A Unified Theory of Proportion
Through many cultures, star polygons were used as sacred
symbols with the star of David and the Sri Yantra Hindu patterns shown
in Figure 1a and 1b as two examples.
The fact that Venus traverses a five-pointed star over an eight-year cycle
in the heavens as seen from the Earth, shown in Figure
1c, was known to ancient civilizations. Also the designs of ancient
sacred geometry use a small vocabulary of proportions such as Ö
2, Ö 3, the golden mean t
= (1+Ö 5)/2 and the silver mean
q
= 1 + Ö 2. I will show that all of these
constants can be related to the edge lengths of star polygons and that
they are ultimately related to a sequence of numbers called silver means
the first of which is the golden mean. These silver means will also be
shown to be generalizations of the imaginary number
A regular star polygon is denoted by the symbol {
Only connected stars are considered to be star polygons.
For example, {6/2} actually corresponds to the connected 3-gon {3/1}. As
a result of the fact that
The edges of a star 3 ® 6 ® 12 ¬ 4 ¬ 2. (1) The edges of the triangle reappear as diagonals of the
hexagon and the 12-gon. Also the edges of the square reappear within the
12-gon. The factor 2 corresponds to the diameter {12/6} in Figure
4 and can be thought of as a polygon with two edges, referred to as
a f (1) + f (2) + f (3) + f (4) + f (6) + f (12) = 12. (2) These are all the star polygons related to the 12-gon. In addition to the single vertex and the digon, five of these have their edges oriented clockwise while five mirror images have retrograde edges. It can be shown that, in general, where the summation is taken over the integers that divide
evenly into Star polygons are connected to number theory in many ways. One striking example is due to H.S.M. Coxeter in which the star polygon gives an elegant proof of Wilson’s Theorem.
Proof: For any ( p–1 or (p–1)!
+ 1 = N._{p}And, as a result, (
Consider Pascal’s triangle.
Starting from the left in Table 1 the
diagonals: 111…, 123…,136…,
Fibonacci-Pascal
TriangleTable 2. ( where ( Letting Letting Letting In general letting
Nx
+ _{k}x1 , _{k}-and whose ratio of successive terms is
When the polynomials in Table 2 have
alternating signs they are denoted by
Nx
– _{k}x1 , _{k}-and whose ratio of successive terms is
Fibonacci and Lucas sequences are intimately connected.
The standard Fibonacci sequence, { L}, is 1
3 4 7 11 18... where _{n}L = _{n}F1 + _{n}-F1.
Adamson has discovered another variant of Pascal’s triangle related to
the Lucas sequence. In fact this Lucas-Pascal Triangle or _{n}+LPT demonstrates
that silver mean constants and sequences are part of an interrelated whole.
Along with the FPT these tables are carriers of all of the significant
properties of the silver means.
To construct the
As before each diagonal becomes a column of the
Beginning with 2 and
for example Setting
We have discovered a simple relationship between the roots
of both the Fibonacci and Lucas polynomials with alternating signs and
the diagonals of regular polygons when the For odd k>1
and edge d_{1} of the n-gon of radius 1 unit where
d = 2sin _{k}kp/n
for k = 1,2,…, (n–1)/2
(4)where the labeling of the
This gives the familiar result that the ratio of the diagonal to the edge of a regular pentagon is the golden mean t .
The numbers have additive properties much as the golden mean and this will be discussed in the next section. For even n-gons
of radius 1 unit where
d = 2sin_{k}kp/n
for k = 1,2,…, (n–2)/2.
(5)For polygons with even
The results for several polygons are summarized in Table
5. The diagonals are normalized to an edge value of 1 unit by dividing
by
n-gonsWe find the curious property that both the sum and product
of the squares of the diagonals of an Not only are the diagonals of regular polygons determined
by Equations 4 and 5, but the areas
From this equation the square is found to have area 2
units while the 12-gon has area 3 units. It can also be determined that
if Notice that the key numbers in the systems of proportions based on various polygons present themselves in Table 5: t–pentagonal system; q and Ö 2 –octagonal; Ö 3, 1+ Ö 3, and 2+Ö 3 –dodecahedral; r and s –heptagonal, and these are pictured in Figure 6.
Since the unique diagonals of an
We can state this result as a theorem:
By the same reasoning as for the 10-gon, the factor tree
of Expression 1 can be used to factor
corresponding to the factoring by the hexagon polynomial
corresponding to factoring by the triangle
or, Finally
The diagonal (edge) of the triangle comes from In what follows the symbol normalized to a unit edge
rather
Similar to t and q
, the diagonals of each of these systems of
h £k.where the diagonals have been normalized to polygons with
edges of
These formulas are applied to the pentagon and the heptagon.
For the pentagon,
The proportional system based on the heptagon is particularly
interesting [Steinbach 1997], [Ogawa
1990]. For the heptagon,
What is astounding is that not only are the products of the diagonals expressible as sums but so are the quotients. Table 6 illustrates the quotient table for the heptagon.
As a result of
The heptagonal system is particularly rich in algebraic
and geometric relationships. The additive properties of r+ s
= rs
(Compare this with t +t 1/r + 1/s
= 1 (Compare this with 1/t + 1/t r s r /s = r– 1 s /r = s– 1 (7) 1/s = s– r 1/r = 1 + r – s The algebraic properties of each system of proportions
are manifested within the segments of the r = 1/r
+ 1/rs + 1/s s = 1/s
+ r /s
Thus we see at the level of geometry that the graphic designer encounters the same rich set of relationships as does the mathematician at the level of symbols and algebra. The following pair of intertwining geometric s
-sequences and corresponding Fibonacci-like integer series exhibit these
additive properties:
The integer series is generated as follows: -
Determine the first five terms
*xyzuv*beginning with 111 -
Let
*y*+*z*=*u*and*u*+*x*=*v*, i.e., 1+1=2 and 2+1=3 to obtain 11123 -
Repeat step 2 beginning with the
*zuv*, i.e., from 123, 2+3=5 and 5+1=6 to obtain 12356 - Continue
s = 1s + 0r + 0 s s s s s … Notice that the first coefficient in the equation for
s
while the second coefficient is the sum of the first two coefficients and
the last coefficient is the same as the first of the previous equation,
e.g., in the equation for s ^{n}^{4}: 6=3+2+1,
5=3+2, and 3=3.
A geometric analogy to the golden mean can be seen by considering the pair of rectangles of proportions r: 1 and s : 1 in Figure 8. By removing a square from each, we are left in both cases with rectangles of proportion r: s although oriented differently. (a)
(b)
In general the equation
from which it follows that we can formally set,
an infinite process. For example, if
Although this infinite compound fraction has no mathematical
meaning, the infinite process can be defined to be the imaginary numbers
± We now come to a set of self-referential statements related
to the
If
These are continued fraction representations of the silver
mean constants of types 1 and 2.
With the aid of Pascal’s triangle, the golden mean and
Fibonacci sequences were generalized to a family of silver means. The Lucas
sequence was then generalized with the aid of a close variant of the Pascal’s
triangle. These generalized golden means and generalized
Kappraff, J., Kappraff, J. , Ogawa, T., Generalization of the golden
ratio: any regular polygon contains self-similarity, in Steinbach, P., Golden fields: A
case for the heptagon,
Steinbach, P., Sections Beyond
Golden, in |