Chapter One: "Neighboring Numbers" This first Chapter will be relatively simple, in that we'll mostly be looking at the "Base Set" of Numbers (1-9) to see how this "ordered" arrangement displays "Familial" characteristics in the "Neighboring Numbers". ("Neighboring Numbers" being (obviously) Numbers which are adjacent to one another.) (The 1-9 "Base Set" is an ordered (but not "Natural") arrangement of the "Base Numbers" Their "Natural" orientation will be seen in an upcoming Chapter.). First, we will start with just the "Base Set" of the Numbers 1-9, laid out in order. (We are assuming that this set is Infinitely repeating, in that an assumed 1 follows the 9, and an assumed 9 precedes the 1, both of which are shown below in parenthesis.) (9) 1 2 3 4 5 6 7 8 9 (1) Even in this basic 1-9 "Base Set" arrangement, we will begin to see a pattern of relationships between the individual Numbers, in that they (behaviorally) separate out into the two basic “Core Groups”, the first consisting of the 1,2,4,8,7,5, and the second consisting of the 3,6,9, and this is explained below. First, we see that "Adding" the Numbers to either side of the 1,2,4,8,7, or 5 brings us one step forward in that particular “Core Group” (in its "Natural" 1,2,4,8,7,5 arrangement), as is seen below. 11(2) 8 14(5) / \ / \ / \ (9) 1 2 3 4 5 6 7 8 9 \ / \ / \ / 4 10(1) 16(7) Above, starting from left to right, we see that "Adding" the Numbers to either side of the 1 yields a 2, "Adding" the Numbers to either side of the 2 yields a 4, and "Adding" the Numbers to either side of the 4 yields an 8. And then continuing from right to left ("Mirroring" our direction), we see that "Adding" the Numbers to either side of the 8 yields a 7, "Adding" the Numbers to either side of the 7 yieds a 5, and "Adding" the Numbers to either side of the 5 yields a 1, completing the pattern and returning us to the 1. And the colors seen above highlight the separation of "Family Groups" in the sums, with the 2,5,8 "Family Group" all being on the top (in red), and the 1,4,7 "Family Group" on the bottom (in green). s t e s t H a d w e n o t u s e
Transcript
Chapter One: "Neighboring Numbers"
This first Chapter will be relatively simple, in that we'll mostly be looking at the "Base Set" of Numbers (1-9) to see how this "ordered" arrangement displays "Familial" characteristics in the "Neighboring Numbers". ("Neighboring Numbers" being (obviously) Numbers which are adjacent to one another.) (The 1-9 "Base Set" is an ordered (but not "Natural") arrangement of the "Base Numbers" Their "Natural" orientation will be seen in an upcoming Chapter.).
First, we will start with just the "Base Set" of the Numbers 1-9, laid out in order. (We are assuming that this set is Infinitely repeating, in that an assumed 1 follows the 9, and an assumed 9 precedes the 1, both of which are shown below in parenthesis.)
(9) 1 2 3 4 5 6 7 8 9 (1)
Even in this basic 1-9 "Base Set" arrangement, we will begin to see a pattern of relationships between the individual Numbers, in that they (behaviorally) separate out into the two basic “Core Groups”, the first consisting of the 1,2,4,8,7,5, and the second consisting of the 3,6,9, and this is explained below.
First, we see that "Adding" the Numbers to either side of the 1,2,4,8,7, or 5 brings us one step forward in that particular “Core Group” (in its "Natural" 1,2,4,8,7,5 arrangement), as is seen below.
11(2) 8 14(5) / \ / \ / \
(9) 1 2 3 4 5 6 7 8 9 \ / \ / \ / 4 10(1) 16(7)
Above, starting from left to right, we see that "Adding" the Numbers to either side of the 1 yields a 2, "Adding" the Numbers to either side of the 2 yields a 4, and "Adding" the Numbers to either side of the 4 yields an 8.
And then continuing from right to left ("Mirroring" our direction), we see that "Adding" the Numbers to either side of the 8 yields a 7, "Adding" the Numbers to either side of the 7 yieds a 5, and "Adding" the Numbers to either side of the 5 yields a 1, completing the pattern and returning us to the 1.
And the colors seen above highlight the separation of "Family Groups" in the sums, with the 2,5,8 "Family Group" all being on the top (in red), and the 1,4,7 "Family Group" on the bottom (in green).
