Phi, the 1, 2, root 5 triangle and the Eight Pointed Star
The Fibonacci Series
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, . . .
Consecutive numbers are added together to progress the series:
0+1=1, 1+1=2, 1+2=3, 2+3=5, 3+5=8, 5+8=13, 8+13=21, . . .
In the Fibonacci Series consecutive numbers create fractions. As the fractions progress along the series they become increasingly closer to a proportion known as The Extreme to Mean Ratio.
The Extreme to Mean Ratio
Any two consecutive numbers in the Fibonacci Series represent a fraction. Each progressive fraction in the series more accurately expresses the Extreme to Mean Ratio. The Extreme to Mean Ratio is expressed as the fraction √5 ± 1 ÷ 2 whose decimal equivalent is irrational; .61803398875 . . . .
If we look at the Fibonacci Fraction 55/89 its decimal equivalent is .6179… (55 ÷ 89 = .6179….) This is a very close approximation of the Extreme to Mean Ratio. If we look at the next fraction in the series, 89/144, we get an even closer decimal equivalent of .61805. As the series grows the fractions more accurately describe the irrational Extreme to Mean Ratio. The series goes on theoretically for ever and can never reach the irrational expression of Phi.
The geometry of the Fibonacci Spiral has a peculiar irregularity in its pattern. The square, representing the number 1 of the sequence, repeats itself. This is the only time in the Sequence that a square of the same size repeats itself. It gives us the form of the double square.
Fibonacci Series:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, . . . .
The 1 repeats itself as a result of the equation 0 + 1. So this anomaly of the repeated 1 is created by the power of 0. The geometric drawings of the repeated 1 (the double square) contains inside itself the irrational answer to the Extreme to Mean Ratio, something that the whole of the Fibonacci series can not achieve.
So what is the power of the replication of the square?
If we think back to the equation 1 + 0 = 1, are the two 1’s the same? In other words is 1 the same as 1 + 0 ? In the Fibonacci Series the 1’s are different. There are two of them. 0, 1, 1, 2, 3, 5, 8, 13, 21 . . . . The catalyst for the repetition seems to be the number zero since 1 + 0 gives us the next number in this additive sequence. The 1’s are arguably the same but they exist in two places in a series where numbers do not repeat. Does square make a quantum leap? Contemplating these squares led me to write a short story A Little Fib. I’m including it here to add a little mystic wonder to the square of the Fibonacci Spiral and as a geometry lesson to explain the fascinating properties of simple geometry.
The design paradigm for my Roman Alphabet is the Eight Pointed Star. Ultimately this star is a more complex expression of the simple square, the double square, and the 1, 2, √5 Triangle. Some will argue that the Eight Pointed Star is formed from four 3, 4, 5 Triangles. By creating the Star in this manner the perimeter square is forgotten. I prefer including the perimeter square. Furthermore, it would be very difficult to draw the Eight Pointed Star using just 3, 4, 5 Triangles. Whereas the Star can be drawn without measure in seconds using the 1, 2, √5 Triangle. You will see how both these triangles are inextricably linked as you follow my tutorials.
The Eight Pointed Star is considered as an elemental tool in the designs of antiquity. It has been written about extensively by Tons Brunes and Malcolm Stewart.The two quotes that follow summarize the importance of the eight pointed star in the shaping of our history.
The symbol he has evolved through the ages was not merely of speculative and geometrical interest: it was employed extensively in practical matters, and from earliest times right up through the centuries to the late Middle Ages it formed the basis of construction for temples, churches and other edifices. It became the background of dimensions in sculpture, and generally filled a tremendously important role in the development of culture. The diagram thus was not a mere object of speculation, but was a practical tool, occupying a place of honor in the Temple. And it was to the eight-pointed star and its manifold facets the Temple brethren turned for the answers to their problems.
I cannot prove what I suspect about the Starcut diagram [the Eight Pointed Star]. I suspect that it was an immensely early graphic device predating Euclid by thousands of years. It is not something that grows from sophisticated conclusions about geometry; it is very simple, though as we have seen its applications are vast. I take it to have been a conceptual foundation for the edifice of explorations, recognitions and practical constructions that, with the distillation of many centuries, emerged as Euclid’s Elements.
With such precedence, the alphabets paradigm of the Eight Pointed Star really raises some serious theological questions.
