Concluding remarks; history of development of Scale Structure Theory
This web page (extending to the horizontal line below) is reproduced from my book (W. Roberts, 2003, pp.159—161) with some minor editing and reformatting for the web.
Music has long inspired my thinking as an artist. In 1995, Darryl Reanney ’s book, Music of the Mind, inspired my first colour-modulation paintings*. From there, I began to explore the possibility of music-like scales for visual art, but from a geometrical perspective. Could geometric or other music-like scales be discovered that applied in the visual arts, and could there be a grammar arising from a resonance and interconnectedness between such scale structures? If so, could it ultimately lead to a new dynamic visual language : of art-in-motion, of visual music, a new form of visual logic, or a geometric number notation? This book is a small and conscious step on that long journey.
From 1997 to 1998 I pieced together various ideas or ‘parts’ (some known, and others possibly new) according to a harmonic principle that joined them to form logical wholes (which I called ‘scale structures’), and which were therefore, in various ways, resonant or symmetrical and echoed the principles of musical scales.
The scales in music, with their divisions of an octave (and also of time) into discrete intervals, are clearly based on mathematical ratios. The scales themselves are merely the backbone—the skeletal structures and connecting principles that relate and syntactically combine the regularities of timing and key signatures (scales) with the spontaneous asymmetries and irregularities of melody, rhythm, chordal progression and artistic expression. Without scales, key signatures and time signatures, music, as we know it today, would not be possible.
Spurred on by an apparent ‘universalism’ of music, I began to actively seek out similar principles that might serve as ‘scale structures’ in a new visual dynamic art (visual music). Just as there exists a spectrum of sound, so too there exist spectrums of colour, tone, and space. I first looked at the 2-dimensional space: the plane. Could this continuum be quantised (or divided) just as the continuum of sound had been divided into step-like intervals (or scales) within the genre of music?
Finding a compelling connection between numbers of the form p2 and the three regular polygons (squares, equilateral triangles and hexagons) which can tile the plane without gaps was a key step... and the first major scale structure of the plane that I explored in some depth. In other words, so-called square numbers (of the form p2) are far more than merely square in their connection to the geometry of the plane.
This led to the consideration of more irregular polygons that can tessellate the Euclidean plane, and from there to the notion of the relative unit. This concept of relative units of area ‘tuned’ to a geometric context seemed natural and musical in principle.
From here, the whole question of the measurement of areas and, more importantly, units of area, was thrown wide open. Relative units of areamatched to geometric figures resonated with the musical principle of whole-number divisions and ratios of vibrating strings on musical instruments which, in turn, related to musical scales.
The application of scale-structure principles such as the regular division of the plane (as extensively explored by MC Escher ) combined with the concept of relative units of area, ultimately led me to two (possibly new) geometric theorems (which I discovered in 1998 and later called the Eutrigon theorem and Co-eutrigon theorem) and that are, in many respects, as fundamental as the Pythagorean Theorem. Taken together, these three theorems form a powerful scale structure of resonant triangles, and, via their respective parametric forms, triads of integers able to form such triangles. Moreover the new theorems, expressed in terms of ‘relative units’, have significant implications to number theory and notations. That is, such crucial notions as ‘the unit’, ‘numbers’ and especially of ‘irrational numbers’ are called into question by this new scale-structure way of thinking.
There exist some curious riddles descended from Classical times, such as the problem of constructing a square whose area is the same as a given circle’s area using only an unmarked straight edge and compasses, and within a finite number of steps. This has since been proven impossible, and yet a bevy of ‘circle squarers’ has persisted to the present day, bent on finding a way of squaring the circle. From the newer perspective of relative units an irony is revealed: although many mathematicians have mocked the circle-squarers as cranks (and worse), the fact must be faced that the well-worn (and accepted) formula for the area of a circle (π.r2) returns its result in square units. The irrational numbers returned from the application of the formula for the area of a circle result from applying ‘square pegs to round holes’. Is not this also a form of trying to ‘square the circle’?
