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Polygonal Numbers

In this section we shall study a class of mathematical object known to the ancient Greeks as ‘polygonal numbers.’ They are a generalisation of geometrically-based numbers, such as the numbers of corners of polygons, their yod populations, etc. Remarkable relationships exhibiting beautiful, mathematical harmonies will be shown to exist between polygonal numbers and parameters of the Tree of Life, including those that are group-theoretical numbers of superstring/bosonic string theories. These links turn out to be too closely attuned to Pythagorean mathematical notions to be plausibly explained away as coincidental.

It is important to realize that mathematics during the time of the Pythagorean School in Croton in southern Italy had not yet been formalized into a subject for academic research. Instead, students were taught mathematics in this fraternity in order to contemplate the divine immanent in nature. The mathematical doctrines of Pythagoras cannot be properly understood without recognising that the modern dichotomy of science and religion did not exist then. Pythagoreans believed that man could realise his divine nature by knowing the universal principle that governs the cosmos (a word coined by Pythagoras himself, meaning “world-order,” a world ordered in a state of mathematical harmony). This principle is Number, which is “the principle, the source and the root of all things.” For the Pythagoreans, the spiritual and scientific dimensions of number were complementary and could not be separated. Numbers were not merely measures of things but possessed a spiritual dimension. They believed that understanding the relationship between these two aspects could lead the student to God.

Pythagoras was not only the first to call himself a philosopher but also a priest-initiate of a mystery religion influenced heavily by Orphism, which taught that the essence of the gods is defined by number. Numbers, indeed, expressed the essence of all created things. According to the Pythagorean Philolaus: “All things which can be known have numbers, for it is not possible that without number anything can either be conceived or known.” The Pythagoreans were the first to assert that natural phenomena conformed to mathematical principles and so could be understood by means of mathematics. In this sense, they may be considered the first physicists. However, their doctrine gradually became distorted into the proposition that not only does number express the essence of things but also that, ultimately, all things are numbers. Unconvinced by the peculiar emphasis Pythagoreans gave to numbers because he was not privy to the secrets of their teachings, Aristotle said of them: “These thinkers seem to consider that number is the principle both as matter for things and as constituting their attributes and permanent state.”

The Pythagoreans thought that numbers had metaphysical characters, which expressed the nature of the gods. The number one (the Monad) represented the principle of unity — the undifferentiated source of all created things. The One was equated with the god Apollo (a = not; pollon = of many). Pythagoras became associated with Apollo because he was named after Pythios, the name of Apollo at Delphi, the most sacred place of his oracle. The Pythagoreans did not even regard the number 1 as a number because for them it was the ultimate principle underlying all numbers. The number 2 (Dyad) represented duality — the beginning of multiplicity, but not yet the possibility of logos, the principle relating one thing to another. The number 3 (Triad) was called “harmony” because it created a relation or harmonia (“joining together”) between the polar extremes of the undifferentiated Monad and the unlimited differentiation of Dyad. These properties correspond in the Tree of Life to the meaning of Kether, Chokmah and Binah. The number 4 corresponds to Daath (“knowledge”) because impersonal, objective knowledge proceeds from principles only by standing outside the duality of subject and object, thereby transcending their relativity.

Polygonal Numbers
Pythagoras was the first to use geometrical diagrams as models of cosmic wholeness and the celestial order. Numbers themselves were represented by geometrical shapes: triangles, squares, pentagons, etc., and we have seen that tetractys-divided, regular polygons, the pentagram and the Star of David provide natural representations of Godname numbers or parameters of the Tree of Life, CTOL, and superstring theory. Any positive integer N can be represented by the corners of a regular, N-sided polygon. The ancient Greek mathematicians, however, generalised such ‘figurative numbers’ by considering nests of n regular polygons nested inside one another so that they share two adjacent sides. Dots denoting the number 1 are spaced at regular intervals along the edges of the polygons, the edge of each polygon having one more dot than the edge of its smaller predecessor. The total number P^N_n of dots in a set of n nested regular polygons with N sides is called a “polygonal number.” The number 1 is the first polygonal number, i.e., P^N₁ = 1. The second polygonal number, which is simply the number of corners of an N-sided, regular polygon is P^N₂ = N, the third is P^N₃, etc. The diagram below shows the first three pentagonal numbers: P⁵₁ = 1, P⁵₂ = 5 and P⁵₃ = 12. Notice that the second pentagonal number is depicted with two dots along each edge of the outermost pentagon and that the third has three dots. In general, the representation of the nth polygonal number P^N_n has n dots along each of the N sides of the outermost, regular polygon.

