12 Coherence Proofs

1In 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?

2 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.

3 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

s_{1} A T^{r_{1}} + s_{2} A T^{r_{2}} + \dots s_{n} A T^{r_{n}}

(12.1)

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

q_{1} B T^{t_{1}} + q_{2} B T^{t_{2}} + \dots q_{n} B T^{t_{n}}

(12.2)

5 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 $C$ serieseach of slightly different lengths.¹

12.1 Two Dimensions

12.1.1 Definition 1

6We call any particular polynomial of the form

s_{1} k T^{r_{1}} + s_{2} k T^{r_{2}} + \dots s_{n} k T^{r_{n}} = f_{i} (𝑘𝑇)

(12.3)

7 $s_{i}$ and $r_{i}$ are integers.

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

f_{i} (𝑘𝑇) + f_{j} (𝑘𝑇) \in F (𝑘𝑇)

(12.4)

f_{i} (𝑘𝑇) - f_{j} (𝑘𝑇) \in F (𝑘𝑇)

(12.5)

(T^{n}) f_{i} (𝑘𝑇) \in F (𝑘𝑇)

(12.6)

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

12.1.2 Definition 2

10A 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

12The lines are labeled;

\begin{array}{l} i^{l^{n}} \\ n & = 1, 2, 3, 4, 5 \\ i & = 1, 2, \dots \end{array}

13 $n$ names the direction - $i$ names the particular line pointing in that direction. The angles between lines are;

\frac{π}{5}, \frac{2 π}{5}, \frac{3 π}{5}, \frac{4 π}{5}

12.1.4 Definition 4

14An intersection is named by any two of the lines which intersect there, such as

j^{l^{n}} \times k^{l^{m}}

(12.7)

12.1.5 Definition 5

15If on a line all intervals between intersections are equal to different $f_{i} (𝑘𝑇)$ , then the intersections “fit” each other.

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

17 A line $g^{l^{p}}$ fits another line $f^{l^{q}}$ if the intersection $g^{l^{p}} \times f^{l^{q}}$ fits all the intersections of $f^{l^{q}}$ .

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

19Lemma 12.1.1.If on a line $l^{n}$ an intersection $I_{n}$ fits an intersection $I_{m}$ then $I_{n}$ fits all $I_{o}$ where $I_{o}$ fits $I_{m}$ .

20Lemma 12.1.2.Any $i^{l^{m}}$ , $j^{l^{n}}$ , $h^{l^{p}}$ , $m \neq n$ , $n \neq p$ , $p$ form a triangle similar to $T_{1}$ or $T_{2}$

12.1.6 Definition 6

22These triangles are called the Golden triangles because their sides are in the divine proportion.

23Lemma 12.1.3.If the two vertices of a Golden Triangle fit each other then all vertices fit each other.

24Lemma 12.1.4.If a line $j^{l^{m}}$ fits a line $h^{l^{n}}$ and if all $l^{n}$ fit some line $i^{l^{m}}$ and if $i^{l^{m}}$ fits all $l^{n}$ without the line $j^{l^{m}}$ , then, all intersections $j^{l^{m}} \times l^{n}$ fit $j^{l^{m}} \times h^{l^{n}}$ and the line $j^{l^{m}}$ fits all $l^{n}$ .

26Lemma 12.1.5.If $j^{l^{m}}$ fits $h^{l^{n}}$ then all $l^{p}$ , $p \neq m$ , $n \times j^{l^{m}}$ fit $𝑗𝑙𝑚 \times h𝑙𝑛$ and if $h^{l^{n}}$ fits all $l^{p}$ without $j^{l^{m}}$ them $j^{l^{m}}$ fits all $l^{p}$ .

27Theorem 12.1.6.If a pattern fits and a line $j^{l^{m}}$ is added and $j^{l^{m}}$ fits some line $j^{l^{n}}, n \neq m$ , then the new pattern including $j^{l^{m}}$ fits.

28A 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 $k^{l^{m}}, k \neq j$ , therefore all the intersections on these lines fit.
2.: All the intersections $j^{l^{m}} \times l^{n}$ fit $j^{l^{m}} \times h^{l^{m}}$ and the line $j^{l^{m}} \times h^{l^{m}}$ and the line $j^{l^{m}}$ fits all $l^{n}$ . (Lemma 4) Therefore all the intersections of the lines $l^{n}$ fit.
3.: All the intersections $l^{p, p \neq m, n} \times j^{l^{m}}$ fit $j^{l^{m}} \times h^{l^{n}}$ . (Lemma 5) Therefore, the intersections of line $j^{l^{m}}$ fit $j^{l^{m}} \times h^{l^{n}}$ . Therefore, the intersections fit on all the lines $l^{m}$ (step 1) and $j^{l^{m}}$ fits all $l^{p, p \neq m, n}$ (Lemma 5) which means that all the intersections on all the lines $l^{p, p \neq m, n}$ fit and our theorem is thus proved.