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Where as the numbers to either side of the 3, 6, or 9 "Add" to the opposing 3, 6, or 9 (3 to 6, 6 to 3, and 9 to itself.), as is shown below (with this complete 3,6,9 "Family Group" highlighted in blue).
6 12(3) 9 / \ / \ / \
1 2 3 4 5 6 7 8 9 (1)
So, with either of the two "Core Groups" (1,2,4,8,7,5, and 3,6,9), "Adding" the "Neighboring Numbers" causes a "move" of one step forward in each respective pattern. The 1st pattern being a repeating "Core Group" of 1,2,4,8,7,5, the 2nd being a repeating 3,6 "Sibling" pattern, and the 3rd pattern being the independent and unchanging 9/0 Unity (which is technically "Self-Sibling/Cousin Mirroring").
(Or, to state the above interrelations another, simpler way, in the (ordered) "Base Set" (1-9), the Numbers to either side of any Number "Added" together will always equal double the center Number (except for the 9, where the "Neighboring Numbers" still equal 9 (because 9 doubled is still 9)).)
Now would be a good time to explain the "Core Groups". The "Core Groups" are essentially two (technically three) intertwined "Doubling Patterns". The first of these "Doubling Patterns" is the 1,2,4,8,7,5 "Core Group", in that 1 doubles to 2, 2 doubles to 4, 4 doubles to 8, 8 doubles to 16(7), 16 doubles to 32(5), and 32 doubles to 64(1), with this pattern repeating on through Infinity (128(2), 256(4), 512(8), 1024(7), 2048(5), etc...).
The other "Core Group" (that of the 3,6,9) contains the other two (intertwined) "Doubling Patterns", the first of which is that the 3 doubles to 6, 6 doubles to 12(3), 12 doubles to 24(6), 24 doubles to 48(3), 48 doubles to 96(6), etc..., with this 3,6 ("Sibling/Cousin") "Doubling Pattern" carrying on through Infinity. And the second ("Self-Sibling/Cousin") "Doubling Pattern" in this 3,6,9 "Core Group" is in the unchanging 9, which will always double to itself (9 doubles to 18(9), 18 doubles to 36(9), 36 doubles to 72(9), etc...).
The above example showed the 3,6,9 "Family Group" acting as a "Core Group", in opposition to the 1,2,4,8,7,5 "Core Group" (as was explained in the Preface). Basically, removing the "Core Group" of 1,2,4,8,7,5 from the "Base Set" of 1-9, leaves the"Family Group" of 3,6,9, which will then act as a "Core Group" (if necessary).
1 2 4 5 7 8 3 6 9
(In the above examples, we "Added" "Neighboring Numbers" together. "Addition" is one of the "Four Functions" that we will be working with throughout these Chapters, with the other three being "Subtraction" (which is the "Sibling Function" of "Addition"), "Multiplication", and "Division" (with these last two "Functions" also being considered "Sibling Functions"). These are just the basic
mathematical functions that we are all familiar with, though they will display some unusual behavior as we work our way through all of this, so much so that they required their own Chapter (Chapter 4.71: "What Are The Four Functions"). But for our current purposes, these "Functions" can just be treated like the familiar mathematical functions.)
So, moving on, (inversely) "Subtracting" the "Neighboring Numbers" of 1,2,4,8,7, or 5 always yields the 2 (if we treat the 9 as a 0), and this is true with the 3 and 6 as well, as is shown below.
2 2 2 2 / \ / \ / \ / \
(0) 1 2 3 4 5 6 7 8 9 \ / \ / \ / \ / 2 2 2 2 (All of the "Subtraction" above is done right to left. "Subtracting" the Numbers from (traditional) left to right yields the same Numbers, only in "Negative Number" form, and we're not ready for "Negative Numbers" yet (they will get their own Chapter soon enough, which I will link here upon its completion).)
Alternately, "Subtracting" the Numbers to either side of the 9 yields a 7, which is the same result that we get for the 1 if we treat the 9 as a 9, as is shown below ("Mirrored" back to traditional left to right "Subtraction").
7 7 / \ / \
(9) 1 2 3 4 5 6 7 8 9 (1)
So, in all cases, "Subtracting" any "Neighboring Numbers" yields a result involving the 2/7 "Siblings" (our first example of "Sibling Mirroring", which will be seen throughout upcoming Chapters, and will be a part of the Chapter involving "Negative Numbers").