It can be shown** that:

P^N_n = ½n[(N–2)n – (N–4)].

Table 1 list for future reference formulae expressing the first ten types of polygonal numbers:

Table 1

Table 2 lists the values of P^N_n for 3≤N≤12 and 1≤n≤13:

Table 2. Values of P^N_n.

Also listed are:

a) the sums s_n of the first ten types of polygonal numbers for given n:

            12
s_n ≡ Σ P^N_n = 5n(11n−7)/2;
           N=3

b) the running sums of s_n:

                 n

Σ_n ≡ Σ  s_r = 5n(n+1)(11n−5)/6;
                r=1

c) the sums b_n of the seven polygonal numbers for given n corresponding to the seven types of regular polygons making up the inner form of the Tree of Life, namely, the triangle (N=3), square (N=4), pentagon (N=5), hexagon (N=6), octagon (N=8), decagon (N=10) & dodecagon (N=12):

b_n ≡ T_n + S_n + P_n + H_n + O_n + D_n + d_n = n(17n−10);

d) the running sums of b_n:

  n

c_n ≡  Σ  b_r = n(n+1)(34n−13)/6;
                           r=1

The difference:

s_n − b_n = h_n + N_n + E_n = ½n(21n−15),

which is the sum of the polygonal numbers corresponding to the 7-, 9- and 11-sided regular polygons not defined by the geometry of the Tree of Life (notice the appearance here of the Godname numbers of Kether (21) and Chokmah (15));

f) the running sums of (s_n − b_n):

                                       n
        Σ_n  − c_n =  Σ  (s_r−b_r) = n(n+1)(7n−4)/2.
                                r=1

“I swear by the discoverer* of the Tetractys,

Which is the spring of all our wisdom,

The perennial fount and root of Nature.”

The author has called the mathematical counterpart of the Pythagorean metaphysical doctrine of the Tetrad the “Tetrad Principle.” This states:

10 = 1 + 2 + 3 + 4.

Another example from bosonic string theory is its prediction that there are 16 higher dimensions beyond 10-dimensional space-time, where 16 = 4². Many more will be encountered later and throughout this website. No claim is made for the converse of this principle, e.g., the fourth (or 16th) member or the first four members of any class of mathematical objects always quantify a parameter governing fundamental processes. The Tetrad Principle is a necessary, but not sufficient, condition for a number to be of significance to the subatomic world. This means that a number which is predicted by a theory as having fundamental significance in the subatomic world and which is consistent with the Tetrad Principle is not necessarily an actual parameter of the physics of this world. The fact that examples to be discussed shortly can be found which show how this principle prescribes parameters of the theories of superstrings and bosonic strings does not indicate in itself that these theories are true. Taken individually, such examples merely demonstrate that they satisfy a necessary criterion for being valid. What, however, turns this criterion into a potent principle is the large degree to which these two theories support it. As the subsection entitled "Godname numbers" reveals, their degree of consistency with the Tetrad Principle so exceeds what chance would lead one to expect that such detailed conformity cannot, plausibly, be dismissed as coincidence. Instead, it is indicative of a powerful principle at work hitherto unrecognised by science that prescribes the mathematics of fundamental phenomena. The Pythagoreans appear to have sensed it by their emphasis on the importance of the number 4 to the study of nature. However, their preoccupation with the Tetrad/tetractys was not taken seriously by scholars because the latter understood it only in a secular sense that encompassed too wide a range of seemingly unrelated phenomena to make the principle plausible.

The tetractys is but one member of an infinite series of tetractyses in which the nth member T_n is generated by replacing each yod in the (n−1)th member by a tetractys, two adjacent yods being substituted by two tetractyses sharing a corner. The Pythagorean Monad — the geometrical point — is the first member T₀ of this series (the 0th-order tetractys), the Pythagorean Triangle (1st-order tetractys) is the second member T₁, the 2nd-order tetractys T₂ is the third, and so on. The Tetrad Principle prescribes properties of higher order tetractyses as well as T₁ because they, too, are representations of spiritually whole systems, albeit more differentiated ones. Because of its importance to our discussion, we illustrate below how the Tetrad Principle prescribes properties of T₂:

1)     number of yods = 85 = 4⁰ + 4¹ + 4² + 4³;

2)     number of yods surrounding its centre = 84 = 1² + 3² + 5² + 7²;

3)     number of corners (○) of 1st-order tetractyses = 15 = 4th Mersenne number = 4th triangular number after 1;

4)     number of sides of 1st-order tetractyses = 30 = 1² + 2² + 3² + 4²;

5)     number of hexagonal yods (•) = 70 = 10×7 = (1+2+3+4)×4th odd integer.

