12 Coherence Proofs

2In a structural system or any pattern the question arises; what is the pattern made of? What are the relationships between different elements of the pattern?

3 In a checkerboard pattern such as that shown in figure 9.100 all distances between neighboring intersections are the same. And the distance between two intersections of any line is simply a multiple of this base distance. This base distance then naturally becomes the unit for building the pattern.

4 In our pattern created by the star with five-fold symmetry, the situation is different. There are many different lengths between intersections. If the growth patterns follow simple rules such as those followed in forming the two-dimensional pattern of figure 9.102, then all distances between intersections can be expressed as simple sums of components whose lengths are equal to powers of the divine proportion times some constant. This is also true in three dimensions - the A and B lines of the 31-zone star forming a growth similar to our 2-dimensional growth intersect each other at points where the distance between any two intersections on an A line equals

r1 r2 rn s1AT + s2AT + ... snAT

(12.1)

6 and the distance between two intersections on a B line equals a polynomial

qBT t1 + qBT t2 + ... q BT tn 1 2 n

(12.2)

8 The building blocks for our system are then a series of lengths related by the divine proportion. An A series, a B series and a C series—each of slightly different lengths.¹

12.1 Two Dimensions

12.1.1 Definition 1

11We call any particular polynomial of the form

s kTr1 + s kT r2 + ... s kTrn = f (kT ) 1 2 n i

(12.3)

13 s_i and r_i are integers.

14 We call the class of all such polynomials F(kT) then;

fi(kT) + fj(kT ) ∈ F (kT )

(12.4)

fi(kT) − fj(kT ) ∈ F (kT )

(12.5)

(T n)f(kT ) ∈ F (kT) i

(12.6)

18 We are interested in the distances between intersections along the lines of certain sets of patterns.

12.1.2 Definition 2

20A pattern is a number of extended lines which point in five different directions in one plane with at least one line pointing in each direction.

12.1.3 Definition 3

22The lines are labeled;

i^lⁿ
n	= 1,2,3,4,5
i	= 1,2,…

23 n names the direction - i names the particular line pointing in that direction. The angles between lines are;

π-2π- 3π- 4π- 5, 5 , 5 , 5

12.1.4 Definition 4

26An intersection is named by any two of the lines which intersect there, such as

jln × klm

(12.7)

12.1.5 Definition 5

29If on a line all intervals between intersections are equal to different f_i(kT), then the intersections ‘‘fit’’ each other.

30 An intersection fits a line if it fits all intersections on that line.

31 A line g^{l^p} fits another line f^{l^q} if the intersection g^{l^p} × f^{l^q} fits all the intersections of f^{l^q}.

32 A pattern fits if all the intersections on all the lines fit.

33 Lemma 12.1.1. If on a line lⁿ an intersection I_n fits an intersection I_m then I_n fits all I_o where I_o fits I_m.

35 Lemma 12.1.2. Any i^{l^m}, j^lⁿ, h^{l^p}, m≠n, n≠p, p form a triangle similar to T₁ or T₂

12.1.6 Definition 6

39These triangles are called the Golden triangles because their sides are in the divine proportion.

40 Lemma 12.1.3. If the two vertices of a Golden Triangle fit each other then all vertices fit each other.

42 Lemma 12.1.4. If a line j^{l^m} fits a line h^lⁿ and if all lⁿ fit some line i^{l^m} and if i^{l^m} fits all lⁿ without the line j^{l^m}, then, all intersections j^{l^m} ×lⁿ fit j^{l^m} ×h^lⁿ and the line j^{l^m} fits all lⁿ.

45 Lemma 12.1.5. If j^{l^m} fits h^lⁿ then all l^p, p≠m, n×j^{l^m} fit jlm×hln and if h^lⁿ fits all l^p without j^{l^m} them j^{l^m} fits all l^p.

47 Theorem 12.1.6. If a pattern fits and a line j^{l^m} is added and j^{l^m} fits some line j^lⁿ,n≠m, then the new pattern including j^{l^m} fits.

48 A pattern fits if all the intersections on all the lines fit. (Definition 5) To prove the new pattern fits we must prove all the new intersections fit the lines they are on.

1.: There are no new intersections on the lines k^{l^m},k≠j, therefore all the intersections on these lines fit.
2.: All the intersections j^{l^m} ×lⁿ fit j^{l^m} ×h^{l^m} and the line j^{l^m} ×h^{l^m} and the line j^{l^m} fits all lⁿ. (Lemma 4) Therefore all the intersections of the lines lⁿ fit.
3.: All the intersections l^p,p≠m,n×j^{l^m} fit j^{l^m}×h^lⁿ. (Lemma 5) Therefore, the intersections of line j^{l^m} fit j^{l^m} × h^lⁿ. Therefore, the intersections fit on all the lines l^m (step 1) and j^{l^m} fits all l^p,p≠m,n (Lemma 5) which means that all the intersections on all the lines l^p,p≠m,n fit and our theorem is thus proved.

12.2 Three Dimensions

12.2.1 Definition 1

51In three dimensions we have a pattern made up of the diameter lines through vertices and face midpoints of the icosahedron.

