Dear Sallys,

Sorry, it’s taken so long to answer, but showing how my model could work in three dimensions took some time. As it is I have another topic to talk about first. The question you asked requires equations to answer so I’ll leave it till last. Then those who don’t like equations can read the less mathematical part of my model.
Recently I have read that time is not a geometric dimension. That is not true in an expanding universe. Look again at the one-dimensional spiral.

Note that every point in the one-dimensional universe is defined by both x and y or bθ. As you recall the equation for an Archimedes spiral

r=bθ, where

where r is the distance of a point to the centre of a circle and θ is an angle. In our case it varies from 0 to 2π.

Typically a circle is defined by:
       _________
 r=\/x²+y²

²

In a sense so is a spiral, but unlike rectangular coordinates:
x=bθ cosθ
y=bθ sinθ,
         __________________________
r=  /b² cos²θ+ b²θ² sin²θ , and since cos²θ+ sin²θ=1 
If the universe is not expanding b is a constant and indeed a point is defined by the angle θ, you can see that x and y vary with θ. For instance if I make b equal to one the equation becomes:
r=θ.
However, if the universe expands causing the radius to expand even when θ is held constant, then b must grow. At this moment in time b is about .65 (measured in billions of light years) and our one dimensional universe looks like (30 degrees around the travel- impossible perimeter is 2.133):
    θ             Length    30 degrees arc     Radius

Now let us look at our one-dimensional universe. Like the one we live in, we define it by what is visible or potentially visible. In the one-dimensional universe, that is the spiral shown in the diagram above. The whole disc, including the circle, can be seen by you, me, and I guess Lem. We live in a three-dimensional world. In the one- dimensional world, planets are defined by the intersection of a vector from the centre and our spiral. As in our world our one-dimensional beings see it as it was some time ago. Also, as is clear from the two spirals below, another planet far from us sees and thus is in a different one-dimensional universe. They see the same stars, but at different times in the stars history. Note they see us much earlier in our history than we see them in their history . That statement is not strictly true because it all changes when we look at the other half of our universe. See below. It is clear that all planets are seen twice except the one that is π radians from our planet. It is seen at the same distance when looking in either direction.

In the universe of those 45 degrees from us, this point is in a different place. This is a clear example that we and they belong to different universes except where our spirals cross. Or when the spirals cross each other’s planets.

Even a planet is not as we usually think of it. It is a vector projecting at an angle (θ) from the centre of the circle. It only becomes a point at the intersection of a vector and a spiral universe.
Now we come to the most interesting part. Suppose the length of the spiral were to be 13.8 billion light years (the distance in billions of light years to the big bang at time zero). Now, suppose we decide to fly to a planet 30 degrees around the unobtainable circle of the expanding universe. As I’ve said before, we can’t just fly around the circle. Only Lemma and we, living in three-dimensions, can see the whole two-dimensional disc that is the greater (not the residents of our one-dimensional planet) universe we are studying. Our universe is the spiral the residents can look down to see the Big Bang. We see the planet she plans to fly to as it was 2.084 billion light years from our feature planet in her universe. She is looking at a long trip and because of the effect of the expansion on the time dimension, the trip will actually turn out to be more than 2.084 billion light years distance if she flies at light speed. The spiral line will actually be about 2.237 billion light years flying at the speed of light. It would take longer than she planned. Flying at half the speed of light, the distance would become about 2.41 billion light years and the time to get there would be 4.82 billion years. At twice the speed of light, it would take about 1.095 billion years to fly the 2.19 billion light year distance. As her speed approaches infinity her route becomes closer and closer to the perimetre of the circle If she could actually fly around the circle, the distance would be 2.133 and it would take her no time because the time distance would not change. This instantaneous travel smacks of gates in the books. The table and graphic below illustrate those distances and times:

A trip of billions of light years exceedes any found in your history, but we can talk about distance in thousnds of light years and use the same numbers altougfht that will misrepresent the age of our universe. It is indeed 13.8 billion years old. Also, the only way light speed was exceded was by jumping into another universe. The example you used in your first letter about Jeevra’s funera does not work. Another dimension is required in which disks are stacked. There would have to be a disk below our universe where the speed of light is slower and several stacked on top of ours. The rate of development would have to be different. It would of course effect how one conceived of the Big Bang,
Now to deal with two and three-dimension universes in four and five space (including dimensions for jumpimg universes). It is easy to visualize our spital becoming a two-dimensional spiral just by rotating the spiral in the first graphic through a third dimension. Rotating the two dimensional spiral through a fourth dimension is more difficult, for me impossible, to visualize. However, there are equations that I think work for the two-dimensional and three dimensional case. I promised you that

_______________

  r=\/x²+y²+z²  

Recalling that cos²θ+ sin²θ=1, the following relations hold for the two-dimensional spiral

x=bθ cosθ
y=bθ sinθ cosφ
z=bθ sinθ sinφ
r=bθ = time distance

For the three-dimensional spiral, the following relations hold:
      __________________
r=\/x²+y²+z²+w²

x=bθ cosθ
y=bθ sinθ cosφ
z=bθ sinθ sinφ cosξ
w=bθ sinθ sinφ sin ξ
r=bθ = time distance

In the two-dimensional case, the position of a planet in space and time is defined by r, (time distance), θ, and φ. In the three dimensional case the position is defined by r, θ, φ, and ξ. In all three cases, however, the universe is a spiral. We look down the spiral to see the beginning.

Ciao,

Bruce

²