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Formulae for the geometrical and yod compositions of the outer & inner forms of n Trees of Life/ntree composed of any order of triangle or polygon
As explained in #8, the triangles that are the basic building blocks of sacred geometry can be considered not only as pure triangles (what we may call "0thorder triangles") but also as triangles divided into their three sectors ("Type A triangles," called here "1storder triangles"). If each sector of the Type A triangle is then divided into its three sectors, it becomes a "Type B triangle," or a "2ndorder triangle." It is composed of (3^{2}=9) 0thorder triangles. The next division into three sectors of each 0thorder triangle in a Type B triangles generates a "Type C triangle" ("3rdorder triangle") with (3^{3}=27) 0thorder triangles, and so on. An nthorder triangle is the result of n successive divisions of a 0thorder triangle into 3^{n} 0thorder triangles:
0thorder triangle 
1storder triangle (Type A) 
2ndorder triangle (Type B) 
3rdorder triangle (Type C) 
4thorder triangle (Type D) 
Table 1 shows how the numbers of corners, sides & triangles inside the parent triangle increase with n:
Table 1

n = 0  n = 1  n = 2  n = 3  n = n 
Number of corners (c)  0  3^{0}  3^{0} + 3^{1}  3^{0} + 3^{1} + 3^{2}  3^{0} + 3^{1} + 3^{2} +...+ 3^{n−1} = ½(3^{n}−1) 
Number of sides (s)  0  3^{1}  3^{1} + 3^{2}  3^{1} + 3^{2} + 3^{3}  3^{1} + 3^{2} + 3^{3} +...+ 3^{n} = (3/2)(3^{n}−1) 
Number of 0thorder triangles (t)  3^{0}  3^{1}  3^{2}  3^{3}  3^{n} 
Number of geometrical elements (g=c+s+t)  1  7  25  79  3^{n+1} − 2 
Geometrical composition
The numbers of corners, sides & 0thorder triangles in N Trees
of Life or in the Ntree are listed in Table 2:
Table 2
N Trees of Life 
Ntree 

Number of corners (C)  6N + 4  6N + 5 
Number of sides (S)  16N + 6  16N + 9 
Number of triangles (T)  12N + 4  12N + 7 
Number of geometrical elements (G)  34N + 14  34N + 21 
The number of corners of the Tt triangles in N Trees/Ntree with T nthorder triangles ≡ C = C + Tc.
The number of sides of the Tt triangles in N Trees/Ntree ≡ S = S + Ts.
The number of 0thorder triangles in N Trees/Ntree ≡ T = Tt.
The number of geometrical elements in N Trees/Ntree ≡ G = C + S + T = C + S + T(c+s+t) = C + S + Tg.
Table 3 lists the calculated geometrical composition of N Trees/Ntree with nthorder triangles:
Table 3
N Trees of Life 
Ntree 

Number of corners (C)  2 + 2(3N+1)3^{n} 
½[3 + (12N+7)3^{n}] 
Number of sides (S) 
−2N + 2(3N+1)3^{n+1} 
½[−(4N+3) + (12N+7)3^{n+1}] 
Number of triangles (T)  4(3N+1)3^{n} 
(12N+7)3^{n} 
Number of geometrical elements (G)  2 − 2N + 4(3N+1)3^{n+1} 
−2N + (12N+7)3^{n+1} 
The cases N = 1 and N =10 are of special interest because 10 overlapping Trees of Life are an expanded version of the single Tree of Life. Table 4 lists their geometrical compositions for 0th & 1storder triangles:
Table 4
N Trees of Life 
Ntree  
Corners 
Sides 
Triangles 
Total 
Corners 
Sides 
Triangles 
Total 

n = 0 (all triangles are tetractyses) 

N = 1  10  22  16  48  11  25  19  55 
N = 10 
64 
166  124  354  65  169  127  361 
n = 1 (all triangles are Type A) 