(Above, we used the second of the "Four Functions" ("Subtraction"), which unlike its "Sibling Function" (of "Addition"), is "Directionally Dependent", meaning that performing the "Function" from left to right will yield a different solution than that yielded by the ("Mirrored") right to left "Direction". The "Cousin Functions" of "Subtraction" and "Division" are both "Directionally Dependent Functions", which means that both "Functions" possess the characteristic of "Locality". While the opposing "Cousin Functions" of "Addition" and "Multiplication" are both "Directionally Independent Functions", in that the direction of the "Function" is irrelevent to the solution. So these "Cousin Functions" possess the characteristic of "Non-Locality". (This is just the familar "commutative law" of traditional mathematics.)
Also, the term "Cousin Functions" refers to the "Function" pairs of "Addition And Multiplication" and
"Subtration And Division", which are both ("Cousin") "Related" via this "Locality"/"Non-Locality" characteristic.)
Next, moving on to the third "Function", we see that "Multiplying" the Numbers to either side of the 1,2,4,8,7, or 5 always yields a member of the 3,6,9 "Family Group", as is shown below (with the two complete 3,6,9 "Family Groups" (reversed ("Mirrored") order on top, "Natural" order on the bottom) highlighted in blue).
And "Multiplying" the Numbers to either side of the 3, 6, or 9 always yields the 8, as is shown below.
8 35(8) 8 / \ / \ / \
1 2 3 4 5 6 7 8 9 (1)
Moving on to the fourth and final "Function", we see that "Dividing" the "Neighboring Numbers" yields a series of "Whole" and "Decimal" Numbers that display "Mirroring" and "Matching" between "Siblings" and "Cousins", in various ways that are all explained below (with colors).
(The "color code" for these Chapters is included in the Preface. In this particular case, the colors will be "arbitrary", in that the colors green, red, and blue will simply be representing "A", "B", and "C", and will not be indicating any specific "Familial" affiliations .)
Below, the "Neighboring Numbers" are all involved in the ("Non-Local") "Function" of "Division", with these first examples all being the results of right to left "Division", and the second examples (below that) being the results of ("Mirrored") left to right "Division".
Above, the "center" Number involved in each "Neighboring Number" "Function" is indicated before each "Function", in parenthesis and highlighted in green. And each pair (top and bottom) of "Functions" is grouped by "Cousins" (green 1/8, 2/5, 3/6, 4/7, and 9). And with these "Cousin" pairings, right to left "Division" causes "Cousin Mirroring" in the solutions (highlighted in blue) (1/8 to
2/5, 2/5 to 3/6, 3/6 to 2/5, 4/7 to (a "Self Mirrored") 7/4 (and the non-"Related" 9 to 8)).
(In the above examples, we see some "Infinitely Repeating Decimal Numbers" (indicated by the "..." at the end), with the specific (non-condensed) Numbers used to arrive at the condensed values all highlighted in red. The specifics of how we actually arrive at the condensed values of "Infinitely Repeating Decimal Numbers" will be the subject of an upcoming Chapter on "Validating The Invalid Functions" (which I will link here upon its completion). (These "Invalid Functions" being most Numbers "Divided" by either the 3,6,7, or 9, which (except in a few cases, such as 6/3) will always yield "Infinitely Repeating Decimal Numbers" that are not (intuitively) condensed.))
(One example of an "Invalid Function" (of "Dividing By The 7") is *'d above, and this is to indicate that this "Infinitely Repeating Decimal Number" is a variation on the 124875 "Core Group", with this variation being 142857 (with the 2/4 and 7/5 juxtaposed). This 142857 "Enneagram Pattern" is the result of "Dividing The 1 By The 7", which is covered in depth in Chapter 7: "Dividing By The 7".)
And below, are the same pairs of "Neighboring Numbers", involved in the same "Division Function", only with the "Direction" being a "Mirrored" left to right.
Above, we see that these "Cousin" pairs yield "Cousin" quotients in the (green) 2/5 and 4/7 "Cousins" (both yielding ("Mirrored") (blue) 3/6 "Sibling/Cousins"), and the ("Mirrored" but non-"Cousin") 9/5 and 5/9 (blue) quotients in the (green) 1/8 and 3/6 "Cousins" (along with another 8 yielded by the 9).
Below, are the two alternate right to left "Division Function" "Sibling" pairs, which display "Matching" in their solutions (highlighted in blue). (The green pairs here are now grouped by "Siblings", not "Cousins".) The other pairs (1/8, 3/6, 9) are all "Sibling/Cousins", so there is no need to repeat those examples below.