The polygonal numbers in Table 2 will now be examined in the context of the Tree of Life and CTOL, the Godname numbers and superstring group-theoretical numbers.

Tree of Life

1) The fourth triangular number is 10 (=1+2+3+4). This is the number of Sephirothic corners of its 16 triangles, where 16 is the fourth square number:

The fourth pentagonal number:

is the number of Paths in the Tree of Life forming the sides of its triangles. This is an example of how the Pythagorean Tetrad defines structural parameters of the Tree of Life. As shown here, the Tree of Life has 70 yods when constructed from tetractyses, where 70 = 7×10, 7 is the fourth odd integer and 10 (=1+2+3+4) is the fourth triangular number. The 1-tree has 80 yods, where 80 = 8×10 and 8 is the fourth even integer. Constructed from Type A triangles, it has 171 more yods, where 171 = 19×9, 19 is the tenth odd integer and 9 is the fourth odd integer after 1. The number of yods other than SLs = 240

where 4! = 1×2×3×4.

The arithmetic mean of the first n triangular numbers T_n is

t_n = (n+1)(n+2)/6.

Therefore, t₁₀ = 22 and t₁₉ = 70. Assigning the first 10 triangular numbers to the 10 yods of the tetractys and the first 19 triangular numbers to the 19 yods of a Type A triangle (see here), we see that the average of the former is the number of Paths in the Tree of Life and that the average of the latter is the number of Paths in four Trees of Life, as well as the yod population of the Tree of Life.

3) Polygonal numbers possess a counterpart to how 35 and 70 can be expressed in terms of 4-dimensional, tetrahedral numbers:

that is, 35 is the fourth pentagonal number after 1 and 70 = P₇;

4) the number of Trees in CTOL = 91 (see here) 

 i.e., 91 is the sixth hexagonal number after 1. A Type B hexagon contains 91 yods (see here).

5) The number of SLs in CTOL is

                          P₁                              55                            T₁₀
                        P₂  P₃                     55 55                       T₁₀  T₁₀
550 =           P₄  P₅  P₆        =      55 55 55     =        T₁₀  T₁₀  T₁₀
                  P₇  P₈  P₉  P₁₀        55 55 55 55          T₁₀  T₁₀  T₁₀  T₁₀.

where T₁₀ = 55,

                                                                                                       T₁
                                                                                                   T₂    T₃

                                   55 + 55 + 55 + 55            =                T₄    T₅    T₆

                                                                                           T₇    T₈    T₉   T₁₀,

and

                                                  ●                                                  S₁

                                             55   55                                           S₂   S₃

                                         55   55   55                 =                 S₄  S₅  S₆

                                       ●    55   55     ●                             S₇  S₈   S₉  S₁₀.

Also,

Notice that T₄ is the Pythagorean tetractys and that H₇ = 91. The Star of David ‘defines' CTOL in terms of the 4th triangular number T₄, the 5th square number S₅, the 6th pentagonal number P₆, etc (notice how the order of the polygonal number increases monotonically by 1 for all six numbers).

6) The number of SLs in the 49-tree representing the cosmic physical plane is

The order of each polygonal number is the number of corners of the corresponding polygon. The triangle, square, pentagon, hexagon and octagon are the first five regular polygons enfolded in the Tree of Life. They have 26 corners and contain 161 yods corresponding to the 161 SLs of the 26-tree that is the counterpart of 26-d space-time predicted by the quantum mechanics of spinless strings. We see how the number of YAHWEH defines both space-time and the cosmic physical plane.

7) The number of yods surrounding the centres of the seven separate regular polygons generated by the Tree of Life is 

= 1¹ + 2² + 3³ + 4⁴.

Once again, the octagon, whose 49 yods measure the number value of EL ChAI, the Godname of Yesod, represents a structural parameter of the set of polygons.

Godname Numbers

1) The fifth octagonal number, O₅ = 65, is the number of ADONAI, the Godname of Malkuth. This is the fourth octagonal number after 1, illustrating how the Tetrad Principle defines this powerful number. 65 is the sum of the first 10 integers after 1:

This shows how the number 10, the Decad, determines the Godname number of the tenth Sephirah. Using the identity:

                                                                                                                   _n

                                                                                                                  Σ P_r = nT_n,
                                                                                                                  ^r=1

	P4					T1
	P3	P3				T2	T2
P2	P2	P2		= 65 =	T3	T3	T3
P1	P1	P1	P1		T4	T4	T4	T4 .

