Walking Amidst The Candlesticks

Each static subgroup is comprised of three elements of the octal group - and each lampstand is of six seven cycles with one unity element. We have seen that the unity element is fixed (upon repeated squaring, this always results with unity sent to unity.) However we do not need to see that unity is always written as the first element in the cycle.

A choice of unity holds fixed a particular subgroup, but we can notice that if we choose one of our eight generators to cycle the seven churches by shifting the arrangement of the underlying octal elements we can show that the static subgroup can march through the seven cycles of the eight C7 groups without a loss of structure. (I.e., the choice of unity is completely arbitrary.)

So, (a,b,d,c,f,g,e) is the same cycle as (b,d,c,f,g,e,a).

Likewise, although there is one fixed subgroup - since unity is not a member of the static subgroup and is held fixed : the remaining three elements are also "static" although they do not form a subgroup in the same octal.

So the inverse cycles of x^3,x^5 and x^6 hold fixed (c,g,e) if a=1 in (a,b,d,c,f,g,e). We note that these elements are the first,second and fourth elements before unity moving in reverse through the cycle.

However, using two generators of our eight lampstands, we see that if we use one cycle to permute the elements of any other, such as:

a b d c f g e
c f a e g d b
e g c b d a f
b d e f a c g
f a b g c e d
g c f d e b a
d e g a b f c

Where the same seven cycle is used to permute the elements in every vertical column. But each row is from a different lampstand, or different C7 group on the same octal!

Each column could be considered an "angel" or messenger travelling from church to church as in the same order - the result is that the remaining seven lampstands or churches neatly dovetail together in seven symbols. However any C7 group could do the same, except for those that are in the same group as the initial horizontal, or the top row here.

In every row we may "walk" through the rows and we hold fixed the octal subgroups rather than their counterparts from their static complement set in the octal. If unity is in one particular column, we have one of each subgroups static in each row.

We can liken the vision of Christ with the seven stars in His right hand walking amidst the seven churches' lampstands as these fixed subgroups, paired with a unity element (any row, any column, marching through rows or the column cycle). The triple element structure of static groups is trinitarian - but at this stage it is mysterious as to whether the figure marching is the chosen unity, the static subgroup or both.

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