N = 1  26  70  48  144  30  82  57  169 
N = 10  188  538  372  1098  192  550  381  1123 
Table 5 lists the geometrical composition of N Trees/Ntree for the cases n = 0, 1, 2, & 3:
Table 5
N Trees of Life 
Ntree  
n 
Corners 
Sides 
Triangles 
Total 
Corners 
Sides 
Triangles 
Total 
0 
6N + 4  16N + 6  12N + 4  34N + 14  6N + 5  16N + 9  12N + 7  34N + 21 
1  18N + 8  52N + 18  36N + 12  106N + 38  18N + 12  52N + 30  36N + 21  106N + 63 
2 
54N + 20  160N + 54  108N + 36  322N + 110  54N + 33  160N + 93  108N + 63  322N + 189 
3 
162N + 56  484N + 162  324N + 108  970N + 326  162N + 96  484N + 282  324N + 189  970N + 567 
Yod composition
The number of hexagonal yods inside each of the T nthorder triangles = 2s +
t.
The number of hexagonal yods lining the outer sides of these nthorder triangles = 2S.
The number of hexagonal yods in N Trees/Ntree ≡ H = 2S + T(2s+t).
The number of yods in N Trees/Ntree ≡ Y = C + 2S + T(2s+t).
Table 6 lists the yod population of N Trees/Ntree with nthorder triangles:
Table 6. Yod population of Ntrees/Ntree with nthorder triangles.

N Trees of Life 
Ntree 
Number of corners (C)  2 + 2(3N+1)3^{n}  ½[3 + (12N+7)3^{n}] 
Number of hexagonal yods (H)  −4N + 16(3N+1)3^{n}  −(4N+3) + 4(12N+7)3^{n} 
Number of yods (Y)  2 − 4N + 2(3N+1)3^{n+2}  ½[−(8N+3) + (12N+7)3^{n+2}] 
Table 7 lists the yod populations (Y) for N Trees/Ntree in the case of n = 0 (tetractys), n = 1 (Type A triangle), n = 2 (Type B triangle) & n = 3 (Type C triangle):
Table 7. Yod populations of N Trees/Ntree for n = 0, 1, 2 & 3.
n 
N Trees of Life 
Ntree 
0 
50N + 20  50N + 30 
1 
158N + 56  158N + 93 
2 
482N + 164  482N + 282 
3 
1454N + 488  1454N + 849 
The General view section of Power of the polygons discusses the geometrical and yod populations of nthorder polygons (these are polygons whose sectors are (n−1)thorder triangles, where n≥1). Table 8 lists the populations of nthorder Ngons in the general case and for n = 1, 2 & 3:
Table 8. Geometrical & yod compositions of the nthorder Ngon.
nthorder Ngon 
n = 1  n = 2  n = 3  
Number of corners, including centre (c_{n}^{N}) 
½(3^{n−1}+1)N + 1 
N + 1  2N + 1  5N + 1 
Number of sides (s_{n}^{N}) 
½(3^{n}+1)N 
2N  5N  14N 
Number of triangles (t_{n}^{N}) 
3^{n−1}N 
N 
3N 
9N 
Number of geometrical elements (g_{n}^{N}) 
c_{n}^{N} + s_{n}^{N} + t_{n}^{N} = (3^{n}+1)N + 1 
4N + 1 
10N + 1 
28N + 1 
Number of hexagonal yods (h_{n}^{N}) 
2s_{n}^{N} + t_{n}^{N} = (4×3^{n−1} + 1)N 
5N 
13N 
37N 
Number of yods (y_{n}^{N}) 
c_{n}^{N} + h_{n}^{N} = (3/2)(3^{n}+1)N + 1 
6N + 1 
15N + 1 
42N + 1 
("1" denotes the centre of the nthorder Ngon).
Table 9 lists the geometrical and yod compositions of the inner form of the Tree of Life:
Table 9. Geometrical & yod compositions of the 7 enfolded nthorder polygons and the (7+7) enfolded nthorder polygons.
7 enfolded polygons 
(7+7) enfolded polygons 

