Magda put this together after Jeevra’s memorial. She said it was for you, but she has done nothing with it so I will send it to you. I hope it will clarify some of the stuff in the box I helped you find and to tell you I did really enjoy hanging with you at the party in Keene.
The turnout at Costa Calma to Jeevra’s memorial was a bit disappointing, but I guess that happens when a woman has outlived all her close friends. Jeevra was the last of the original band of rebels that crewed the Ark. Those that did not rebel and chose to remain in stop boxes had been resettled on Mission, not without bitterness. My mother was the second to last to go and she had died over 20 years ago. So, there were just four mourners. There were Sallys and I, twins; there was Artemy, who had travelled with Jeevra and us; there was a little man that Sallys, Jeevra, and I had met in Antarctic years ago. Although he had every right to be there, Sallys and I were not glad he was. We had betrayed his trust and feared his wrath.
Before the ceremony he said nothing to Sallys and I, but greeted Artemy in a friendly manner.
Then the funeral was over. We were all sad, but not crushed. I felt that this had been like the service of that famous Origin country singer, Maybelle Carter, although we had not stood around in a circle nor were there as many people. In fact Jeevra was the mother of us all, of every Progeny or I guess homo sapien that had ever lived.
At that point the little man, Lemma, approached us. Artemy quickly separated herself from us, clearly suspecting what was about to happen and although she was not guiltless, she did not share our sin.
“We’re sorry,” Sallys said in a small voice.
“Love causes one to do unwise things including breaking the pact with Avita and Magda,” replied Lemma looking at me.
“I was sure that William would not tell anyone and he didn’t. He had kept my letter and then he died so suddenly.” I fought tears.
And of course Artemy being Artemy came back, “ But if it hadn’t been for me, Bruce would never have even found the box. I didn’t know what went on in Antarctica at that point, but I never checked to see if there was anything sensitive.”
Lemma could only smile. “How did you come to be? You’re not only programmed to live forever, but also to be loyal forever. Who did you find to be loyal to between Magda and the twins?”
“Avita,” Artemy answered promptly. “But everyone was loyal to her because she was the kindest and most likeable person in the world. I loved Lokoshim too. She might even have been smarter than Jeevra and that really says a lot.”
Lemma shook his head. “How can I really be angry? All sorts of people are speculating in a crude way what I told you two and I think it was important for the story of the galaxy to be known. Come let’s sprinkle the ashes in the ocean. May I go first?”
Of course we agreed. Sallys and I would have agreed to anything then to keep him from being mad at us.
We fetched the urn and a cup. Lemma walked to the water with both. He scooped out cupful, paused, and then sprinkled the ashes in the water. He turned to me and handed the urn and cup to me. I stepped to the edge. Sallys and Artemy stepped up with me. We each took our turn pausing in thought and then sprinkling the ashes. When we finished there were some left. We turned to offer Lemma another turn. He was gone. Sallys laughed. “Of course,” she said. “Just like in front of that impregnable building. I hope it doesn’t start snowing.” It didn’t.
Later we three friends sat around a campfire near and talked about what had been wrought on a little island, called Big Island, and what that had meant for our galaxy. We had chosen the Island of Fuerteventura because it was reasonably close to where Big Island used to be. We talked about our meeting with Lemma and that he had been kinder than I deserved. We talked a lot about how Jeevra and her strange math described the universe.
My sister Sallys broke a rather long silence. “The only good thing is that Jeevra didn’t read the last book your boyfriend’s grandson wrote. I’m not sure that she’d have been as kind as Lemma was.”
“I did dig up some stuff for him,” Artemy interjected and so did you, Sallys. He probably would have written about it sooner or later if he hadn’t died so soon after you dumped him, Maglie.”
“I didn’t exactly dump him. I still loved him, but he loved Harriet as well as me and there was a future with her, but not really with me.”
“I know, but I had to poke you,” laughed Artemy. “However, to change the subject just a little, why did Lokoshim‘s amazing algorithm about what the Origin scientists now call space -time not get into the box?”
“I never heard of any amazing algorithm,” I argued.
“Well, Metira did a rather complete write up about it,” Artemy replied somewhat hotly.
“No wonder,” said Sallys. “No one in my generation paid any attention to Metira’s fantasies.”
