Polygons as the geometrical expression of numbers
A socalled "ngon" (a polygon with n corners or sides) can be divided, firstly, into its n sectors. It is what we call a "Type A polygon" (the term also applies to a polygon with its sectors turned into tetractyses). Then each triangular sector can be divided into three sectors, i.e., each sector can be turned into a Type A triangle. This is a "Type B polygon" (as it also is when its 3n triangles are turned into tetractyses). Such division can proceed indefinitely, generating Type C polygons, each of whose sectors is a Type B triangle, then Type D polygons, each of whose sectors is a Type C triangle, etc. Discussion here will be mostly confined to Type A and Type B polygons. The way in which the Type C triangle embodies the dimension 248 of the rank8, exceptional Lie group E_{8} used in superstring theory is discussed here.
The regular polygons are the simplest way of symbolizing
integers by the number of their corners or sides, starting with the integer 3 as the number of corners or sides of
the equilateral triangle. However, there are three more subtle ways in which polygons embody numbers, whether
regular or not:
1. Number of points, lines & triangles needed for their
construction.
There are seven possible combinations of these three geometrical elements or building blocks: 1. point, 2. line, 3. triangle, 4. point+line, 5. point+triangle, 6. line+triangle, 7. point+line+triangle. A polygon embodies in its geometry the seven corresponding numbers of such combinations. If the numbers of various combinations of geometrical elements surrounding the centre of the polygon are considered as well (of which four new ones exist), a polygon embodies 11 numbers. A Type A ngon is composed of (n+1) corners and 2n sides of n triangles, i.e., (4n+1) geometrical elements, where "1" refers to its centre. A Type B ngon has (2n+1) corners and 5n sides of 3n triangles, i.e., (10n+1) geometrical elements.
Table 1 displays the geometrical composition of the seven types of Type A and Type B regular polygons making up the inner Tree of Life:
Table 1. Geometrical composition of the seven Type A or Type B polygons in the inner Tree of Life.
Triangle 
Square 
Pentagon  Hexagon  Octagon 
Decagon 
Dodecagon 
Total  
A 
B 
A  B  A  B  A  B  A  B  A  B  A  B  A  B  
Number of corners  3+1=4  6+1=7  4+1=5  8+1=9  5+1=6  10+1=11  6+1=7  12+1=13  8+1=9  16+1=17  10+1=11  20+1=21  12+1=13  24+1=25  48+7=55  96+7=103 
Number of sides  6  15  8  20  10  25  12  30  16  40  20  50  24  60  96  240 
Number of triangles  3  9  4  12  5  15  6  18  8  24  10  30  12  36  48  144 
Number of geometrical elements  12+1=13  30+1=31  16+1=17  40+1=41  20+1=21  50+1=51  24+1=25  60+1=61  32+1=33  80+1=81  40+1=41  100+1=101  48+1=49  120+1=121  192+7=199  480+7=487 
(The number "1" in some cells denotes the centre of a polygon; as usual, the gematria number values of the ten Sephiroth in the four Kabbalistic Worlds (see here) are in bold). The seven Type A polygons have 192 geometrical elements surrounding their centres. 384 geometrical elements surround the 14 centres of both sets of seven polygons. This 192:192 division is characteristic of holistic systems. For example, as discussed here, the table of 64 hexagrams used in the ancient Chinese divination system of I Ching contains 384 lines & broken lines (192 in each diagonal half). Surrounding the centres of the seven Type B polygons are 240 sides and 240 corners & triangles, i.e., 480 geometrical elements. This 240:240 division is another feature of holistic systems. Its counterpart in E_{8}×E_{8} heterotic superstring theory is the set of 240 roots of each group E_{8} in the symmetry group E_{8}×E_{8}. It manifests in each set of seven separate Type A polygons as their 240 hexagonal yods (see table below) when their 48 sectors are tetractyses (see also here & here).
2. Numbers of yods needed to construct their sectors from tetractyses.
A Type A ngon has (6n+1) yods made up of (n+1) corners & 5n hexagonal yods. A Type B ngon has (15n+1) yods made up of (2n+1) corners & 13n hexagonal yods. Notice the appearance in the algebraic expression for its yod population of the number value 15 of YAH (יה), the older version of the Godname assigned to Chokmah.
Table 2 displays the yod populations of the seven regular polygons comprising the inner Tree of Life:
Table 2. Yod composition of the seven Type A or Type B polygons.
