared for liftoff.
"Hey, Rachel, you coming?"
Rachel stood and watched them with her lip curled in a defiant sneer. James shook his head and closed the door. Lights began flashing and the ship looked as though it would soon engage.
"Wait! Waaaaaait!" Rachel sprinted and lunged forward, jumping horizontally and diving into the ship. She held onto the outside of the vehicle and banged on the wall repeatedly. James quickly pushed open the door. "Get IN!"
In a heartbeat, she was inside and clinging to James like a fetus. James slammed the door shut just before the ship took off at incredible speed. All three of them felt the inertial effect of their speed pushing them into the seats. It continued this way for their entire journey, and they felt as though they were nowhere and everywhere at once. In fact, when they stopped moving, they really were. Rachel regained her composure at that point and pushed James aside, jumping over the seat into the back compartment.
"My boy, do you know where we aren't?"
"Where are we?"
"That's the thing. We aren't in the physical universe. Let me explain. A black object is black because it absorbs all light, right?"
"Right."
"And a black hole is invisible because it sucks everything in, including light. Therefore, this region of space we are in is black because everything that starts here is unable to ever leave here. Nothing inside this space can diverge outward. This, my boy, is the realm in which we may find some of the answers we seek. This - is the Mandlebrot set."
Chapter 3