Number of corners (including centres of polygons) 
C_{n} = ½(35 + 47×3^{n−1}) 
C = 2C_{n} −2 = 33 + 47×3^{n−1} 
Number of sides 
S_{n} = ½(35 + 47×3^{n}) 
S = 2S_{n} − 1 = 34 + 47×3^{n} 
Number of triangles 
T_{n} = 47×3^{n−1} 
T = 2T_{n} = 94×3^{n−1} 
Number of geometrical elements 
G_{n} = C_{n} + S_{n} + T_{n} = 35 + 47×3^{n} 
G = C + S + T = 2G_{n} − 3 = 67 + 94×3^{n} 
Number of hexagonal yods 
H_{n} = 35 + 188×3^{n−1} 
H = 2H_{n} − 2 = 68 + 376×3^{n−1} 
Number of yods 
Y_{n} = ½(105 + 47×3^{n+1}) 
Y = 2Y_{n} − 4 = 101 + 47×3^{n+1} 
Notice that the number 101 is the 26th prime number and that the number 47, which is the number of sectors of the seven enfolded polygons, is the 15th prime number. The number 26 of YAHWEH, the complete Godname of Chokmah, and the number 15 of YAH, the shorter form of this Divine Name, arithmetically determine the yod population of the inner Tree of Life, whatever the order of its 14 polygons. Notice also that the number 67 of Binah appears in the formula for the geometrical population of the inner Tree of Life. As was revealed in #41, it is the number of yods below Binah of the 1tree when its 19 triangles are 1storder triangles.
Let us call the seven enfolded nthorder polygons P_{n}. Table 9 indicates that the number of corners in P_{n} ≡ C_{n} = ½(35 + 47×3^{n−1}). Therefore, the number of corners in P_{n+1} = C_{n+1} = ½(35 + 47×3^{n}). But, according to Table 9, this is the number of sides (S_{n}) in P_{n}. We find that the number of corners in a given P_{n} is equal to the number of sides in P_{n−1}, or conversely, that its number of sides is equal to the number of corners (C_{n+1}) in P_{n+1}:
C_{n+1} = S_{n}.
The number of yods in P_{n} outside the shared root edge = ½(105 + 47×3^{n+1}) − 4 = ½(97 + 47×3^{n+1}). The number of Haniel, the Archangel of Netzach, is 97.
The number of corners & triangles in P_{n} = ½(35 + 47×3^{n}). According to Table 9, this is equal to the number of sides (S_{n}). This means that the number (G_{n}) of corners, sides & triangles in P_{n} is twice its number of sides:
G_{n} = 2S_{n,}
so that
G_{n} = 2C_{n+1}.
The property G_{n} = 2S_{n} has its counterpart in a polygon of any order, namely, the number of corners, sides & triangles surrounding its centre is twice the number of their sides.* For the (7+7) enfolded polygons, the number of corners & triangles = 33 + 47×3^{n}, which is equal to the number of sides outside the root edge, whatever the order of polygon. These remarkable properties are clearly displayed in Table 10, which lists the geometrical and yod composition of the inner Tree of Life for n = 1, 2, 3 & 4:
Table 10
7 enfolded polygons 
(7+7) enfolded polygons 

