ARTICLE 65
by
Stephen M. Phillips
Flat 4, Oakwood House, 117119 West Hill Road. Bournemouth. BH2 5PH. England.
Website: http://smphillips.mysite.com
Abstract
Formulae are derived for the populations of geometrical
elements and yods in a polygon divided any number of times into triangles. The
populations of N overlapping Trees of Life, the Ntree and their inner form as N
sets of (7+7) enfolded, regular polygons are calculated. The ratio of the yod
and geometrical populations of a set of seven enfolded polygons is found to be
3/2, independent of the number of times they are divided. This is the tone ratio
of the perfect fifth, the primary note in a musical octave. For any number n of
their divisions into triangles, the corresponding ratio for the (7+7) enfolded
polygons is within 0.1% of this fraction, converging towards it in the
asymptotic limit as n approaches infinity. The integers 1, 2, 3 & 4
symbolised by the four rows of the Pythagorean tetractys determine various
populations of yods and geometrical elements in the inner Tree of Life. ADONAI,
the Divine Name assigned to Malkuth, is found to prescribe the population of
yods and geometrical elements in the combined outer & inner forms of N Trees
and the Ntree.

1
Table 1. Gematria number values of the ten Sephiroth in the four Worlds.

SEPHIRAH

GODNAME

ARCHANGEL

ORDER OF
ANGELS

MUNDANE
CHAKRA

1 
Kether
(Crown)
620 
EHYEH
(I am)
21 
Metatron
(Angel of the
Presence)
314 
Chaioth ha Qadesh
(Holy Living
Creatures)
833

Rashith ha Gilgalim
First Swirlings.
(Primum Mobile)
636 
2 
Chokmah
(Wisdom)
73 
YAHWEH, YAH
(The Lord)
26, 15

Raziel
(Herald of the
Deity)
248 
Auphanim
(Wheels)
187 
Masloth
(The Sphere of
the Zodiac)
140 
3 
Binah
(Understanding)
67 
ELOHIM
(God in multiplicity)
50

Tzaphkiel
(Contemplation
of God)
311

Aralim
(Thrones)
282

Shabathai
Rest.
(Saturn)
317 

Daath
(Knowledge)
474 




4 
Chesed
(Mercy)
72 
EL
(God)
31 
Tzadkiel
(Benevolence
of God)
62 
Chasmalim
(Shining Ones)
428

Tzadekh
Righteousness.
(Jupiter)
194 
5 
Geburah
(Severity)
216

ELOHA
(The Almighty)
36

Samael
(Severity of God)
131

Seraphim
(Fiery Serpents)
630

Madim
Vehement Strength.
(Mars)
95 
6 
Tiphareth
(Beauty)
1081

YAHWEH ELOHIM
(God the Creator)
76 
Michael
(Like unto God)
101

Malachim
(Kings)
140

Shemesh
The Solar Light.
(Sun)
640 
7 
Netzach
(Victory)
148

YAHWEH SABAOTH
(Lord of Hosts)
129

Haniel
(Grace of God)
97 
Tarshishim or
Elohim
1260

Nogah
Glittering Splendour.
(Venus)
64 
8 
Hod
(Glory)
15

ELOHIM SABAOTH
(God of Hosts)
153

Raphael
(Divine
Physician)
311

Beni Elohim
(Sons of God)
112

Kokab
The Stellar Light.
(Mercury)
48 
9 
Yesod
(Foundation)
80

SHADDAI EL CHAI
(Almighty Living
God)
49, 363

Gabriel
(Strong Man of
God)
246

Cherubim
(The Strong)
272

Levanah
The Lunar Flame.
(Moon)
87 
10 
Malkuth
(Kingdom)
496

ADONAI MELEKH
(The Lord and
King)
65, 155

Sandalphon
(Manifest
Messiah)
280 
Ashim
(Souls of Fire)
351

Cholem Yesodeth
The Breaker of the
Foundations.
The Elements.
(Earth)
168 
The Sephiroth exist in the four Worlds of Atziluth, Beriah, Yetzirah
and Assiyah. Corresponding to them are the Godnames, Archangels, Order of
Angels and Mundane Chakras (their physical manifestation). This table gives
their number values obtained by the ancient practice of gematria, wherein a
number is assigned to each letter of the alphabet, thereby giving a number
value to a word that is the sum of the numbers of its letters.