Above, we see that these two "Sibling" pairs display "Matching", with this "Matching" displaying "Sibling Mirroring" of its own, with two sets of the 3/6 "Siblings" as solutions.
And these same ("Sibling") "Division Functions", when reversed, display "Matching" behavior (in their "Matching"), as is shown below.
And that concludes our look at the "Neighboring Numbers" of the "Base Set" of 1-9. We will now continue on in the same ("Neighboring") vein, moving on to the "Neighboring Numbers" of the various sub-groups ("Family Groups", "Core Groups", etc...).
It should be noted at this point that all of these various "Groups” ("Base", "Core", and "Family") "Add" to a condensed 9, as is shown below.
"Base Group": 1+2+3+4+5+6+7+8+9=45 and 4+5=9
"Core Group": 3+6+9=18 and 1+8=9
"Core Group": 1+2+4+5+7+8=27 and 2+7=9
"Family Group": 3+6+9=18 and 1+8=9
"Family Group": 1+4+7=12 and 1+2=3
"Family Group": 2+5+8=15 and 1+5=6
The "Base Group", both "Core Groups", and the 3,6,9 "Family Group" all "Add" up to the 9 (in red), while the 1,4,7 and 2,5,8 "Family Groups" "Add" up to 3 and 6 respectively (in arbitrary green and blue), which then need to be "Added" together in order to make 9. This is an example of the 1,4,7 and 2,5,8 "Family Groups" being a bit more connected than the other, more independent 3,6,9 "Family Group". This is also the same phenomenon that the 4/7 and 2/5 (traditional) "Cousins" exhibit, where they seem to be connected a bit closer, and therefore are a bit more restricted than the other "Self-Cousins" and "Sibling/Cousins". (All varieties of "Cousins" will be the subject of the next Chapter, Chapter Two: "Cousins and Decimals").
Moving on, these same "Neighboring Number Functions" will now be performed on the 1,2,4,8,7,5 "Core Group". First, in its "ordered" arrangement of 1,2,4,5,7,8, and then again in its "Natural" arrangement of 1,2,4,8,7,5. (Why the 1,2,4,8,7,5 arrangement is the "Natural" arrangement will become clearer in Chapter 2: "Cousins And Decimals".)
First, we see that "Adding" the "Neighboring Numbers" of any Number in the 124578 "Core Group" will give us that Number’s "Cousin", as is shown below (with the "Cousins" highlighted in (arbitrary) colors). 5 11(2) 8 / \ / \ / \
Above, we see our second example of the "Cousin" relationship of the Numbers. In this above case, the 1 and 8 (in red) are each acting as their own "Cousin" (which they tend to (but are not obliged to) do). While the 2 and 5 (in green) and the 4 and 7 (in blue) are acting as each other's "Cousin" (which they are obliged do). We will see much more of these "Cousin" relationships as we progress through all of this (again, the next Chapter is titled "Cousins and Decimals").
Moving on to the next "Function", we see that "Subtracting" the "Neighboring Numbers" of any Number in the (ordered) 124578 "Core Group" will yield either a 3 or a 6, as is shown below (in blue). (With the "Subtraction" performed from right to left in some cases, and left to right in others to prevent "Negative Numbers".)
6 3 3 / \ / \ / \
(8) 1 2 4 5 7 8 (1) \ / \ / \ / 3 3 6
Next, "Multiplying" the "Neighboring Numbers" in the (ordered) 1,2,4,8,7,5 "Core Group" will yield a 1,4,7 "Family Group" pattern, as is shown below.
16(7) 10(1) 40(4) / \ / \ / \
(8) 1 2 4 5 7 8 (1) \ / \ / \ / 4 28(1) 7
Above, we see that "Multiplying" the "Neighboring Numbers" in the (ordered) 1,2,4,8,7,5 "Core Group" yields Numbers involving the 1,4,7 "Family Group" (with one complete "Family Group" on top, and another on the bottom, both (out of order) in green). (This is another example of the 1,4,7 "Family Group" acting independently of the 2,5,8 "Family Group".)
Next, "Dividing" these "Neighboring Numbers" yields Numbers that are a bit more interesting than those that we've seen so far. These Numbers involve the 4/7 "Cousins", as well as the result of "Dividing The 1 By The 7" (the "Enneagram Pattern", or variation on the .124875 ("Core") pattern, which was seen earlier, and to which Chapter 7: "Dividing By The 7" is dedicated entirely to).