The sum of the first five nonagonal numbers:

defines the number of the complete Godname ADONAI MELEKH of Malkuth.

2) The remarkable mathematical properties of the Pythagorean tetractys, Tetrad and Decad are illustrated by the equation

which relates the number of ADONAI and the Godname numbers of the first four Sephiroth to the first four sets of ten polygonal numbers.

3) The inverse of the summation in (2) is

where the number at the centre of this 2nd-order tetractys array of integers: 

is the number of Paths in the Tree of Life. 1870 = 187×10, where 187 is the number value of Auphanim, the Order of Angels assigned to Chokmah. The number 26 of its Godname YAHWEH forms the base of this array.

4) The number of the Godname YAHWEH ELOHIM of Tiphareth is

that is, it is the sum of the first four polygonal numbers of the fourth order. It illustrates how the Tetrad defines this number. Notice that T₄ + S₄ = 26, the number of YAHWEH, and that H₄ + P₄ = 50,  which is the number of ELOHIM. The Tetrad also determines the latter because 50 is the sum of the first four triangular numbers and the first four square numbers:

50 = (T₁ + T₂ + T₃ + T₄) + (S₁ + S₂ + S₃ + S₄).

5) Compare (4) with

which relates the tenth triangular, square, pentagonal and hexagonal numbers — the first four types of polygonal numbers — to the number 49 of the Godname EL ChAI of Yesod.

6) The number of SHADDAI EL ChAI, the complete Godname of Yesod, is

This summation of the first six octagonal numbers after 1 relates the octagon with 49 yods prescribed by EL ChAI to the sixth Sephirah of Construction.

7) The eighth, non-trivial hexagonal number is H₉ = 153, which is the number of the Godname ELOHIM SABAOTH of Hod, the eighth Sephirah, counting from Kether.

8) The seventh, non-trivial heptagonal number is h₈ = 148, which is the number of Netzach, the seventh Sephirah, counting from Kether.

9) The sum of the first seven orders of polygonal numbers corresponding to the seven regular polygons of the inner Tree of Life is

 ₇

Σ  (T_n + S_n + P_n + H_n + O_n + D_n + d_n) = 2100.

ⁿ⁼¹

This illustrates the potency of the Godname number 21 of Kether. Notice that the number 26 of YAHWEH forms the base of this tetractys array.

10) The sum of the first seven orders of non-trivial polygonal numbers (i.e., numbers larger than 1) corresponding to the seven regular polygons of the inner Tree of Life is

 ₈

Σ (T_n + S_n + P_n + H_n + O_n + D_n + d_n) = 3108

ⁿ⁼²

This demonstrates the beautiful symmetry inherent in these numbers based upon the Pythagorean Tetrad, which defines the number 7 as the fourth odd integer. 3108 is the number of geometrical elements making up the 91 Trees of Life in CTOL (for proof, see here). This parameter of CTOL is prescribed in a very remarkable way by EL ChAI, the Godname of Yesod with number value 49, because 3108 is the sum of (7×7=49) polygonal numbers. The fact that it is also the sum of the fourth powers of the first four odd integers demonstrates par excellence the Tetrad Principle.

11) The sum of the first (10×10) polygonal numbers is

              _{10     12}

Σ₁₀ =  Σ   Σ  P^N_n = 9625 = 25×385

              ^n=1   N=3

_11  12

Σ  Σ P^N_n = 12750 = sum of 15 integers 15, 25, 35, 45… 155 = 10×85×15,

^n=2 N=3

where

13) The sum of the first (10×5=50) polygonal numbers is

  _{10     7}
 Σ    Σ  P^N_n = 2750 = 55×50,

 ^{n=1   N=3}

where

(the tenth triangular number and the tenth Fibonacci number) is the arithmetic mean of the first 50 polygonal numbers. This illustrates how the parameter 55 of CTOL, which has 550 (=10×55) SLs (see here), is prescribed by the Godname ELOHIM with number value 50. The 55th SL from the bottom of CTOL is the 496th SL from its top, where 496 is the number value of Malkuth and the dimension of both E₈×E₈ and SO(32), the two gauge symmetry groups describing the unified forces between the two types of heterotic superstring (see under "Superstring gauge symmetry group" here). The number 55, therefore, determines the number at the very heart of superstring theory that was shown by physicists Michael Green and Gary Schwarz to be the dimension of a symmetry group that generates interactions between superstrings that are free of quantum anomalies.