Triangle 
Square 
Pentagon  Hexagon  Octagon  Decagon  Dodecagon  Total  
A  B  A  B  A  B  A  B  A  B  A  B  A  B  A  B  
Number of corners  3+1=4  6+1=7  4+1=5  8+1=9  5+1=6  10+1=11  6+1=7  12+1=13  8+1=9  16+1=17  10+1=11  20+1=21  12+1=13  24+1=25  48+7=55  96+7=103 
Number of hexagonal yods  15  39  20  52  25  65  30  78  40  104  50  130  60  156  240  624 
Number of yods  18+1=19  45+1=46  24+1=25  60+1=61  30+1=31  75+1=76  36+1=37  90+1=91  48+1=49  120+1=121  60+1=61  150+1=151  72+1=73  180+1=181  288+7=295  720+7=727 
As pointed out earlier, each set of seven polygons has not only 240 sides and 240 corners & triangles when they are Type B polygons but also 240 hexagonal yods when they are Type A. The 240:240 division has now extended from one set to both sets. The number 240 is a parameter of all holistic systems (see here; scroll down to "240 = 72 + 168").
3. Sums of consecutive integers (always starting with 1) assigned to yods in the polygon (always starting at its centre).
A Type A ngon has (3n+1) yods inside its boundary and 3n yods (i.e., one less) on its boundary. So it contains (6n+1) yods. As this is an odd integer, the sum of the first (6n+1) integers that can be assigned to its yods:
1 + 2 + 3 + ... + (6n+1) = (3n+1)(6n+1),
contains (3n+1) odd integers and 3n even integers (i.e., one less). It is, therefore, mathematically possible, as well as natural, to assign the odd integers up to (6n+1) to internal yods, starting with the integer 1 at its centre, and to assign, starting with 2, the even integers up to 6n to yods on the sides of the polygon. The sum of the former is:
1 + 3 + 5 + ... + (6n+1) = (3n+1)^{2}
and the sum of the latter is:
2 + 4 + 6 + ... + 6n = 3n(3n+1).
The sum of all (6n+1) integers that can be assigned to the yods in a Type A ngon is (3n+1)(6n+1). Hence, a Type A polygon embodies three numbers that are these three sums. Notice that the arithmetic means of the odd integers, the even integers and all the integers are the same, namely, 3n + 1, which is the number of internal yods in the Type A ngon. The only one of the seven regular polygons making up the inner form of the Tree of Life for which this is a Godname number is the decagon (n = 10). In this polygon, each mean is 31, which is the number of EL, the Godname of Chesed. This acquires significance in view of the fact that the Type A decagon with its ten sectors turned into tetractyses and containing 61 yods is the polygonal counterpart of the 61 SLs up to the highest Chesed of ten overlapping Trees, which is a representation of the Tree of Life in which each Sephirah is replaced by a Tree of Life. A Type B polygon also embodies three numbers, although, having a yod population of (15n+1) that can be either even or odd and not having the same number of yods on its boundary as in its interior, assignment of odd integers to internal yods and even integers to boundary yods is no longer possible, so that the interior integers must be 1, 2, 3, ... (12n+1) and the boundary integers must be (12n+2), (12n+3), ... (15n+1).
The table above shows that the seven separate Type B polygons contain 727 yods. This number is the 129th prime number, where 129 is the gematria number value of ELOHIM SABAOTH, the Godname of Hod (see here). This is a spectacular example of how the Godnames of the ten Sephiroth prescribe the properties of the inner Tree of Life (in the case of the seven Type A polygons, see pages 2534 in Sacred geometry/Tree of Life).
The number of yods surrounding the centres of the seven Type B polygons = 720 = 10×72. This is the number of yods that surround the centre of a decagon with 2ndorder tetractyses as sectors, each sector having 72 such yods:
720 yods surround the centre of a decagon with 2ndorder tetractyses as its sectors. 
This demonstrates the archetypal power of the Decad symbolised by the 10corner decagon to determine the properties of holistic systems.
It needs to be emphasized that, as we are dealing here not with ordinary geometry but with sacred geometry. Because the seven polygons of the inner Tree of Life are constructed from tetractyses — the template of this archetypal geometry — the numbers thereby generated by this transformation are not arbitrary or insignificant. Instead, they turn out to be numbers that characterize all sacred geometries in a global sense, as demonstrated many times in this website. In other words, they are universal parameters of sacred geometry. As such, their presence helps to distinguish what is true sacred geometry, namely, the representation of the divine archetypes, from what is merely the kind of axiomatic geometry that has been studied by mathematicians ever since the days of Euclid and taught to school children for hundreds of years.
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 polygons, P_{3} is the Type C polygon, etc. A sector of the first three orders of polygons is shown below:



The number of corners per sector of the nthorder polygon P_{n} (n = 1, 2, 3, 4, etc) increases with n as 1, 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 will be 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.
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}.
For future reference, Table 3 lists the numbers of corners, sides & triangles per sector for the first four orders of any polygon:
Table 3. Geometrical composition per sector for the first four orders of a polygon.