n = 1 
n = 2 
n = 3 
n = 4 
n = 1 
n = 2 
n = 3 
n = 4 

Number of corners (including centres of polygons)  41  88  229  652  80  174  456  1302 
Number of sides  88  229  652  1921  175  457  1303  3841 
Number of triangles  47  141  423  1269  94  282  846  2538 
Number of geometrical elements  176  458  1304  3842  349  913  2605  7681 
Number of hexagonal yods  223  599  1727  5111  444  1196  3452  10220 
Number of yods  264  687  1956  5763  524  1370  3908  11522 
In the case of the seven enfolded polygons, for n = 1, 41 + 47 = 88; for n = 2, 88 + 141 = 229; for n = 3, 229 + 423 = 652; for n = 4, 652 + 1269 = 1921. In the case of the (7+7) enfolded polygons, for n = 1, 80 + 94 = 175 − 1; for n = 2, 174 + 282 = 457 − 1; for n = 3, 456 + 846 = 1303 − 1; for n = 4, 1302 + 2538 = 3841 − 1. Notice that the numbers of geometrical elements for the seven enfolded polygons (green cells) is always twice the number of sides (blue cells).
The number of yods in P_{n} with C_{n} corners and S_{n} sides of T_{n} triangles ≡ Y_{n} = C_{n} + 2S_{n} + T_{n}. As, according to Table 9,
C_{n} + T_{n} = S_{n},
Y_{n} = 3S_{n}.
The number of corners, sides & triangles in P_{n} ≡ G_{n} = C_{n} + S_{n} + T_{n} = 2S_{n}. Therefore,
Y_{n}/G_{n} = 3/2.
The numbers in Table 10 confirm this for n = 1, 2, 3 & 4:
Y_{n}/G_{n} = 3/2
264/176 = 687/458 = 1956/1304 = 5763/3842 = 3/2.
Y_{n} = 3S_{n}
264 = 3×88, 687 = 3×229, 1956 = 3×652, 5763 = 3×1921.
G_{n} = 2S_{n}
176 = 2×88, 458 = 2×229, 1304 = 2×652, 3842 = 2×1921.
What is familiar to musicians as the tone ratio 3/2 of the perfect fifth (namely, the arithmetic mean of the tone ratios of the first and last notes of an octave — see here) is also the ratio of the yod and geometrical populations of the seven enfolded polygons, whatever their order. This is a remarkable property. The 0thorder triangle and the tetractys that corresponds to it have an analogous property, for nine yods line its boundary, which comprises three corners and three sides, i.e., six geometrical elements, and 9/6 = 3/2. Notice that for the root edge, the ratio of the number of yods (4) to the number of geometrical elements (3) is 4/3, which is the tone ratio of the perfect fourth! As the number of yods surrounding the centre of an nthorder Ngon = y_{n}^{N} − 1 = (3/2)(3^{n}+1)N and the number of geometrical elements surrounding its centre = g_{n}^{N} − 1 = (3^{n}+1)N, the ratio of these two numbers for any nthorder Ngon is 3/2, whatever the values of n and N. In view of this property for individual polygons, it is not remarkable that the ratio of the sums of these two quantities for all seven separate, nthorder polygons is also 3/2. Indeed, it is true for any set of separate polygons, not just for those that make up the inner Tree of Life. What is nontrivial is that this same ratio applies to the populations of all yods and geometrical elements in the seven regular polygons when they are enfolded. 31 yods disappear when the separate polygons become enfolded, so that the yod population of 295 is reduced to 264. Twentythree geometrical elements disappear, so that the 199 geometrical elements become 176 elements. The 288 yods surrounding the centres of the seven separate polygons become the 257 yods that surround the centres of the seven enfolded polygons, whilst the 192 geometrical elements surrounding centres of the seven separate polygons become 169 geometrical elements surrounding their centres when enfolded. Despite this disappearance, the fraction 3/2 survives the enfoldment, this time as the ratio of the total populations of yods and geometrical elements instead of as the ratio of numbers of yods and elements surrounding centres of the seven separate polygons (notice that 288/192 = 3/2 but 257/169 ≠ 3/2, whereas 264/176 = 3/2)! This would not happen for any set of polygons. In fact, the property exists only for a set of enfolded polygons that includes either a pentagon & decagon or a triangle, pentagon, hexagon & decagon (as in the case of the inner Tree of Life).**
As Y_{n} = 3S_{n}, the number of yods in two separate sets of seven enfolded polygons ≡ Y′ = 2Y_{n} = 6S_{n} and their number of geometrical elements ≡ G′ = 2G_{n} = 4S_{n}. Therefore,
Y’/G′ = 3/2
and
G′/Y_{n} = 4/3.
This is the tone ratio of the perfect fourth in music. The ratio Y′/Y_{n} = 2 corresponds to the octave. These three ratios are the ratios generated by the tetractys in the historical context of Pythagoras' contribution to music theory that is wellknown to students of music and science:

What is the corresponding relation between the yod (Y) and geometrical (G) populations of the (7+7) enfolded polygons? As
Y = 2Y_{n} − 4
and
G = 2G_{n} − 3,
we find that
2Y = 3G + 1.
This compares with the relation
2Y_{n} = 3G_{n}
for the seven enfolded, nthorder, regular polygons making up half of the inner form of the Tree of Life. As Y = 2Y_{n} − 4, G = 2G_{n} − 3, Y_{n} = 3S_{n} and G_{n} = 2S_{n},
Y = 6S_{n} − 4
and
G = 4S_{n} − 3.
The total number of sides in the (7+7) enfolded polygons ≡ S = 2S_{n} − 1 = 34 + 47×3^{n}. Therefore,
Y = 3S − 1
and
G = 2S − 1,
so that
Y/G = (3S−1)/(2S−1).
This may be simplified to:
Y/G = 3/2 + ½(2S−1)^{−1}.
As n→∞, S→∞ and Y/G→3/2. The ratio of the yod & geometrical populations of the complete inner form of the Tree of Life is larger than the fraction 3/2 but reduces as n→∞ from its maximum value of 524/349 when n = 1 to 3/2 as its asymptotic limit for an infinitely populated inner Tree of Life. This amounts only to a decrease of about 0.095%. Even when n = 1, the ratio is only about 0.1% above 3/2. But, for the seven enfolded polygons, the ratio remains exactly 3/2 for all values of n.
In terms of their number (C) of corners of their T triangles, G = C + S + T. Therefore,
C + S + T = 2S − 1,
so that
S = C + T + 1.
If we define S′ as the number of sides outside the root edge of the (7+7) enfolded, nthorder polygons, then S′ = S − 1. This means that
S′ = C + T.
We find that the number of external sides is equal to the number of their 0thorder triangles and their corners. This is a remarkable, geometrical property of the inner form of the Tree of Life. Therefore, as G = C + S + T,
G = S + S′.
The number of geometrical elements in the inner Tree of Life is the sum of the number of sides of all its triangles and the number outside the root edge shared by the (7+7) enfolded polygons. The figures in Table 10 illustrate this:
349 = 175 + 174.
913 = 457 + 456.
2605 = 1303 + 1302.
7681 = 3841 + 3840.
"Intrinsic yods" are discussed in many pages of this website. They are defined as all the yods in the inner form of the Tree of Life except the topmost corners of the two hexagons, which are shared with the lowest corners of the two hexagons belonging to the inner form of the next higher, overlapping Tree of Life. Intrinsic yods are those unshared yods that solely make up the inner form of each successive Tree. "Intrinsic geometrical elements" are, similarly, discussed many times. They are all those elements, apart from the topmost, shared corners of the pair of hexagons, that uniquely shape the inner form of each overlapping Tree because they are unshared with polygons enfolded in the next higher Tree of Life. Two other sets of yods and geometrical elements may be defined: 1. the yods or geometrical elements outside the root edge that create the form of the inner Tree of Life, and 2. the intrinsic yods or geometrical elements outside the root edge. Table 11 lists the relations between the populations of these types of yods and geometrical elements in the seven enfolded polygons and the (7+7) enfolded polygons (n.b. the subscript n is dropped from symbols because the relation: 2Y_{n} = 3G_{n} holds whatever the value of n, although both populations are, of course, functions of n. But, for the purpose of Table 11, the change of symbols: Y_{n}→y & G_{n}→g will be made, with Y & G still referring to the (7+7) enfolded polygons).
Table 11
7 enfolded polygons 
(7+7) enfolded polygons 

Number of yods Number of geometrical elements Relation: 
y 
Y = 2y − 4 
Number of yods outside root edge 
y′ ≡ y − 4 
Y′ ≡ Y − 4 
Number of intrinsic yods Number of intrinsic geometrical elements Relation: 
y_{0} ≡ y − 1 
Y_{0} ≡ Y − 2 
Number of intrinsic yods outside root edge Number of intrinsic geometrical elements outside root edge Relation: 
ŷ ≡ y_{0} − 4 
Ŷ ≡ Y_{0} − 4 
Notice in Table 11 that the relations between types of populations contain only the Pythagorean integers 1, 2, 3 & 4 expressed by the four rows of the tetractys. In particular, these integers are the constant that appears in the relations for the complete inner form of the Tree of Life. Their presence is indicating that these four classes of yods and geometrical elements are special. Much of the analysis in the pages of this website is devoted to them because they generate numbers and patterns that manifest in the group mathematics of E_{8}×E_{8} and in the remoteviewing accounts of superstrings given over a century ago by Annie Besant & C.W. Leadbeater (see Occult Chemistry), as well as in the seven musical diatonic scales and in the correspondences between the inner form of the Tree of Life and other sacred geometries, as described in Correspondences and Wonders of correspondences.
The number of yods that line the S_{n} sides of the T_{n} triangles in P_{n} ≡ B_{n} = C_{n} + 2S_{n}. As G_{n} = 2S_{n},
B_{n} = C_{n} + G_{n} = ½(105 + 329×3^{n−1}).
The number of boundary yods in the (7+7) enfolded nthorder polygons = 2B_{n} − 4 = 101 + 329×3^{n−1}. Table 12 lists the boundary yods for n = 1, 2, 3 & 4:
Table 12
7 enfolded polygons 
(7+7) enfolded polygons 