We have briefly re-examined the so-called square numbers (those of a form p x p, or more simply, p2) from a scale structure point of view, and found that they should never have been called ‘the squares’ since the language has literally locked us in to a predominantly square approach to problems in analytic geometry. These special numbers are every bit as much triangular or hexagonal, and even strongly relate to other less ‘regular’ polygons such as scalene triangles (if seen in terms of relative units).
Underlining its pivotal and foundational role in geometry, Euclid’s Elements placed the equilateral triangle at the very beginning of his classic treatise on the subject. William Dunham (W Dunham, 1994, p. 79) sums up the enormity of the task which faced Euclid, and of the equilateral triangle’s cornerstone-role in geometry:
With a huge body of geometry to deduce from a tiny collection of definitions, postulates, and common notions, where does one start? This is the sort of initial challenge known to freeze mathematicians (and authors) in their tracks. But if, as the Chinese tell us, a journey of a thousand miles begins with a single step, then Euclid’s journey through geometry began with an equilateral triangle. The very first proposition of the Elements was the construction, upon a given line segment, of just such a figure.
Yet this most fundamental and symmetric triangular shape of Euclidean geometry, the equilateral triangle, has been typically and traditionally subjected to ‘the square plug of mensuration’. Surely such a fundamental and symmetric polygon, and moreover, one of only three regular polygons that can tile the plane without gaps ought to have a rational value for its area. But no, at least from the old square perspective, an equilateral triangle of unitary side-length returns an irrational number for its area: √3/4 square units*†. But as we have seen, ... this irrational number resulted from an ‘irrational approach’ rather than from any inherent irrationality of the geometry per se. When we instead applied relative units-of-area matched to the geometry (the equitriangular unit, or etu) we found its area is simply 1 unit, that is, 1 etu. What could be simpler or more rational? This led us in turn to new rational meanings and harmonious geometric interpretations of the Cosine Rule for the resonant 60° and 120° angles.
The irresistible beauty of relativity
The equilateral triangle is one of the most fundamental and regular of all polygons, and yet an equilateral triangle of unitary side-length returns an irrational number (if expressed in traditional square units of area)!
By contrast, if its area is expressed in terms of relative units (etu), matched to the geometry of the triangle rather than the square, the area is not only a rational number, but in fact even simpler: one whole unit (an etu). This single example is enough to call into question, and indeed undermine, the very notion of what constitutes an ‘irrational number’*‡.
The new theorems also reopen the chapter of trigonometry and of the ‘trigonometric identities’. The Eutrigon and Co-eutrigon theorems include the area of the triangle in question and not merely the relative proportions of the sides and areas of the perimetrically–constructed squares as in the Pythagorean Theorem. This is significant because now new links can be formed among not only the sides and angles of triangles, but also their areas (as expressed in terms of the new relative units).
Towards new number notations
Moreover, it is likely that the triangular and hexagonal properties of the p2 numbers (traditionally known simply as ‘the squares’) will play a crucial role in the development of new and more powerful number notations.
'Negativity' and 'positivity' of number reflected in the new Eutrigon and Co-eutrigon Theorems
The numberline with its linear spread of numbers to the left and right of zero seems to make supreme sense of the concepts of positivity and negativity as properties of numbers. But what about numbers as shapes on the Euclidean plane? Fascinatingly, there is a covert but relativistic parallel: the algebraic form of the Eutrigon Theorem is the same as the Co-eutrigon theorem (except for a change in sign from positive to negative or vice versa)!
You can see this from a comparison of the two equations for each (given in etu)
Eutrigon theorem ab = a2 + b2 – c2
Co-eutrigon Theorem ab = c2 – a2 – b2
The areas ab are the same in absolute terms (for the same leg lengths a and b) but the shapes of eutrigons and co-eutrigons differ, and so here the shapes of their areas connects (or relates) to the sign (positive or negative) of the lower-dimensional number-line model.
| back to top |
next > |