“Well, you should have to this. It was quite cool particularly for an uneducated girl from Origin. Shall I tell you how it goes?”
“Do,” I said. I was afraid Sallys would sneer at it and I wanted to hear it.
I was right. Sallys groaned, “ OK then.”
Artemy, as usual unfazed, turned, snarled at Sallys, and laughed, making Sallys flinch.
“So here goes,” Artemy started. “Lokoshim told Magda and peripherally the rest of us that she had a ingenious idea. She drew a big circle in the ground where we were on some island in what’s now called the Red Sea and told us that the ring around the circle was a one-dimensional universe. One-dimensional people lived on it, but were invisible cause they were so small. She asked Magda how old our universe was and Magda replied that she thought it was about 13.8 billion light years. That is of course the distance light travels in 13.8 billion years.”
“’OK.’ Lokoshim said. ‘The circumference is 27.6 billion light years. That’s why the planets and people are so little. Till, this dot here,’ She put a dot on the circle. ‘This dot is Origin.’”
“Why don’t stop trying to tell it as Lokoshim did and just tell us the meat of it yourself.” A somewhat exasperated Sallys said.
“Ok, then,” grumbled Artemy. “I suppose no one has any chalk.”
“Actually, I always have at least four or five pieces of colored chalk,” I offered.
“You do?” Artemy exclaimed, surprised.
“Yeah, she always does,” Sallys said. “She likes to draw on things.”
“Then, hand it over,” Artemy demanded laughing.
I gave her four pieces. She walked over to a nearby tent pad and drew a rather amazingly good circle with a dot in the middle, a dot on one side, and a line from the side dot through the middle and to the other side.
“The little dot on the side is Origin or Earth as they call it; the center dot is the Big Bang location that made our one-d universe expand. The circumference is about 27.6 light years long. Note that I am leaving out the billion; the line is just over 8.78 light years long. The line from the dot or the radius would be half of that or about 4.39 light years long.
“Now you guys are looking at it as Lemma would from his 3-d perspective. You can see ‘now’ all at once. A billion years ago you would have seen much the same thing, but the circle would have been smaller. Suppose though that you were on a one-dimensional Origin and looking at a planet say 2.76 billion light years away around the circle. Since the light would take 2.76 billion years to get to you, you would be seeing it as it was 2.76 billion years ago The circle is was on would be further in, in fact it would be .439 billion light years closer to the center. If you drew a curved line to that star, it would be 2.622 light years long. It would look like…”
She drew lines to make the circle look like a pie with ten pieces. Then he drew a curved line on the circle from Origin to a spot one ten of the way around and one tenth of the way into the center of the circle.
“Whoa,” I exclaimed. “How do you know that’s the proper curvature?”
“There is an equation, I think,” Sallys replied.
“Yes, but it easiest to assume that I took a trip going by steps of .0000276 light years and I connected the points by straight lines. Visibly the curve would look just like that,” Artemy replied.
“You are sneaking integration past us. Those are all delta lengths.” I exclaimed.
“Of course,” she laughed. “I took a million little delta steps, both around the circle and into the circle. Now let me show you how I continue in much the same way. Each time I will go another tenth around the circle.”
She started to draw furiously ending up with nine more circles.
“See,” she said. “The lines grow into a spiral as I go further away from the starting point. They also get shorter from pie line to pie line.”
Sallys had been calculating furiously and put up a hand for us to be still and watch. She drew a table of arc lengths with the chalk.
|Circle Arcs||Circle lengths||Spiral Lengths|
|0.0 – 2.76||2.76||2.622|
|2.76 – 5.522||2.76||2.346|
|5.52 – 8.28||2.76||2.070|
|8.28 – 11.04||2.76||1.794|
|11.04 – 13.80||2.76||1.518|
|13.80 – 16.56||2.76||1.242|
|19.32 – 22.08||2.76||0.690|
|22.08 – 24.84||2.76||0.414|
|24.84 – 27.60||2.76||0.138|
“See,” she said. “The sum of the arcs making up the spiral is 13.8 billion and 13.8 billion light years is the age of the universe. So, you can look down the spiral and see the big bang..”
“That is exactly what Lokoshim showed us. She also pointed out that you could see the beginning by looking the other way. There are of course only two ways in a one-dimensional universe.” Artemy drew again and produced another spiral.