n = 1 
n = 2 
n = 3 
n = 4 
n = 1 
n = 2 
n = 3 
n = 4 

Number of boundary yods  217  546  1533  4494  430  1088  3062  8984 
In particular, the number of boundary yods shaping the seven enfolded, 4thorder polygons outside their root edge = 4494 − 4 = 4490 = 449×10, where 449 is the 87th prime number and 87 is the number value of Levanah, the Mundane Chakra of Yesod.
As the top corners of both hexagons in the (7+7) polygons enfolded in a given Tree of Life coincide with the lowest corners of the hexagons enfolded in the next higher, overlapping Tree of Life, there are (430−2=428) yods lining the (7+7) enfolded 1storder polygons that are intrinsic to them. This is the number value of Chasmalim, the Order of Angels assigned to Chesed. Table 12 indicates that 217 yods line the seven enfolded polygons. (217−1=216) intrinsic yods line the 88 sides of the 47 tetractyses making up each set of enfolded 1storder polygons with 36 corners, where 216 is the number value of Geburah and 36 is the number value of ELOHA, its Godname.
Geometrical and yod compositions of the combined outer & inner forms of N Trees of Life/Ntree with nthorder polygons
When the outer form of the Tree of Life combines with its inner form, the number of geometrical elements or yods in the combination is not the sum of the respective numbers of both forms. This is because certain elements and yods coincide when they combine. They are indicated in the diagram below as blue yods and lines. Remembering that the plane occupied by the (7+7) enfolded polygons is that formed by the outer Pillars of the Tree of Life, each nthorder hexagon has six yods (two of which are corners) and two vertical sides of sectors that are shared with the Pillars of Judgement and Mercy and the horizontal Path joining Chesed and Geburah. Each enfolded nthorder triangle that occupies one of the sectors of each hexagon has one corner and one centre that coincide with yods in the outer Tree. It means that 16 yods (including six corners of triangles) and four sides are shared by the outer and inner forms of the Tree of Life, so that the combination needs 16 yods and 10 geometrical elements (six corners & four sides) to be subtracted from the sums of the respective, separate populations.
For mathematical consistency, when the inner form of the Tree of Life is regarded as composed of 14 nthorder polygons, the triangles making up its outer form must be regarded only as nthorder. It would be inconsistent to combine 1storder polygons with 0thorder triangles in the outer Tree as shown in the diagram, which is intended merely to display what yods and what geometrical elements become shared during the combination of the outer and inner forms — whatever the common order of their polygons and triangles. This consistency must be maintained when the outer and inner forms of N overlapping Trees or the Ntree are considered. What also must be considered is the fact that the two blue hexagonal yods on the ChesedGeburah Path remain hexagonal yods, whatever the order of the triangles in the outer Tree of Life, whereas the blue yod (with which they coincide) at the centre of each triangle in the (7+7) polygons remains a corner of a 0thorder triangle, whatever the order of these polygons. It means that there are two unique yods in the combined outer & inner forms of the Tree of Life (and for every overlapping Tree) that are hexagonal yods with respect to its outer form but which are corners with respect to its inner form. Their dual status must be taken into account in calculating the hexagonal yod population of the combined Trees. Finally, the topmost corners of the two hexagons enfolded in a given Tree of Life coincide with the lowest corners of the two hexagons enfolded in the next higher Tree. This means that the number of yods shared by the outer and inner forms of N overlapping Trees of Life is 14N + 2, where "2" denotes the topmost corners of the two hexagons enfolded in the Nth Tree. It also means that the number of geometrical elements shared by both forms is 8N + 2. Both these quantities need to be subtracted from the sum of the separate populations of yods and geometrical elements to avoid doublecounting when the two forms are combined.
Using Tables 3, 6 & 8, the calculated yod and geometrical compositions of the combined outer and inner forms of N Trees/Ntree when the former have nthorder triangles and the latter have nthorder polygons are shown in Table 13:
Table 13. Numbers of corners, geometrical elements & yods in combined forms of N Trees/Ntree.
Combined outer & inner forms 