n 
Corners (c_{n}) 
Sides (s_{n}) 
Triangles (t_{n}) 
Total (g_{n}) 
Type A 
1 
1 
2 
1  4 
Type B 
2 
2 
5 
3  10 
Type C 
3 
5 
14  9  28 
Type D 
4 
14 
41  27  82 
Table 4 lists the yod populations of the first four orders of the seven types of polygons making up the inner Tree of Life:
Table 4. Yod populations of the first four orders of the seven types of polygons in the inner Tree of Life and their sums.

n 
Y_{n}(3) 
Y_{n}(4) 
Y_{n}(5) 
Y_{n}(6) 
Y_{n}(8) 
Y_{n}(10) 
Y_{n}(12) 
Y_{n} 
Type A 
1 
1 + 18 = 19 
1 + 24 = 25 
1 + 30 = 31  1 + 36 = 37  1 + 48 = 49  1 + 60 = 61  1 + 72 = 73  7 + 288 = 295 
Type B 
2 
1 + 45 = 46 
1 + 60 = 61 
1 + 75 = 76  1 + 90 = 91  1 + 120 = 121  1 + 150 = 151  1 + 180 = 181  7 + 720 = 727 
Type C 
3 
1 + 125 = 126 
1 + 168 = 169  1 + 210 = 211  1 + 252 = 253  1 + 336 = 337  1 + 420 = 421  1 + 504 = 505  7 + 2016 = 2023 
Type D 
4 
1 + 369 = 370 
1 + 492 = 493  1 + 615 = 616  1 + 738 = 739  1 + 984 = 985  1 + 1230 = 1231  1 + 1476 = 1477  7 + 5904 = 5911 
Seven enfolded polygons
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 in the seven separate nthorder polygons
= ∑C_{n}(m) = 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}.
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}.
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}.
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.
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.
For reference, Table 5 lists the formulae for the geometrical and yod populations of the nthorder mgon:
Table 5. Geometrical & yod compositions of the nthorder mgon.
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 
Also for reference, Table 6 lists the formulae for the geometrical and yod populations of the inner Tree of Life composed of nthorder polygons:
Table 6. 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}) 
33 + 47×3^{n−1} 
Number of sides 
S_{n} = ½(35 + 47×3^{n}) 
34 + 47×3^{n} 
Number of triangles 
T_{n} = 47×3^{n−1} 
94×3^{n−1} 
Total number of geometrical elements 
G_{n} = 35 + 47×3^{n} 
67 + 94×3^{n} 
Number of hexagonal yods 
H_{n} = 35 + 188×3^{n−1} 
68 + 376×3^{n−1} 
Total number of yods 
Y_{n} = ½(105 + 47×3^{n+1}) 
101 + 47×3^{n+1} 
The number of yods in the (7+7) enfolded polygons outside the root edge = 97 + 47×3^{n+1}, where 47 = 21 + 26 (sum of Godname numbers of first two Sephiroth) and 97 = 21 + 26 + 50 (sum of Godname numbers of first three Sephiroth). The number value of Haniel, the Archangel of Netzach, is 97. The number of yods in the seven enfolded polygons that line the S_{n} sides of their T_{n} tetractyses = Y_{n} − T_{n} = ½(105 + 329×3^{n−1}). The number of yods that line sides of the 2T_{n} tetractyses in the (7+7) enfolded polygons = 101 + 329×3^{n−1}. 101 is the number value of Michael, the Archangel of Tiphareth, which immediately precedes Netzach. The total number of yods in the (7+7) enfolded polygons = 101 + 47×3^{n+1}.
From the formulae in Table 5, the following relationship between G_{n}(m) and Y_{n}(m) emerges:
Y_{n}(m) = ½(3G_{n}(m) −1).
Writing Y_{n}(m) = Ŷ_{n}(m) + 1 and G_{n}(m) = Ĝ_{n}(m) + 1, where "1" denotes the centre of the mgon, the number of yods Ŷ_{n}(m) surrounding this centre is related to the number of geometrical elements Ĝ_{n}(m) that surround it by:
2Ŷ_{n}(m) = 3Ĝ_{n}(m) = 3m(1+3^{n}),
i.e., Ŷ_{n}(m)/Ĝ_{n}(m) = 3/2. This simple result helps to reduce the amount of calculation needed to determine the yod and geometrical populations of polygons of any order, as well as any combination of separate polygons. In particular, the number of yods that surround the centres of the seven types of separate nthorder polygons making up the inner Tree of Life = ∑Ŷ_{n}(m) = (3/2)(1+3^{n})∑m = 72(1+3^{n}) = 72 + 8×3^{n+2}. It means that the number of geometrical elements surrounding the centres = ∑Ĝ_{n}(m) = (2/3)×∑Ŷ_{n}(m) = (2/3)×(72 + 8×3^{n+2}) = 48 + 16×3^{n+1}. As a check, the number of yods surrounding the centres of the seven separate 1storder polygons = 72 + 216 = 288, and the number of geometrical elements surrounding them = 48 + 144 = 192, both of which calculations are correct (see the last A columns in Tables 1 & 2).