(Numbers in this table referred to in the article will be written in
boldface).
2
1.
Geometrical composition of N Trees/Ntree with nthorder
triangles Triangles are the basic building blocks of geometry.
But in sacred geometry they can be considered not only as pure triangles (what we may call
"0thorder triangles") but also as triangles divided into their three sectors (what in previous
articles have been called "Type A triangles" but which will here be called "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 triangle in a Type B
triangles generates a "Type C triangle" ("3rdorder triangle"), 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

2ndorder triangle

3rdorder triangle

4thorder triangle

Table 2 shows how the numbers of corners, sides & triangles inside the parent triangle
increase with n:
Table 2

n = 0

n = 1

n = 2

n = 3

n = n

Number of corners (c)

0

3

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
Table 3 lists the numbers of corners, sides & 0thorder triangles in N
Trees of Life or in the Ntree:
Table 3

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 4 lists the calculated, geometrical composition of N Trees/Ntree with
nthorder triangles:
Table 4

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 5 lists their
geometrical compositions for 0th & 1storder triangles:
Table 5

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

3
Table 6 lists the geometrical composition of N Trees/Ntree for the orders n
= 0, 1, 2, & 3:
Table 6

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 7 lists the yod population of N Trees/Ntree with nthorder
triangles:
Table 7. 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 8 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 8. 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

2. The geometrical & yod
compositions of polygons
Let us define a 1storder polygon P_{1} as a polygon divided
into its sectors, a 2ndorder polygon P_{2} as a P_{1} whose sectors are
each divided into three sectors, etc. P_{1} is the Type A polygon, P_{2} is
the Type B polygon, P_{3} is the Type C polygon, etc. A sector of the first three
orders of polygons is shown below:



1storder polygon

2ndorder polygon 
3rdorder polygon

The number of corners per sector of the nthorder polygons P_{n} (n = 1, 2, 3, 4, etc.)
increases with n as 1,
4
2, 2 + 3^{1}, 2 + 3^{2}, etc. The number of sides increases
as 2, 2 + 3^{1}, 2 + 3^{1} + 3^{2}, 2 + 3^{1} +
3^{2 +} 3^{3}, etc. The number of triangles increases as 1,
3^{1}, 3^{2}, 3^{3}, etc. Including its centre,
P_{n} has c_{n} corners, s_{n} sides &
t_{n} triangles, where:
c_{n} = 1 + 3^{0} + 3^{1} +... +
3^{n−2} = (1+3^{n−1})/2,
s_{n} = 1 + 3^{0} + 3^{1} + +... +
3^{n−1} = (1+3^{n})/2,
and
t_{n} = 3^{n−1}.
Therefore,
c_{n} + t_{n} =
(1+3^{n−1})/2 + 3^{n−1} = (1+3^{n})/2 = s_{n},
i.e., the number of corners & sides per sector of P_{n} is
equal to the number of sides per sector of the triangles in each sector. This means that the
number of corners & triangles surrounding the centre of P_{n} is equal
to the number of sides of its triangles. This division of the geometrical elements
surrounding the centre of a polygon into two equal sets is highly significant,
having important implications that are revealed in the discussion elsewhere in this website
of how sacred geometries encode the group mathematics and structure of
E_{8}×E_{8} heterotic superstrings.
The number of corners, sides & triangles per sector ≡
g_{n} = c_{n} + s_{n} + t_{n} =
2s_{n} = 1 + 3^{n}.
Including its centre, the number of corners in an nthorder mgon ≡
C_{n}(m) = 1 + mc_{n} = 1 + m(1+3^{n−1})/2.
The number of sides ≡ S_{n}(m) = ms_{n} =
m(1+3^{n})/2.
The number of triangles ≡ T_{n}(m) = mt_{n} =
3^{n−1}m.
The number of geometrical elements in an nthorder mgon ≡ G_{n}(m)
= C_{n}(m) + S_{n}(m) + T_{n}(m) = 1 + mg_{n} = 1 +
m(1+3^{n}).
The number of hexagonal yods in an nthorder mgon ≡ h_{n}(m) =
2S_{n}(m) + T_{n}(m) = m(1+3^{n−1}+3^{n}).
Including the centre of the mgon, the number of yods ≡ Y_{n}(m) =
C_{n}(m) + h_{n}(m) = 1 + m(3+3^{n+1})/2.
For reference, Table 9 lists the formulae for the geometrical and yod
populations of the nthorder mgon:
Table 9
Number of
corners (including centre of mgon)