N Trees of Life  Ntree  
Number of corners 
2 + 27N + (65N+6)×3^{n−1} 
½[3 + 54N + (130N+21)×3^{n−1}] 
Number of geometrical elements  2 + 55N + 2(65N+6)×3^{n}  2 + 55N + (130N+21)×3^{n} 
Number of hexagonal yods  54N + 8(65N+6)×3^{n−1}  54N − 3 + (520N+84)×3^{n−1} 
Number of yods 
2 + 81N + (65N+6)×3^{n+1} 
½ [162N − 3 + (130N+21)×3^{n+1}] 
Notice that ADONAI, the Godname of Malkuth with number value 65, prescribes the yod population of the combined outer & inner forms of N Trees of Life, as well as their number of corners and geometrical elements, the number 130 being the 65th even integer. Here is remarkable evidence that the ancient Hebrew Divine Names determine through their gematria number values the very mathematical nature of the cosmic blueprint called the "Tree of Life."
The increase in yods from N Trees to (N+1) Trees = 81 + 65×3^{n+1}. For n = 1, the increase is 666. This is the 36th triangular number, showing how ELOHA, the Godname of Geburah with number value 36, measures how many yods are needed to build the outer and inner forms of successive Trees of Life. The average number of hexagonal yods that surround the axes of the five Platonic solids in their faces and interiors is 666 (see bottom of page here).
Table 14 lists the populations of geometrical elements and yods in the Tree of Life and the 1tree for 1st, 2nd, 3rd & 4thorder triangles/polygons:
Table 14
Combined outer & inner forms  
Tree of Life 
1tree 

n = 1 
n = 2 
n = 3 
n = 4 
n = 1 
n = 2 
n = 3 
n = 4 

Number of corners  100  242  668  1946  104  255  708  2067 
Number of geometrical elements  483  1335  3891  11559  510  1416  4134  12288 
Number of hexagonal yods  622  1758  5166  15390  655  1863  5487  16359 
Number of yods  722  2000  5834  17336  759  2118  6195  18426 
The combined outer & inner forms of the Tree of Life with 1storder triangles and polygons comprise 483 geometrical elements. Of these, three make up the root edge, so that 480 elements are outside it. This number is a characteristic parameter of holistic systems, e.g., the 480 hexagonal yods in the (7+7) separate polygons of the inner Tree of Life (see #21) and the 480 hexagonal yods in the faces of the first four Platonic solids constructed from tetractyses (see here). In the case of the E_{8}×E_{8} heterotic superstring, it is the number of roots of E_{8}×E_{8} (see under heading "Superstring gauge symmetry group" here).
As the 1tree shares 10 geometrical elements with its inner form (namely, the three corners and two sides lining the Pillars of Mercy & Judgement), the combined forms comprise 510 geometrical elements, of which 500 (=50×10) are unshared by either form. This shows how ELOHIM, the Godname of Binah with number value 50, prescribes the combined, outer & inner forms of the 1tree.
Of the 622 hexagonal yods in the combined forms of the Tree of Life made up of 1storder triangles and polygons, two belong to the root edge. 620 hexagonal yods are outside it. This number is the number value of Kether (see #7). Of the total of 722 yods, two hexagonal yods lie on the root edge, leaving 720 yods that comprise 100 corners and 620 hexagonal yods. This 100:620 division of the number 720 is identical to that displayed by the decagon when its sectors are 2ndorder tetractyses:


Its 100 tetractyses have 720 yods surrounding its centre. They comprise 100 corners and 620 hexagonal yods. This embodiment of the gematria number value 620 of Kether (the first of the 10 Sephiroth) by two 10fold representations of God, one Kabbalistic, the other of a Pythagorean character, demonstrates the mathematical parallels between different sacred geometries when they are truly such. See here for further analysis of how the combined Trees of Life embody this number, as well as other Kabbalistic numbers. Other examples of the embodiment of the number 720 by sacred geometries are:
the number of sides of triangles in an nthorder Ngon ≡ s_{n}^{N} = ½(3^{n}+1)N,
and the number of triangles in an nthorder Ngon ≡ t_{n}^{N} = 3^{n−1}N.
Therefore,
c_{n}^{N} + t_{n}^{N} = ½(3^{n−1}+1)N + 3^{n−1}N = ½(3^{n}+1)N = s_{n}^{N}
and
c_{n}^{N} + s_{n}^{N} + t_{n}^{N} = 2s_{n}^{N}.
** Proof: consider a set P of m different, separate nthorder polygons with N corners. According to Table 8, the number of yods in an nthorder rgon = y_{n}^{r} = (3/2)(3^{n}+1)r + 1.
Number of geometrical elements = g_{n}^{r} = (3^{n}+1)r + 1.
Number of yods in P ≡ Y = ∑y_{n}^{r} = (3/2)(3^{n}+1)N + m.
Number of geometrical elements in P ≡ G = ∑g_{n}^{r} = (3^{n}+1)N + m.
The ratio (Y−m)/(G−m) of the numbers of yods and geometrical elements surrounding the centres of the m separate nthorder polygons is always 3/2. Suppose that p yods and q geometrical elements outside the sides that merge into the root edge vanish when the m polygons become enfolded. These are not the total numbers that vanish because the 4(m−1) yods and 3(m−1) geometrical elements making up the (m−1) sides that merge into the root edge of the enfolded polygons disappear as well.
Number of yods in m enfolded nthorder polygons ≡ Y_{n}(m) = Y − p − 4(m−1) = (3/2)(3^{n}+1)N + 4 − 3m − p.
Number of geometrical elements in m enfolded polygons ≡ G_{n}(m) = G − q − 3(m−1) = (3^{n}+1)N + 3 − 2m − q.
Therefore, Y_{n}(m)/G_{n}(m) = 3/2 if 3q − 2p = 1. Yods and geometrical elements can disappear only for two pairs of polygons: trianglehexagon (let us call it "pair 1") and pentagondecagon ("pair 2"). This is because the triangle takes the place of one sector of the hexagon, sharing six yods and four geometrical elements, and a corner of the pentagon coincides with the centre of the decagon, so that one yod and one geometrical element disappears when they become enfolded. Either p = 1 (P includes only pair 2), 6 (P includes only pair 1) or 7 (P includes pairs 1 & 2) and q = 1 (P includes pair 2), 4 (P includes pair 1) or 5 (P includes pairs 1 & 2). The relation between p and q:
3q − 2p = 1
derived above does not hold for p = 0 = q, so that P must include at least one of the two pairs of polygons that share yods and geometrical elements. Therefore, Y_{n}(m)/g_{n}(m) = 3/2 only if P includes:
It is concluded that Y_{n}(m)/G_{n}(m) = 3/2 only if P includes either pair 2 (but not pair 1) or both pairs, i.e., m≥2. In the case of the inner Tree of Life, the 14 enfolded polygons have 70 corners that correspond to the 70 yods in the outer Tree of Life constructed from tetractyses. This means that either set of seven enfolded polygons has 36 corners. If we insist that the 2m enfolded polygons must have 70 corners to match the 70 yods, then the m enfolded polygons must have 36 corners. Separate, they have N corners; when enfolded, they have [N − 2(m−1)] corners because (m−1) sides disappear into the root edge. Therefore,
36 = N − 2(m−1) = N − 2m + 2,
so that
N = 34 + 2m.
As m≥2, N≥38. P cannot be just pair 2, because they have only 15 corners. Nor can it be just pair 1 & pair 2, because they have only 24 corners. This means that m>2 and N>38. As the polygons must include the pentagon and the decagon, the remaining (m−2) separate polygons must have more than (38−15=23) corners. If they include both pairs of polygons, m>4 and N>(34+8=42), so that the remaining (m−4) separate polygons must have more than (42−24=18) corners.
The 1tree has 80 yods (see #40). They are matched by the 80 corners of the 94 sectors of the (7+7) enfolded polygons. If we insist that the 2m enfolded polygons must have sectors with 80 corners, then each set of m enfolded polygons must have sectors with 41 corners. Suppose that the centres of A polygons coincide with their corners when enfolded. As P must include at least pair 2, A = 1 if P includes only pair 2 (m>2) and A = 2 if it includes both pairs (m>4). Separate, the sectors of the polygons have (N+m) corners; when enfolded, they have [N − 2(m−1) + m − A] corners. Therefore,
41 = N + 2 − m − A,
so that
N = 39 + m + A.
But
m + A > 3 (P includes only pair 2)
> 6 (P includes pairs 1 & 2).
This means that either N>42, so that the remaining (m−2) polygons must have more than (42−15=27) corners, or N>45, so that the remaining (m−4) polygons must have more than (45−24=21) corners. If we insist that the 2m enfolded polygons have 70 corners and that their sectors have 80 corners, then P must include the pentagon and the decagon and the remaining (m−2) polygons must have more than 27 corners, or else it must include the triangle, pentagon, hexagon & decagon and the remaining (m−4) polygons must have more than 21 corners. In the case of the inner Tree of Life, the three remaining polygons (square, octagon & dodecagon) have 24 corners and so, as expected, the seven enfolded polygons display the property: Y_{n}(m)/G_{n}(m) = 3/2.
As
N = 34 + 2m = 39 + m + A,
m − A = 5.
As A = 1 or 2, m = 6 or 7 and m + A = 7 or 9. This means that N = 46 or 48. The latter possibility is displayed by the seven polygons making up the inner Tree of Life.
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