The Root Rectangles
p.32
All things unfold out of, and are found within Unity.
It now seemed to be the appropriate time and place to describe how the root rectangles unfold out of Unity, and are also found within Unity. Unity at this time represented by the square.
The drawings continue the theme of the relationships of parts to the whole, though at the time of drawing I did not fully understand that this was indeed a major and significant thread running through the whole book.
The diagram is a very beautiful one to draw out. Firstly a square is drawn with sides of 1 unit (the unit can be of any measure: a centimetre, a foot, a mile.) If the side of a square is 1, the diagonal across it is √2. Then taking a point at the corner, the compasses open along the diagonal, and draw an arc to meet the extension of the line at the top of the square. A perpendicular line is dropped, and a rectangle formed. If the diagonal of the square is √2, then the long side of the rectangle must also be √2.
A similar process is followed to find the subsequent root rectangles: root √3, √4, √5…….and so on. Noticing that √4=2 is a double square.
Finding the root rectangles within the square, a similar process is followed. A square is drawn with sides of 1 unit; and then a diagonal, and an arc joining the apposing corners. The crossing points of diagonal and arc deliver the comparable set of root rectangles to the drawing above, only this time, they are 1/√2, /1/√3, etc, as they are found within the square.
p.33
Robert Lawlor writes in his book Sacred Geometry: ‘In this geometric demonstration of the relationship between proportion and progression, we are reminded of the alchemical axiom that everything in creation is formed from a fixed immutable component (proportion) as well as a volatile, mutable component (progression).
Lawlor continues ‘science errs in attempting to attach fixed, absolute laws and definitions to the changing world of appearances. The history of science shows us perpetually discarding or revising one world model after another. Because of the disturbingly unstable quality of scientific knowledge, not only our physicists, but also our philosophers, artists and society as a whole have become relativists. But the unchanging, generative principles remain, and our contemporary rejection of them is taking place only because we have sought for the permanent in the empirical world instead of its true abode, the metaphysical’.
The oscillation between whole numbers and square roots represent the relationship between the knowable and the unknowable: the rational and the irrational(an irrational number is that which runs to an infinite number of non-recurring figures after the decimal point).
This page extends the introduction of the square roots of numbers seen in the previous page, and emphasises their significance and importance in a philosophical context. Many cultures have realised the more mystical aspects of number in the making of their artefacts and sacred buildings all over the world.
From One to the Many. (The side of a square is called its root.) In expansion and diminution, the square and its diagonal describe how One becomes Two, and subsequently the proliferation of Number, through geometric progression ( through the function of v2). Though these relationships would seem to be paradoxical, if the drawing shown on p33 is examined carefully the paradox will be understood and found to be true. Though the squares increase and decrease in size, their root/diagonal relationship remains constant.
Implicit to Explicit: the simple act of drawing the diagonal gives rise to √2, not because the Square has been divided in half, but because Square 2 is implicit, since the diagonal of Square 1 is the root ( the foundation) of Square 2. The half produces the double-through√2.
The Square and √2: The Principle of Generation and Transformation: Growth from the root
p.30
The square, as well as the circle, is also a symbol representing Unity; but it enters the world of measure and orientation: each of the four sides can be measured, and if a mid-point is drawn at the four sides, this defines orientation - north, south, east, west. However, if the side of the square is 1 unit, the diagonal is √2: an irrational number. (an irrational number is that which runs for and unlimited digits after the decimal place).
As Robert Lawlor says, ‘By definition the square is four equal straight lines joined at right angles. But a more important definition is that the square is the fact that any number, when multiplied by itself, is a square…. When we cross a vertical with a horizontal….we see that this crossing generates a square surface: a tangible measurable entity comes into existence as a result of crossing. The principle can be transferred symbolically to the crossing of any contraries such as the crossing of male and female which gives birth to an individual being, or the crossing of warp and weft gives birth to a cloth surface,—So the crossing is an action- principle which the square perfectly represents.’ Thus the square came to represent the Earth, the rational, measurable dimension, and the irrational number √2, the irrational number of spirit and transformation lay within it.
Also 4 is the first number where product and sum are equal. For instance the more laborious 1+1+1+1=4 of addition, rather than 1x4 =4 of multiplication.
At the top of the page, the drawings show the four lines of orientation arising from the circle and square. the second shows the square arising from the double circles; the third drawing gives double squares arising from the double circles and how the diagonal of the smaller square is also 1 unit of the double circles. Lastly, in the fourth diagram, bringing the 3 red points upwards to join together forms the first Platonic solid - the Tetrahedron: another dimension: the third dimension born from two dimensions.
At the bottom of the page, diminishing squares and circles can be made using alternating whole number and irrational numbers. This practice was often used by medieval builders, as it relies on related proportions and not through individually measured up units.
p.31
The understanding of the square and the diagonal continues at the top of this page; and extends to the describing of the first Pythagoras Triangle: the square on the hypotenuse is equal to the sum of the square on the other two sides: 1+1=(√2x√2)=2. Continuing along the top line reaffirms the power of the diagonal for infinite expansion or diminution.
The main drawing brings the power and importance of the diagonal into a deeper, more psychological setting. The Square Root of Two is the Root ( the foundation) of Square Two. Thus, that which breaks the unity of the original square is itself the root / the foundation of the larger square. It can also mean in terms of our daily lives - that which destroys our present worldview is itself the foundation of a larger version. It can happen many times as we grow in understanding; and often, when our worldview seems to have been torn apart, it is helpful to remember that the very event that breaks it is the gateway to the larger vision.