C_{n}(m) = 1 + m(1+3^{n−1})/2

Number of
sides

S_{n}(m) = m(1+3^{n})/2

Number of
triangles

T_{n}(m) = 3^{n−1}m

Total number of
geometrical elements

G_{n}(m) = 1 + m(1+3^{n})

Number of hexagonal
yods

h_{n}(m) =
m(1+3^{n−1}+3^{n})

Total number of
yods

Y_{n}(m) = 1 + 3m(1+3^{n})/2

("1" denotes the centre of the nthorder mgon).
Seven separate polygons
The seven types of regular polygons present in the inner form of the Tree of
Life have 48 corners. The number of geometrical elements
in the seven separate nthorder polygons = ∑G_{n}(m) = 7 + ∑mg_{n} = 7
+ g_{n}∑m = 7 + 48g_{n} = 7 +
48(1+3^{n}) = 55 + 48×3^{n}.
"55" is the number of corners of their
48 sectors.
The number of hexagonal yods in the seven separate nthorder polygons ≡
H_{n} = ∑h_{n}(m) = (1+3^{n−1}+3^{n})∑m =
48(1+3^{n−1}+3^{n}) =
48 + 64×3^{n}.
The number of yods in the seven polygons ≡ Y_{n} =
∑Y_{n}(m) = ∑[1 + m(3+3^{n+1})/2 = 7 + 24(3+3^{n+1}) = 7 +
72(1+3^{n}) = 79 + 8×3^{n+2}.
Seven enfolded polygons
Corners The number of corners per sector of an (n−1)thorder
polygon = c_{n−1} = (1+3^{n−2})/2. Hence, the number of corners in a
(n−1)thorder triangle = 1 + 3c_{n−1} = 1 + 3(1+3^{n−2})/2 =
(5+3^{n−1})/2. The number of corners of triangles in the seven separate, nthorder
polygons = ∑C_{n}(m)
5
= 7 + ½(1+3^{n−1})∑m = 7 + 24(1+3^{n−1}) = 31 +
8×3^{n}.
When the polygons are enfolded, a corner of the pentagon coincides with the
centre of the decagon, a corner of the triangle coincides with the centre of the hexagon and
the nthorder triangle replaces a sector (an (n−1)thorder triangle) of the nthorder
hexagon. Also, single sides of five other polygons coincide with the root edge, so that the
(5×2=10) corners that these sides join disappear. The number of corners of triangles in the
seven enfolded nthorder polygons ≡ C_{n} = 31 +
8×3^{n} – 1 – 10 − (5+3^{n−1})/2 = (35+47×3^{n−1})/2.
The number of corners in the (7+7) enfolded polygons =
2C_{n} − 2 = 33 + 47×3^{n−1}.
Sides
The number of sides of triangles in an (n−1)thorder polygon =
ms_{n−1} = m(1+3^{n−1})/2. Hence, the number of sides in an
(n−1)thorder triangle = 3(1+3^{n−1})/2. The number of sides in the seven separate
nthorder polygons = ∑S_{n}(m) = ∑ms_{n} = ½(1+3^{n})∑m =
24(1+3^{n}). The number of sides in the seven enfolded polygons ≡
S_{n} = 24(1+3^{n}) – 5 − 3(1+3^{n−1})/2 = (35 +
47×3^{n})/2. The number of sides in the (7+7) enfolded polygons =
2S_{n} − 1 = 34 + 47×3^{n}.
Triangles
The number of triangles in an (n−1)thorder polygon = T_{n−1}(m) =
3^{n−2}m. Hence, the number of triangles in an (n−1)thorder triangle =
3^{n−1}. The number of triangles in the seven separate polygons = ∑T_{n}(m)
= ∑3^{n−1}m = 48×3^{n−1} = 16×3^{n}. The number of
triangles in the seven enfolded polygons ≡ T_{n} =
16×3^{n} − 3^{n−1} = 47×3^{n−1}. The number of triangles
in the (7+7) enfolded polygons = 2T_{n} = 94×3^{n−1}.
Geometrical elements
The number of geometrical elements in the seven enfolded polygons ≡
G_{n} = C_{n} + S_{n} + T_{n} = 35 +