“Wow.” I shouted. “You can see every star twice at two different times in its history.”
“All except the one 180 degrees around the circle,” Artemy noted. “Both are 13.8 billion light years around the circle and…” she started adding and finally said, “then 10.35 light years along the spiral. An interesting theory of the shape of the universe don’t you think?”
“More than a theory,” I suggested. “If the circle really represents the expansion of the universe, space-time dictates that a spiral is the shape it must be. But, but this is only one-dimensional. What about two or three dimensions?”
“Actually, two dimensions is not a problem,” Sallys replied. “Put a coin on edge and rotate it a miniscule amount, but in your mind’s eye leave it as it was at the same time. Of course both coins, the mind’s-eye one and the rotated one will allow for a spiral like we just drew. Right?”
We both nodded. Then Artemy said, “It’s going to be like the tiny delta-sized curves. We keep rotating it by tiny amounts, all the time keeping the earlier ones in mind. Finally, we will have a whole sphere cover in our mind’s eye. Right?”
“Oh,” I continued. “Then no matter in which direction we look there will be a 13.8 line of sight right to the center. And, yes, all the lines of sight will cross at a point 10.35 billion light years away.. Wow, so we have two dimensions. What about three?”
“I can’t prove this, but I think a rotation like we just did but through the fourth dimension will do the same thing. Aside from it being logical that the dimensions would be alike in this way, there is another compelling thing. It is true that if I multiply the volume inside a cover by the length of a circle having the same radius, I will have the bounding area of a sphere (a cover) two dimensions greater no matter how many dimensions the hypersphere has.”
“You, mean,” Artemy asked, “that if you multiply the length of the circumference of a simple circle times the volume a hypersphere of 1000 dimensions, you will get the area, really the volume, of the 1001 dimensional cover of a 1002 dimensional hypersphere?”
“I do mean that,” Sallys replied. “And that means that the rotating operation implied by that is continuous across all dimensions of hyperspheres. Still, I can’t picture what that 3-d universe would look like. The only thing I’m virtually sure of is that wherever your planet is, it will appear to be the center of everything.”
“What about the inner universes that we go to to effectively exceed light speed?” I asked.
“Two points on that,” Sallys replied. The first is easy and the second may or may not be true. Give me the chalk. Here it is for a one-dimensional univers
There it was. Of course. “What is the second more dubious point?” I asked.
“Well, you know the speed of light is supposedly 11.09 times faster for every layer you go in?” Sallys said.
“Yes,” Artemy replied.
“It’s not necessary to produce the effect we get. All it needs is for the parallel universe to have a circumference of 27.6/11.09 or about 2.5 billion light years. That means if you can jump directly into the smaller circle and travel for one light year when you jump out you will have covered 11.09 light years in the larger circle. Here, let me show you. Maglie, do you have a small ruler?”
“Of course,” I replied. “It’s a fold up 30 cm one. That OK? If I’m guessing right you’ll need a roll of string too. I have that and scissors.”
Artemy laughed. “You are going to draw circles, I guess. There are going to have to be some big ones.”
“Very big.” Sallys replied. “If we make the radius of the smallest circle 3 cm, then the next should be 11.09 times 3 cm or 33.27 cm. The next will be just under 3.68 meters and the largest will be about 40.8 meters”
“Why will that be the largest? Aren’t we going to count the universe the space angels lived? Let’s see,” I calculated. “That would be over 450 meters or almost half a kilometer for the radius. I guess that would be a little hard for the campground to accommodate. Besides I think my string is only about 50 meters.”
“Question answered, Magda,” laughed Artemy.
“Let’s do the others then right now,” I suggested.
There was no problem with first three circles, but we had to ask permission to invade two other campsites. When we told them sort of what we were doing, they got right into it and cleared stuff out of the way, laughing and shouting enthusiastically. We couldn’t really use the chalk so we stuck sticks in the ground every 10 cm or so. It took awhile because 2342 (we could use the chalk on their pads) sticks turned out to be a lot of sticks. It took a couple of hours with us and the seven other campers working.
We all sat down and drank a couple of beers before we started again. We told them a well-rehearsed false story about our background. They were all under 30. Needles to say, we didn’t say how old we were or what Artemy could do. They had already told us that they came from London and Dublin, so our St. Paul in the US, where we had spent some time seemed safe, much safer than saying we spent a lot of our time on a space ship or living on a planet called New Terre. We did tell them we had shortly before scattered the ashes of a friend.
