Hi Bruce,

      This is Artemy writing. I have been looking at your and our astrophysics articles. I have decided that without breaking what we three call  “Magda’s law” there is stuff I can remind you of and also tell you. Although I don’t see a breach in “Magda’s law” I’m sure Maglie and Sallys would. They are very sensitive about betraying Lemma because of their first ‘betrayal”.

       You and Sallys devised (or stole from Lokoshim) a one-dimensional universe. It was an Archimedes spiral drawn within a circle of radius 4.076 units (I shall use units instead of billions of light years from here on). The spiral was then 13.8 units long, which represents the age of the universe. In fact there were two spirals one in each direction:

Your scientists and ours offer another way of looking at the universe. Suppose the radius is 13.8 units. In that case, the spiral will be a 46.68 units long, approximately the length of the so-called observable universe. In the later case, the universe would be 598.24 square units in size. Remember, we determined that every body in the universe was not a point but rather a vector in time. This does not imply that every object vector can be traced to the origin or exists now. Also notice that I have made it so that some vectors cross the object vector twice generating two sightings of the “same” objects at different times. Finally, there is the “NOW” universe. It is 86.708 units long; it is the circumference of the circle.       

            We have identified four universes. First we have the vector representing the age of the universe which we can see only through our own explorations of history; the observable universe, 2×46.68 units long; the ”NOW” universe, 86.708 unit universe containing all objects as they presumably are now, but only visible at any one time to a three-dimensional being; and the TOTAL universe being all the objects at all time, also only visible to a three-dimensional being You do know how we came up with an Archimedes spiral. I know. We drew a circle of radius one (circumference 2π) and drew a radius line every 36 degrees or1/10^th of the way around the circle (That is an angle of .2π . We drew nine more circles inside all 1/10^th of the radius apart. We connected straight lines from one circle to the next, as shown below. The result resembled an Archimedes spiral (r=bθ, θ is the angle of rotation of the radius aroud ; a is a constant You, Bruce then made the angles between the vectors smaller and smaller. The straight lines continued to converge to the centre in the same manner as the Archimedes spiral. So, you fitted one. The result is also shown below in the figure and table.

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Of course, the problem was that we had not been working in three dimensions.  However, that did not stop us. So, now to deal with two and three-dimension universes in three and four space. It is easy to visualize our spiral 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.

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            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, (time distance), θ, φ, and ξ.  In all three cases, however, the universe is a spiral. We look down the spiral towards the beginning. To see it all we must be five dimensional. The length of the spiral is given by:

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Here θ  is equal to 6.283 (2π).  The constant b is .159 (1/2π) for a radius of ‘1’ and 2.219 for a radius of 13.8. Looking at the equation of the length what is immediately obvious is that there must be a second derivative meaning acceleration. That is true although the rate of expansion of the circle does not accelerate.

            The table below shows the difference scores between points on the spiral for two different diametres. The first difference scores drop as the spiral gets closer to the origin. The second shows the acceleration, which has three stages. It bobs around randomly, then it decreases and finally it increases near the origin. Finally, the third set of difference scores are near zero until near the origin. 

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This would be true for any planet in the NOW universe, or for any other planet in our Observable Universe. Of course their Universe will not be the same as ours because from their perspective they will be at place where θ=2*π, but b is less than 2.219. (See above.) and so will L (also see above). In addition the star we see at say b=1.775 will of course have an observable universe. It will perceive itself as at θ=2*π. It’s universe will not be congruent to ours from that point on.

            Here is an interesting asiide. Many of your physists speak of aten dimensional universe. They curl up the extra dimensons because we can’t see them. Of course we can’t. We’d have to be at least 11-dimensional beings. We can’t even truly see three dimensions. We use off-set two-dimensional pictures. Instead suppose our four dimensions (including time) were the result of something like below. There are 10-dimensions (a,b,c,…j). A linear transformation converts them into four linear orthogonal combinations (W, X, Y, Z). using four orthogonal vectors. The second matrix is a constant that renders the radius of the sphere equal to 13.8.  The mi’s  are the ten coordinates in the original system.

the coordinates of a 4-d point.

A 3-dimensional cover of a 4-sphere ball is defined by all points fitting

                                                  r42  =  W2 + X2 + Y2 + Z2 += 13.82 

Note that each of  new coordinates are linear combinations of all ten dimensions. This new sphere can have a spiral defined upon it.

Love,

Artemy