47×3^{n}. The number of geometrical elements in the (7+7) enfolded polygons =
2G_{n} − 3 = 67 + 94×3^{n}, where 67 is the
number value of Binah. As the topmost corner of each hexagon coincides with the bottom of a
hexagon enfolded in the next higher Tree of Life, the number of geometrical elements that
are intrinsic to the inner form of each Tree = 65 +
94×3^{n}, where 65 is the number value of ADONAI, the Godname of
Malkuth.
Yods
The number of hexagonal yods in the seven enfolded polygons ≡
H_{n} = 2S_{n} + T_{n} = 35 +
47×3^{n} + 47×3^{n−1} = 35 + 188×3^{n−1}.
The number of hexagonal yods in the (7+7) enfolded polygons =
2H_{n} − 2 = 68 + 376×3^{n−1}.
The number of yods in the seven enfolded polygons ≡ Y_{n} =
C_{n} + H_{n} = (35 + 47×3^{n−1})/2 + 35 +
188×3^{n−1} = ½(105 + 47×3^{n+1}).
The number of yods in the (7+7) enfolded polygons = 2Y_{n} − 4
= 101 + 47×3^{n+1}. Amazingly, 101 is the 26th prime
number and 47 is the 15th prime number, where 26 is the number value of
YAHWEH, the Godname of Chokmah, and 15 is the number of YAH, the shorter version
of the Godname. It demonstrates how this wellknown Name of God prescribes through its
gematria number the yod population of the inner form of the Tree of Life.
The number of yods surrounding the 14 centres of the (7+7) enfolded polygons
= 87 + 47×3^{n+1}, where 87 is the number value of
Levanah, the Mundane Chakra of Yesod.
Table 10 lists the geometrical and yod compositions of the 7 enfolded
nthorder polygons and the (7+7) enfolded nthorder polygons:
Table 10. Geometrical & yod compositions of the inner form of the Tree of
Life.

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
6
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. It is the number of yods below Binah of the 1tree
when its 19 triangles are 1storder triangles [1].
Let us call the seven enfolded nthorder polygons P_{n}. Table 10
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 10, this is the
number of sides 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}, whilst its number of
sides is equal to the number of corners 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 10, 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 [2]. 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 11, which lists the geometrical and yod
composition of the inner Tree of Life for n = 1, 2, 3 & 4:
Table 11

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.
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 10,
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 11 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
7
tone ratios of the first and last notes of the 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) [3].
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:

The tone ratios of the octave, perfect 5th & perfect 4th appear in the
proportions of various populations of yods and geometrical elements in the two
separate halves of the inner Tree
of Life (the case of 1storder polygons is chosen as an example).