Above, we have "Divided" all the "Neighboring Numbers" from left to right, which has yielded results that involve the 4/7 "Cousins" (in green). We also have the result of .571428..., which is a "shifted" (142857) "Enneagram Pattern") represented above with a * ("shifted" in that it begins on the 5th Number in the pattern (which is 5)).
Below, the "Direction" of this "Division" is reversed (to right to left).
.25(7) 2.5(7) 1.6(7) / \ / \ / \
(8) 1 2 4 5 7 8 (1) \ / \ / \ / 4 1.75(4) *
Above, we see that "Dividing" the Neighboring Numbers in the "Mirrored" "Direction" of right to left, yields a "Mirrored" result, with 4's in place of 7's, and 7's in place of 4's (again, all in green). And the * in this case is shifted one step to the right, and represents the same "Enneagram Pattern" as it did in the 1st example, which this time it shows up in its "Natural" arrangement of .142857.
Next, these same "Four Functions" will be performed on the 1,2,4,8,7,5 "Core Group", in its "Natural" arrangement.
First, we "Add" the "Neighboring Numbers" together, as is shown below.
7 10(1) 13(4) / \ / \ / \
(5) 1 2 4 8 7 5 (1) \ / \ / \ / 5 11(2) 8
And we see above that "Adding" together the "Neighboring Numbers" in the 1,2,4,8,7,5 "Core Group" yields a new 1,2,4,8,7,5 "Core Group", split neatly into one each of the 1,4,7 and 2,5,8 "Family Groups" (1,4,7 represented on top in green, and 2,5,8 represented on the bottom in red). (This is a "Mirror" of the behavior yielded by "Adding" the "Neighboring Numbers" of the "ordered" 124578 arrangement, which saw the 2,5,8 "Family Group" show up on top, and the 1,4,7 "Family Group" on
the bottom).
Next, we will "Subtract" these "Neighboring Numbers", as is shown below.
3 6 3 / \ / \ / \
(5) 1 2 4 8 7 5 (1) \ / \ / \ / 3 3 6
And above, "Subtracting" the "Neighboring Numbers" of the "Core Group's" "Natural" arrangement yields essentially the same result as the "ordered" arrangement did, that result involving the 3/6 "Sibling/Cousins" (in blue).
Next, we will "Multiply" these same "Neighboring Numbers", as is shown below.
10(1) 16(7) 40(4) / \ / \ / \
(5) 1 2 4 8 7 5 (1) \ / \ / \ / 4 28(1) 7
Above, "Multiplying" the "Neighboring Numbers" yields variations on the 1,4,7 "Family Group" (in green), just as they did in their "ordered" arrangement earlier in the Chapter.
Next, we "Divide" these "Neighboring Numbers", first from left to right, as is shown below.
2.5(7) .25(7) 1.6(7) / \ / \ / \
(5) 1 2 4 8 7 5 (1) \ / \ / \ / .25(7) * 7
Above, "Dividing" the "Neighboring Numbers" from left to right yields all (green) 7's, except for the *, which again represents (the "shifted" "Enneagram Pattern of) .571428... .
Next, we will "Divide" these "Neighboring Numbers" in the "Mirrored" right to left "Direction", which is shown below.
.4 4 .625(4) / \ / \ / \
(5) 1 2 4 8 7 5 (1) \ / \ / \ / 4 1.75(4) *
And above, we see "Mirroring", in that the (green) 4's "Mirror" the previous 7's, as well as the "Decimals" "Mirroring" to "Whole Numbers" (and vice versa). Also, as happened previously (in the "ordered" arrangement), the * has shifted over to the far right, and has presented itself in the "Natural" .142857... arrangement.
Next, to finish off our look at "Core Groups", we will move on to performing these same "Four Functions" on the "Neighboring Numbers" of the 3,6,9 "Core Group" (which will also function as a first look at one of the "Family Groups", due to the fact that the 3,6,9 "Core Group" and the 3,6,9 "Family Group" are one and the same).
First, "Adding" the "Neighboring Numbers" of any Number in the 3, 6, 9 "Core Group" yields either 3, 6, or 9, as is shown below.
12(3) 15(6) 9 / \ / \ / \
3 6 9 3 6 9 3 \ / \ / 9 12(3)
Above, we see that "Adding" the "Neighboring Numbers" together yields either a 3,6,or 9 (all highlighted in blue).