12.2 Three Dimensions

12.2.1 Definition 1

29In three dimensions we have a pattern made up of the diameter lines through vertices and face midpoints of the icosahedron.

30 The lines through the vertices are the $A$ lines.

31 The lines through the face midpoints lines are $B$ lines.

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

33 We name a line

\begin{array}{l} i^{A^{n^{1}}} \\ n & = 1, 2, 3, 4, 5, 6 \\ i & = 1, 2, \dots \end{array}

\begin{array}{l} j^{B^{m^{1}}} \\ m & = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \\ j & = 1, 2, \dots \end{array}

34or generally

\begin{array}{l} h^{X_{i}^{p^{1}}} \\ p & = 1, 2 \dots 10 \\ h & = 1, 2, \dots \\ X & = A, B \end{array}

12.2.2 Definition 2

35All lines of a 3D pattern are connected directly or through other intersections to the original pattern of 16 lines.

12.2.3 Definition 3

36Intersections on $A$ lines “fit” if all the intervals between them can be expressed as polynomials

s_{1} A T^{r_{1}} + s_{2} A T^{r_{2}} + \dots s_{n} A T^{r_{n}} = f_{i} (𝐴𝑇)

(12.8)

37 $s_{i}, r_{i} =$ integers and $A =$ an interval of length.

12.2.4 Definition 4

38Intersections on $B$ lines “fit” if all intervals between them can be expressed as polynomials.

s_{1} B T^{r_{1}} + s_{2} B T^{r_{2}} + \dots s_{n} B T^{r_{n}} = f_{i} (𝐵𝑇)

(12.9)

39 $s_{i}, r_{i} =$ integers and $B =$ an interval of length.

12.2.5 Definition 5

B = \frac{\cos (\frac{π}{6})}{\cos (\frac{π}{10})} A

(12.10)

12.2.6 Definition 6

40A 3D pattern “fits” if all the intersections on all the lines fit.

12.2.7 Definition 7

41An $R$ 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

\frac{π - 𝜃 - ϕ}{2} or \frac{π - 𝜃 - ϕ}{2}

(12.11)

12.2.8 Definition 8

42A $T$ plane is a plane perpendicular to an $A$ line. A $T$ plane is named by this $A$ line

T^{A^{n}}

43Lemma 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.

44Lemma 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)

45Unsure: What does 0.A resolve to? (if the interval is on the $A$ line perpendicular to the $T$ plane).

46Lemma 12.2.3.The projection of the interval $B$ along any $B$ line onto a $T$ plane

= B \sin (\frac{π - 𝜃 + ϕ}{2}) = k 𝑜𝑟 = B \sin (\frac{π - 𝜃 - ϕ}{2} = T^{- 1_{k}})

(12.14)

48Lemma 12.2.4.The projection onto any $T$ plane of any interval along any line $k^{X_{i}^{r^{1}}}$ between any two intersections which fit in the 3D pattern is a polynomial

s_{1} k T^{r_{1}} + s_{2} k T^{r_{2}} + \dots + s_{n} k T^{r_{n}} = f_{i} (𝑘𝑇)

(12.15)

\begin{array}{l} s_{i}, r_{i} & = integers \\ k = A \sin (𝜃) & = B \sin (\frac{π - 𝜃 + ϕ}{2}) \end{array}

49Lemma 12.2.5.The angles between the lines projected on a $T$ plane are

\frac{π}{5}, \frac{2 π}{5}, \frac{3 π}{5}, \frac{4 π}{5}

(12.16)

51Lemma 12.2.6.3D Patterns which fit project onto all $T$ planes 2-dimensional patterns which fit.

52Lemma 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.

53For an intersection

i^{X_{k}^{n^{l}}} \times i^{X_{j}^{m^{l}}}

(12.17)

54not 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.

55At most, two different lines intersect $i^{X_{k}^{n^{l}}}$ . The two intersections

i^{X_{k}^{n^{l}}} or g^{X_{t}^{p^{l}}}

56lie. Therefore, there is at least one $T$ plane in which both 3D intersections project as 2-dimensional intersections.

57Lemma 12.2.8.If some pairs of intersections

i^{X_{k}^{n^{l}}} \times h^{X_{j}^{m^{l}}} and i^{X_{k}^{n^{l}}} \times g^{X_{f}^{p^{l}}}

(12.18)

58did not fit in our 3D pattern, that is, we cannot find any polynomial $f_{1} (X_{k} T)$ 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.

59Lemma 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.

60Theorem 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.

61Proof.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. □

62 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.

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

T^{n} = T^{n - 1} + T^{n - 2}

(12.19)

64all the terms of any such polynomial can descend by subdivisions into a polynomial with only two terms-

f_{i} (𝐴𝑇) = 𝑟𝐴 T^{n} + 𝑠𝐴 T^{n - 1}

(12.20)

65There are many interesting side-lights to these investigations. One of which is that it is impossible to divide any one of our building blocks $A T^{n}$ into equal pieces.

66 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.