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}
8
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;
2. the intrinsic yods or geometrical elements outside the root edge. Table
12 below list the relations between the populations of these types of yods and geometrical
elements in the seven enfolded
9
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 12, 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 12

7 enfolded
polygons

(7+7) enfolded
polygons

Number of yods
Number of geometrical elements
Relation

y
g
2y = 3g

Y = 2y − 4
G = 2g − 3
2Y = 3G + 1

Number of yods outside root
edge
Number of geometrical elements outside root edge
Relation

y′ ≡ y − 4
g′ ≡ g − 3
2y′ = 3g′ + 1

Y′ ≡ Y − 4
G′ ≡ G − 3
2Y′ = 3G′ + 2

Number of intrinsic yods
Number of intrinsic geometrical elements
Relation

y_{0} ≡ y − 1
g_{0} ≡ g − 1
2y_{0} = 3g_{0}+ 1

Y_{0} ≡ Y − 2
G_{0} ≡ G − 2
2Y_{0} = 3G_{0} + 3

Number of intrinsic yods outside
root edge
Number of intrinsic geometrical elements outside root edge
Relation

ŷ ≡ y_{0} − 4
ĝ ≡ g_{0} − 3
2ŷ = 3ĝ + 2

Ŷ ≡ Y_{0} − 4
Ĝ ≡ G_{0}− 3
2Ŷ = 3Ĝ + 4

Notice in Table 12 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 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 13 lists the boundary
yods for n = 1, 2, 3 & 4:
Table 13

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 any
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 these (7+7)
1storder polygons that are intrinsic to them alone. This is the number value of
Chasmalim, the Order of Angels assigned to Chesed. Table 13 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.
3. Geometrical & yod populations
of the combined outer/inner N Trees and Ntree
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
10
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 same 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 = 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 = 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 4, 7 & 10, 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 14:
Table 14. 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."
11
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 15 lists the populations of geometrical elements & yods in the
Tree of Life and the 1tree for 1st, 2nd, 3rd & 4thorder triangles/polygons:
Table 15

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 here) 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 here). Of the 722 yods, two hexagonal 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:


The 720 yods in the combined Trees of Life
other than the two hexagonal yods in the root edge comprise 100 corners and
620 hexagonal yods.

The 720 yods surrounding the centre of a decagon
with 2ndorder tetractyses as sectors comprise 100 corners of 1storder
tetractyses and 620 hexagonal yods.

Its 100 tetractyses have 720 yods surrounding its centre. They comprise 100 corners and
620 hexagonal yods. This embodiment of the number 620 of Kether (the first of the
10 Sephiroth) by two 10fold
12
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 five Platonic solids constructed from tetractyses, which have 720 hexagonal
yods in their faces (see here);

the 720 yods surrounding the centres of the seven separate, 2ndorder polygons of
the inner Tree of Life (see Fig. 7 here);

the 720 corners of triangles surrounding the axis of the disdyakis triacontahedron
that make up its Type A triangular faces and internal triangles formed by its edges
and face sector sides (see here);

the 720 sides & triangles surrounding the axis in each half of the first four
Platonic solids with their faces divided into sectors when their interior triangles
formed from edges and sector sides are Type A (see here);

the 720 edges of the 600cell (see below Table 1a here).
References
1. See here.
2. Proof: an nthorder Ngon is one whose N sectors are (n−1)thorder
triangles. Using Table 9, the number of corners surrounding the centre of an nthorder Ngon
≡ c_{n}^{N} = ½(3^{n−1}+1)N,
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}.
3. Proof: consider a set (P) of m different, separate nthorder polygons
with N corners. According to Table 9, 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:
13

only pair 2. Then p = 1 = q, which satisfies the relation.

pairs 1 & 2. Then p = 7 and q = 5, which obeys the relation.
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 here). 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 N = 46 or 48. The
latter possibility is the set of seven polygons with 48 corners called the
inner Tree of Life.
14