And below, "Subtracting" these "Neighboring Numbers" yields either a 3 or a 6.
6 3 3 / \ / \ / \
3 6 9 3 6 9 3 \ / \ / 3 6
Above, "Subtracting" the "Neighboring Numbers" of the 3,6,9 "Family Group" yields 3's and 6's (in blue).
Next, there is a variation on this 3/6 "Sibling" pattern, with "Division" in the left to right "Direction", which is shown below.
.3 1.5(6) 2 / \ / \ / \
3 6 9 3 6 9 3 \ / \ / 2 .3
Above, "Dividing" these "Neighboring Numbers" in the left to right "Direction" yields results that involve the 3/6 "Sibling/Cousins" (in blue) and the 2 (in red). The 3's are represented both times above only once, though they are "Infinitely Repeating Decimal Numbers" (.333...). (Again, we will look at both "Infinitely Repeating" and "non-repeating" "Decimal Numbers" in future Chapters.)
Next, "Dividing" these "Neighboring Numbers" in the "Mirrored" (right to left) "Direction", yields a "Mirrored" result, as is shown below.
3 .6 .5 / \ / \ / \
3 6 9 3 6 9 3 \ / \ / .5 3
Above, we can clearly see "Mirroring", in that the two (blue) 3's are represented with "Whole Numbers", where they were represented with "Decimal Numbers" in the previous (left to right) example. And here the (blue) 6 is represented with an "Infinitely Repeating Decimal Number", where it was a "Whole Number" in the previous example. There is also "Cousin Mirroring" involving the two (red) 5's "Mirroring" their "Cousins" (2's) in the previous example. And, where the 2's in the first example were "Whole Numbers", the 5's in the second example are represented by ("non-repeating") "Decimal Numbers". (These are all good examples of some of the various forms of "Mirroring", which will be seen in a variety of forms throughout these Chapters (eventually requiring a Chapter of their own, which I will link here upon completion).)
Next, to round out this Chapter, we will take a look at the "Neighboring Numbers" of the other two (opposing) "Family Groups", which are shown below. (Since we have already covered the (third) 3,6,9 "Family Group" as a "Core Group", we'll start with the (first) 1,4,7 "Family Group". (All "Family Groups" are presented as "Infinitely Repeating" patterns.)
First, "Adding" the "Neighboring Numbers" of the 1,4,7 "Family Group" yields the opposing (2,5,8) "Family Group", as is shown below.
11(2) 5 8 / \ / \ / \
(7) 1 4 7 1 4 7 (1) \ / \ / \ / 8 11(2) 5
Above, we see that "Adding" the "Neighboring Numbers" of the 1,4,7 "Family Group" yields Numbers involving the 2,5,8 "Family Group" (one complete ("Naturally" ordered) "Family Group" on top, and another ("shifted") on the bottom, both in red). (This is another example of the 1,4,7 "Family Group" being connected with the 2,5,8 "Family Group".)
Next, we "Subtract" these "Neighboring Numbers", as is shown below.
3 3 6 / \ / \ / \
(7) 1 4 7 1 4 7 (1) \ / \ / \ / 6 3 3
Above, "Subtracting" the "Neighboring Numbers" yields (blue) 3's and 6's.
Next, we "Multiply" these "Neighboring Numbers", which is shown below.
28(1) 4 7 / \ / \ / \
(7) 1 4 7 1 4 7 (1) \ / \ / \ / 7 28(1) 4
Above, "Multiplying" these "Neighboring Numbers" yields Numbers involving the 1,4,7 "Family Group" (one complete ("Naturally" ordered) "Family Group" on top, and another ("shifted") on the bottom, both in green). (This is another example of the 1,4,7 "Family Group" showing up on its own, without needing the 2,5,8 "Family Group".)
Next, we "Divide" these "Neighboring Numbers", first in the left to right "Direction", as is shown below.
1.75(4) 4 * / \ / \ / \
(7) 1 4 7 1 4 7 (1) \ / \ / \ / * 1.75(4) 4
Above, "Dividing" these "Neighboring Numbers" from the left to right "Direction" yields nothing but (green) 4's and *'s (.142857 "Enneagram Patterns"), with the *'s located on the top right and bottom left. (The 4's are represented with "Whole Numbers".)
Next, we "Divide" these "Neighboring Numbers" in the ("Mirrored") right to left "Direction", which is shown below.