52 The lines through the vertices are the A lines.

53 The lines through the face midpoints lines are B lines.

54 There can be other lines parallel to the original 16 lines. They are given the name of the line they are parallel to.

55 We name a line

i^{A^n¹}
n	= 1,2,3,4,5,6
i	= 1,2,…

j^{B^m¹}
m	= 1,2,3,4,5,6,7,8,9,10
j	= 1,2,…

57 or generally

h^{X_i^p¹}
p	= 1,2…10
h	= 1,2,…
X	= A,B

12.2.2 Definition 2

60All lines of a 3D pattern are connected directly or through other intersections to the original pattern of 16 lines.

12.2.3 Definition 3

62Intersections on A lines ‘‘fit’’ if all the intervals between them can be expressed as polynomials

r1 r2 rn s1AT + s2AT + ... snAT = fi(AT )

(12.8)

64 s_i,r_i = integers and A = an interval of length.

12.2.4 Definition 4

66Intersections on B lines ‘‘fit’’ if all intervals between them can be expressed as polynomials.

s1BT r1 + s2BT r2 + ... snBT rn = fi(BT )

(12.9)

68 s_i,r_i = integers and B = an interval of length.

12.2.5 Definition 5

cos(π) B = ----6--A cos( π-) 10

(12.10)

12.2.6 Definition 6

72A 3D pattern ‘‘fits’’ if all the intersections on all the lines fit.

12.2.7 Definition 7

74An R section is a plane containing two A lines and two B lines. The angle between the two A lines is 𝜃; the angle between the two B lines is ϕ; the angle between an A and a B line is

π-−-𝜃-−-ϕ π −-𝜃-−-ϕ 2 or 2

(12.11)

12.2.8 Definition 8

77A T plane is a plane perpendicular to an A line. A T plane is named by this A line

An T

79 Lemma 12.2.1. There are five R sections perpendicular to each T plane. Each A line is contained in five different R sections. Each B line is contained in three different R sections.

81 Lemma 12.2.2. The projection of the interval A along any A line onto a T plane

= A sin(𝜃)

(12.12)

12.2.9 Definition 9

k = sin(𝜃) or = 0.A

(12.13)

84Unsure: What does 0.A resolve to? (if the interval is on the A line perpendicular to the T plane).

86 Lemma 12.2.3. The projection of the interval B along any B line onto a T plane

π − 𝜃 + ϕ π − 𝜃 − ϕ = B sin (---------) = k or = B sin(--------= T−1k) 2 2

(12.14)

90 Lemma 12.2.4. The projection onto any T plane of any interval along any line k^{X_i^r¹} between any two intersections which fit in the 3D pattern is a polynomial

r r r s1kT 1 + s2kT 2 + ...+ snkT n = fi(kT )

(12.15)

s_i,r_i	= integers
k = Asin(𝜃)	= B sin()

93 Lemma 12.2.5. The angles between the lines projected on a T plane are

π 2π 3π 4π -,---,---,--- 5 5 5 5

(12.16)

97 Lemma 12.2.6. 3D Patterns which fit project onto all T planes 2-dimensional patterns which fit.

99 Lemma 12.2.7. All intervals between intersections of lines of the 3D pattern appear as intervals between intersections in the projections on some T planes.

100 For an intersection

Xnl Xml i k × i j

(12.17)

101not to appear as an intersection in the pattern projected on a T plane, the intersecting lines must lie in one of the five perpendicular R sections.

102 At most, two different lines intersect i^{X_k^{n^l}}. The two intersections

nl pl iXk or gXt

103 lie. Therefore, there is at least one T plane in which both 3D intersections project as 2-dimensional intersections.

104

105 Lemma 12.2.8. If some pairs of intersections

nl Xml nl Xpl iX k × h j and iX k × g f

(12.18)

106did not fit in our 3D pattern, that is, we cannot find any polynomial f₁(X_kT) that equals the interval between them, then also for some T plane we cannot find a polynomial f_i(kT) which equals the interval between their projected intersections.

107

108 Lemma 12.2.9. Any misfits between intersections of the 3D pattern project as misfits between intersections on some T plane. Therefore, a 3D pattern which projects on all T planes as a 2D Pattern which fits is a 3D pattern which fits.

109

110 Theorem 12.2.10. If a 3D pattern fits and a new line is added k^{X_i^{n^l}} which intersects and fits some other line h^{X_j^{n^l}} then the new 3D pattern including k^{X_i^{n^l}} fits.

111

112 Proof. In each T plane the projection of the line k^{X_i^{n^l}} fits the projection of the line h^{X_j^{m^l}}. In each T plane the projection of the 3D pattern without k^{X_i^{n^l}} fits. Therefore, the projection is with the line k^{X_i^{n^l}} fits on each T plane and the theorem is proved. □

113 The coherence proof demonstrates that if one builds a structure using the A and B lines of the 31 zone star (the C lines may be used only within the forms defined by the A and B lines) and always follows the rule that new parts are added at intersections of existing parts or at points along existing parts which can be reached by subdividing a large part into component small parts, then no matter how far or intricately one builds, two extensions of two entirely different limbs of the same structure can always be locked back together in a perfect fit with a combination of our simple parts.

114 We have shown that the intervals between intersections are all equal to certain polynomials in T. Because

T n = Tn−1 + Tn−2

(12.19)

115all the terms of any such polynomial can descend by subdivisions into a polynomial with only two terms-

f(AT ) = rAT n + sAT n−1 i

(12.20)

116There are many interesting side-lights to these investigations. One of which is that it is impossible to divide any one of our building blocks ATⁿ into equal pieces.

117 There are much shorter proofs of the coherence of this system, but the short proofs don’t lead one through so many characteristics of the structure.

118