Then it came time to find more twigs and make two lines of sticks from the circle to the center of the circle. We drew everything we could draw with chalk, with chalk.
Now Sallys started to instruct us all. “See the distance between our two straight lines on the big circle. They are just a bit over 27.6 meters apart where they contact the circle.” She went and measured the two contact points on the next circle in. She measured the distance between them and announced, “See, now they are about 2.3 meters apart, just about 11.09 times shorter than on the big circle. On the next circle in, the distance between them is about 208 millimeters. Finally, the distance on the smallest is just under two millimeters. Each distance was 11.09 times smaller then the one out side it. Now assume we can jump along the line from the big circle to the next one in and travel at 2.3 meters a second. We then jump out along the line to the bigger circle and find we are 27.6 meters from our starting point, although we really only travelled 2.5 meters a second on the inner circle.”
“But can we travel along the line at infinite speed?” ask Joni, the 14-year-old camper boy.
“Ah,” Artemy answered. “That is the sticking point. For this to work, you have to assume you can travel the straight line at some maximum speed.”
I had been working furiously on my calculator and was ready. “You’d have to travel at 7.36 meters a second along the straight line both ways to have an advantage to going in.”
“But,” said the boy, “I thought you said you were going to go effectively 11.09 times faster by going in.”
“Yes,” replied Sallys, “And that does require an instantaneous leap down the line from circle to circle. You could compensate for a finite speed down the line, but that would require the circles to much further apart if you wanted to maintain the 11.09 ratio.”
“The little red-headed Irish girl, probably eleven spoke up. “What you really want is to go faster than light isn’t it?”
Artemy laughed. “Another Lokoshim.”
The girl looked puzzled and a little frightened. Me too.
Artemy gave me a hard look and then, “A very bright friend first dreamed up these circles quite a long time ago,” Artemy told her. “By the way do you know what 11.09 is?”
“I know it is the fifth power of the golden ratio at least.” She answered. “Is there more”
“Only if our friend’s theory is right about travelling faster than light,” I replied. “You did that awfully fast,” I added.
“Not as fast as you might think,” Artemy commented.
The girl looked at her with real fear and I knew. Artemy looked at Sally, I saw she knew too, and me, and she grinned. “Surprise,” she said. Looking around I saw that no one else got it. I guessed that was good.
Suddenly the girl wasn’t looking frightened. I knew she had just had a talk with Artemy who of course would have been unaffected. “Someday, if you’re good, I promise,” Artemy told her.”
The girl walked back to her parents, standing about five meters away looking puzzled. “I’m a math whiz, of course,” she laughed and they looked OK.
I was not OK. I hoped that Artemy’s bribe would be enough. I was encouraged that her family didn’t know. The girl could not have used it in too disruptive a manner. I wondered about school and sports. Mostly, I was bothered that someone not on Big Planet could stop people, as she had done. How many others were there?
The rest of the afternoon and evening went along quite normally. When it got, late everyone retired to their tents, but the little girl, Kristana–-a worrisome name—reminded Artemy as she left, “You must keep your promise.”
“I will if you keep yours,” Artemy replied.
“I will, I promise,’” Kristana replied. “I don’t want to end up like my great-grandma.” Artemy just nodded.
Poor Artemy. Sally and I virtually pounced on her when we got into our tent. “So, tell us,” I demanded.
Artemy laughed. “She stopped everyone but me for about 10 minutes the first time and about 15 minutes the second time. The first time she was calculating and didn’t even notice that I wasn’t stopped. The second time we discussed some history and she made some promises. Firstly, her great grandmother was burned as a witch in Dublin. She doesn’t know anything about ancestors further back. Her mother can slightly time stop. Kristana isn’t stopped so she sees others stopped for a few seconds. Her mother doesn’t even know she does it. More interesting is that almost all the women in her family have lived to be over 100. She doesn’t know anyone outside of her family that can stop anyone or has lived to be over 90. Still, we now know the feyness exists on this world still.”
We talked late into the night about feyness, universes, Jeevra, and what we were told by Lemma. When we finally awakened the next morning, both the campsites of our friends from the day before were empty and as it turned out the celebration of Jeevra’s long life was too.