14) The inverse of the summation in (13) is the sum of the (5×10=50) polygonal numbers:

 _{5      12}

Σ     Σ  P^N_n = 1250 = 25×50

^{n=1   N=3}

                                                   _{7        9}
                                                  Σ      Σ  P^N_n = 1764 = 49×36.
                                                  ^{n=1    N=3} 

The number 36 of ELOHA, the Godname of Geburah, is the arithmetic mean of the first 49 polygonal numbers, where 49 is the number value of EL ChAI, the Godname of Yesod.

16) The sum of the Godname numbers of the seven Sephiroth of Construction is

                                                                                     ⁸

           539 = 31 + 26 + 76 + 129 + 153 + 49 + 65 =   Σ O_n.

                                                                                    _n=2

Parameters of Superstring & Bosonic String Physics

(1) The dimension of superstring space-time is

10 = T₄,

the number of compactified dimensions of bosonic strings beyond superstring space-time is

16 = S₄,

and the total number of compactified dimensions for bosonic strings is

22 = P₄.

These numbers demonstrates the Pythagorean principle (referred to here as the "Tetrad Principle") of a 4-fold pattern in these numbers characterizing the nature of matter, for they are the fourth triangular, square & pentagonal numbers.

2) The dimension of the superstring symmetry group E₈ is

 

 

i.e., the sum of the first four orders of polygonal numbers corresponding to the three regular polygons (7-, 9- & 11-sided polygons) which are not generated by the Tree of Life is equal to the number of non-zero roots of E₈. Once again, the Pythagorean Tetrad is seen to generate a group-theoretical parameter of this superstring symmetry group. The number of ADONAI expresses the sum of the first four orders of the first ten types of polygonal number.

4)  The dimension of the anomaly-free gauge symmetry groups E₈×E₈ and SO(32) of heterotic superstrings is

 

              _n
   (1/n)  Σ  P_r = T_n.
             ^r=1

Therefore, 496 is the arithmetic mean of the first 31 pentagonal numbers:

                     ₃₁

496 = (1/31) Σ  P_n,

                     ⁿ⁼¹

so that

₃₁

Σ  P_n = 31×496 = 31×(1+2+3+... + 31).

ⁿ⁼¹

 

 

Using the identity:

H_n = 4T_n-1 + n,

    H₁₆ = 4T₁₅ + 16

    = 480 + 16 = 496,

which compares with the fact that the 496 roots of E₈×E_8, one of the two anomaly-free, gauge symmetry groups of superstrings, consist of 480 non-zero roots and 16 zero roots. H₁₆ is the 15th hexagonal number after the trivial hexagonal number: H₁ = 1. This shows how the number 15 of the Godname YAH of Chokmah determines the dimension 496 of the two symmetry groups at the heart of superstring theory.

5) The 26th triangular number is the sum of the sixth polygonal numbers of the first six types:

 

6) The sum of the tenth polygonal numbers of the first seven types is

 ,

where 133 is the dimension of E₇, the rank-7 subgroup of E₈. Notice that

is the sum of 100 (= 1³ + 2³ + 3³ + 4³) odd integers comprising these 10 (=1+2+3+4) square numbers. The sum of the tenth set of the first 10 polygonal numbers is

 

7) This is the 4th-order tetractys T₄:

It has

hexagonal yods surrounding its centre that belong to lower-order tetractyses corresponding to Sephiroth of Construction because each hexagonal yod in the 1st-order tetractys gets replaced in the 2nd-order tetractys by a 1st-order tetractys with seven hexagonal yods, and so on, so that the numbers of such yods in higher-order tetractyses increases according to the geometric series 7¹, 7², 7³, 7⁴, etc. The 5th-order tetractys T₅ has 16800 hexagonal yods surrounding its central set of seven hexagonal yods. The fourth-order tetractys, T₄, therefore expresses through its 2400 hexagonal yods the number of space-time components of the gauge fields corresponding to the 240 non-zero roots of E₈, namely, 240×10 = 2400, whilst T₅, the fourth higher-order tetractys beyond the ordinary tetractys T₁, embodies the number (16800) of coils in the 10 helical whorls of the UPA/subquark superstring paranormally described by C.W. Leadbeater and Annie Besant (see here). In the former case, the Tetrad Principle prescribes a dynamical parameter of superstrings; in the latter, it determines the tetractys representation of their structural parameter.