* .25(7) 7 / \ / \ / \
(7) 1 4 7 1 4 7 (1) \ / \ / \ / 7 * .25(7)
Above, we see that "Dividing" the "Neighboring Numbers" in the "Mirrored" (right to left) "Direction" yields "Mirrored" results, with (green) 7's instead of 4's (an example of (4/7) "Cousin Mirroring"), in the location of the *'s ("Perfect Mirror" on top, "Weak Mirror" on the bottom), the order of the pattern of the non-condensed value of the * ("shifted" .571428 "Enneagram Pattern"), and the use of ("non-repeating") "Decimal Numbers" to represent the 7's.
(The concepts of "Perfect Mirror" and "Weak Mirror" seen above will be more thoroughly explained as we progress through these Chapters, and will eventually be part of an upcoming Chapter on "Mirroring", which I will link here upon its completion. But for our immediate purposes, an "exact" (or "ideal") "Mirroring" would be a "Perfect Mirror", and an "incomplete" ("semi") "Mirroring" would be considered a "Weak Mirror". (In the above examples, the top * moving from far right to far left was a "Perfect Mirror" due to the "Polarity" (opposition) of the two (far) positions, while the bottom * moving from far left to center was a "Weak Mirror", in that it could have (ideally) "Mirrored" even farther away, but (for whatever reason) did not.)
And finally, we will move on to the only remaining "Family Group", that of the 2,5,8, which is shown below.
First, we will "Add" the "Neighboring Numbers" of the 2,5,8 "Family Group", which is shown below.
13(4) 7 10(1) / \ / \ / \
(8) 2 5 8 2 5 8 (2) \ / \ / \ / 10(1) 13(4) 7
Above, "Adding" these "Neighboring Numbers" yields Numbers involving the 1,4,7 "Family Group", with one complete ("shifted") "Family Group" on top, and another ("Naturally" ordered) on the bottom, both in green.
Next, we "Subtract" these "Neighboring Numbers", as is shown below.
3 3 6 / \ / \ / \
(8) 2 5 8 2 5 8 (2) \ / \ / \ / 6 3 3
Above, as was the case with the 1,4,7 "Family Group", "Subtracting" the "Neighboring Numbers" of the 2,5,8 "Family Group" yields 3's and 6's (in blue). (The fact that both of these "Family Groups" (as well as the 3,6,9 "Family Group") yield the 3/6 "Sibling/Cousins" in these situations is a very (very) important characteristic, and the reasons for this will become clearer in upcoming Chapters.)
Next, we "Multiply" these "Neighboring Numbers", as is shown below.
40(4) 10(1) 16(7) / \ / \ / \
(8) 2 5 8 2 5 8 (2) \ / \ / \ / 16(7) 40(4) 10(1)
Above, we see that "Multiplying" these "Neighboring Numbers" again yields Numbers involving the 1,4,7 "Family Group" (with complete ("shifted") "Family Groups" on the top and bottom, in green). (The "shift" is "Mirrored", in that the top "Family Group" is shifted one step to the right, while the bottom "Family Group" is shifted one step to the left.)
And finally, we "Divide" these "Neighboring Numbers", starting from the left to right "Direction", as is shown below.
Above, "Dividing" these "Neighboring Numbers" from left to right yields exclusively (green) 7's. At this point, we might expect that "Dividing" in the "Mirrored" "Direction" would yield exclusively 4's, which we will find out below.
.625(4) .4 4 / \ / \ / \
(8) 2 5 8 2 5 8 (2) \ / \ / \ / 4 .625(4) .4
And, as is seen above, "Dividing" in the "Mirror" right to left "Direction" did indeed yield exclusively 4's (in green). And again, the "Mirroring" also involves the "Decimals"/"Whole Numbers" as well.
The fact that all (except "Subtraction") of the "Functions" of the 2,5,8 "Family Group" involved only elements of the 1,4,7 "Family Group" is another indicator of how the 2,5,8 "Family Group" generally doesn't stray too far off on its own. It tends/prefers to be "connected" with the 1,4,7 "Family Group" whenever possible, as has been mentioned previously.
And that about covers the "Neighboring Numbers" of the one "Base Set", the two "Core Groups", and the three "Family Groups". The next Chapter (Chapter Two: "Cousins and Decimals") will deal with "Cousins", in terms of what they are, and how they relate to "Decimals".