8) The sum of the first seven orders of the first 10 types of polygonal numbers is 3360:

₇

 ∑ (T_n + S_n + P_n + H_n + h_n + O_n + N_n + D_n + E_n + d_n) = 3360.

 ⁿ⁼¹

This is a primary structural parameter of the subquark superstring that is embodied in sacred geometries (see The holistic pattern) because its 10 whorls make five complete revolutions around its axis of spin, i.e., each of their revolutions on average comprises 16800/5 = 3360 circular turns of their helices. Amazingly, this is the number of yods in the seven enfolded polygons of the inner Tree of Life with 2nd-order tetractyses as their sectors (for proof, see Article 24, ref. 3) :

This extraordinary property is illustrated by the 2nd-order tetractys array shown above with polygonal numbers assigned to its 70 hexagonal yods. The seven orders of each type of polygonal number are assigned to the hexagonal yods corresponding to Sephiroth of Construction. Polygonal numbers of the type corresponding to the seven enfolded polygons of the Tree of Life are assigned to the 7×7 hexagonal yods of the seven tetractyses arranged at the corners and centre of a hexagon. The dodecagonal numbers are assigned to the central tetractys because both the dodecagon as the seventh of the regular polygons of the inner Tree of Life and the central yod of a tetractys correspond to Malkuth, the seventh Sephirah of Construction. Polygonal numbers corresponding to the three types of polygon not generated by the Tree of Life (7-, 9- & 11-cornered polygons) are assigned to hexagons in the tetractyses at the corners of the 2nd-order tetractys. The sum of these (3×7=21) numbers is

₇

∑ (h_n + N_n + E_n) = 1260

ⁿ⁼¹

1260 is the number of Tarshishim, the Order of Angels assigned to Netzach, the seventh Sephirah. It is remarkable that the number 126 is the sum of the values of the four types of combinations of the letters of EHYEH (Hebrew: אהיה; English: AHIH), the Godname of Kether, whose number value is 21:

A = 1, H = 5, I = 10

It is also remarkable that it is the arithmetic mean of the first 26 triangular numbers and therefore prescribed by YAHWEH, the Godname of Chokmah with number value 26:

                      26
126 = (1/26) Σ T_n.
                               ⁿ⁼¹

The sum of the remaining (7×7=49) polygonal numbers in the hexagonal array of seven hexagons is

₇

∑ (T_n + S_n + P_n + H_n + O_n + D_n + d_n) = 2100

ⁿ⁼¹

where

This is an example of the Tetrad Principle expressing a structural parameter of the superstring through the 4×4 square array of the number value of a Godname, in this case, EHYEH.

This polygonal counterpart of the inner Tree of Life demonstrates the beautiful harmony between number and sacred geometry. The number 3360 is the number of circular turns in one revolution of the 10 whorls of the UPA/subquark superstring. The polygonal representation of this structural parameter of the subquark state of a superstring reveals its archetypal character in a remarkable way. How it is embodied in sacred geometries is discussed here [see under heading "3360 = 1680 + 1680 (or 336 = 168 + 168)"]. See also here.

* A reference to Pythagoras.

** Proof: the representation of P^N_n consists of the superposition of (n−1) regular polygons with 2, 3, 4, ...n dots along each of their N edges. The rth polygon has r dots per edge. The total number of dots in (n−1) separate polygons is

                                        _n−1

Σ Nr = ½nN(n−1).

 ^r=1

Of these, there are

2 + 3 + 4 +... + n−1 = ½n(n+1) − 1

dots along one set of edges and

         1 + 2 + 3 +... + n−1 = ½n(n−1)

dots along the set of adjacent edges. The total number of shared dots when the polygons are superimposed is

½n(n+1) − 1 + ½n(n−1) = n²  − 1.

The total number of unshared yods is

½nN(n−1) − (n²−1) = ½n[[(N−2)n − N] + 1.

Superimposed, the (n−1) polygons have n dots along one shared edge and (n−1) dots along the adjacent shared edge. The total number of dots in the representation is

P^N_n = (n−1) + n + ½n[(N−2)n − N] + 1

= ½n[(N−2)n − (N−4)].

 

*** Proof: the n-tree has (12n+7) triangles with (6n+5) corners and (16n+9) sides. Number of corners, sides & triangles in the n-tree = 34n + 21.

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