Symmetry and Harmonics

(This is an introductory text I wrote in 1998 as preparation for a collective project we did with teachers and students at the “Interfaculty Image and Sound”, now called “Artscience” )

Symmetry and Harmonics
Joost Rekveld, Rotterdam, 24-11-1998


figure 1: mirror cabinet of Z.Traber, 1675

harmonic images

Pure intervals are pure for a physical reason: if the ratio of frequencies of two soundwaves can be described in small numbers, the minima and maxima of these waves overlap nicely. This makes a chord sound at rest. If the ratio of frequencies is more complex, these overlaps form a more complex pattern and the tones do not blend properly or a difference tone appears. Similar phenomena occur if images are generated using vibrations. Several physicists from the 19th century did experiments in this direction.
Ernst Chladni wrote his book ‘Entdeckungen im Reich des Klanges’ in 1787. It is the first general treatise on acoustics. He illustrated it with diagrams of the vibrations of thin metal plates (fig. 2). For these experiments he covered the plates with a thin layer of sand and made them vibrate by striking them with a bow. The vibrations displaced the sand toward the locations on the plate where the waves in the metal formed ‘knots’. Chladni analized these sandpatterns, classified them according to shape and tried to understand the relationship with their corresponding pitch. He concluded that a vibrating plate generates a set of tones (fundamental and harmonics) that corresponds with the harmonic series produced by a vibrating string.


figure 2: sound figures by Chladni, 1787

In 1967 Hans Jenny published his book ‘Cymatics, The Structure and Dynamics of waves and vibrations’, in which he describes the results of his continuation of the research of Chladni. He elaborated on the principle Chladni used by using modern technology and by extending his research to many different kinds of materials and vibrations. He did a lot of work on patterns in vibrating fluids and he developed his ‘Tonoscope’. This was a device through which the human voice could form patterns in sand, without any form of amplification (fig. 3). According to Jenny his research uncovered aspects of an underlying structure of reality. Because the cosmos is based on vibrations, hands-on research and sensual confrontation with these vibrations is the only way to develop concepts that can shed light onto this ‘ultimate reality’. Also Jenny speculated about links between the forms of vowels in the hebrew and sanskrit alphabets and the patterns generated in his ‘Tonoscope’ by the corresponding sounds.


figure 3:the vowel ‘A’ in sand, Jenny, 1967

Charles Wheatstone was one of the first fysiologists to study our senses by measuring their properties. He is best known for his invention of the stereoscope. One of his more obscure inventions is the kaleidophone. He developed it in 1827, triggered by the success of the kaleidoscope which David Brewster had invented shortly before. The kaleidophone is a curious assembly of rods with shiny metal objects at their extremities, mounted on a wooden board (fig. 4). These rods can be set into vibration by hitting them or bowing them like a string. If this happens near to a small lightsource (like the flame of a candle), the reflections of this source in the moving metal object generate many different kinds of patterns (fig. 5). Also it was possible to generate more complex patterns by connecting several reflecting objects to the same rod.


figure 4: the kaleidophone, 1827


figure 5: kaleidophone patterns

The patterns generated by the kaleidophone were described in more detail by the mathematician and physicist Jules Lissajous, later in the nineteenth century. He studied the curves created by the combination of two perpendicular vibrations. In order to visualize these curves he designed an instrument consisting of two little mirrors mounted on tuning forks (fig. 6). If the tuning forks sound in a ‘pure’ interval, the resulting image is static and symmetric (fig. 7). If the interval is not ‘pure’, the result is a chaos of lines. Smaller interferences between the two frequencies translate into subtle movements of the generated patterns.


figure 6: the set-up by Lissajous, 1857


figure 7: Lissajous figures

The research into these patterns was continued by Friedrich Schoenemann, who designed a system to be able to draw Lissajous-figures. It consisted of two wooden plates that could swing independently from eachother (fig. 8 ). On one piece of wood was a piece of paper, on the other a pencil. By setting both pieces of wood in motion, very complex patterns could be reached (fig. 9).


figure 8: the set-up by Schoenemann, 1875

Lissajous-patterns have found many applications in the twentieth century. Toys as the ‘Sprirograph’ are based on the same principle. Templates for the complex patterns of lines on banknotes were made in a similar fashion. The abstract filmmakers Mary Ellen Bute and Norman McLaren made a number of films in which they used oscilloscopes to generate moving images. The principle of Lissajous (including the little mirrors) also found an application in simple laser-effects.


figure 9: ‘Rhythmogramm 3782/247’ by Heinrich Heidersberger

arabic ornaments and ‘traces regulateurs’

An interesting parallel with the above sound figures emerges in the analysis Javier Sanchez Gonzales made of Islamic geometrical patterns. According to him these are based on an underlying grid that is determined by two factors:
– an equal subdivision of a circle, giving a number of directions
– a ratio of distances. For optimum symmetry these distances are connected with the angles generated by the subdivision of the circle.
The underlying grid is constructed by drawing lines in the chosen ratio of distances in every direction given by the circle. Subdivions which are frequently used are those in 8, 10, 16, 20, 24 and 32, because these result in pleasant distance ratio’s, like the golden section in the case of 10 and 20. Also these subdivisions can be constructed with a compass.


figure 10: a grid with a division of the circle in 8

The grid in figure 10 is for instance based on a subdivision of the circle in 8. The ratio of distances is 1 : V2. The result of this is that in the whole plane a kind of ‘echo’s’ appear of the original center. These echo’s are similar to the ‘knots’ in the sound figures of Chladni.


figure 11: turkish ornament from Konya

Things become even more complex if modulations are made within the fundamental grid to other grids based on other subdivisions of the circle. An example of this is figure 11, from the Aksaray Sultan Han in Konya, Turkey. This relief contains subdivisions in 12, 16 and 10. The complex patterns of ‘echo’s’ generated in this way are mathematically exactly the same as the patterns that occur when several tones sound together. Interesting as well is that this type of modulation around a ‘local’ centre is very similar to the type of modulation used in the classical music of the Middle East.
These ratio’s based on subdivisions of the circle were common not only in oriental cultures. In the twenties the german architect Moesel published a study about the use of proportions in the architecture of the Middle Ages and of western antiquity. In his analysis he used a system very similar to the above (fig. 12). According to him this kind of systems were used by the classical and gothic architects he studied.


figure 12: analysis of groundplan by Moesel

katoptrics, kaleidoscopes and theatrical mirrors

In 1815 David Brewster invented the kaleidoscope as a side-effect of experiments he was doing with mirrors in a laboratory. He was fascinated by their mutual reflections and after some experimenting he came up with the basis of the kaleidoscope. In its most simple form the kaleidoscope is a triangular shape consisting of three oblong mirrors (fig. 13). The objects seen through it are seemingly reflected an infinite number of times.


figure 13: a basic kaleidoscope by Brewster, 1819

The kaleidoscope was one of the first of a whole series of ‘philosophical toys’ that appeared in the course of the nineteenth century. These were made in rather large numbers for the upper middle class. Most of these toys were based on peculiarities of the human eye or in some other way generated images about which it was interesting to speculate. ‘If we reflect further on the nature of the designs thus composed, and on the methods which must be employed in their composition, the Kaleidoscope will assume the character of the highest class of machinery, which improves at the same time that it abridges the exertions of individuals. There are few machines, indeed, which rise higher above the operations of human skill. It will create in an hour, what a thousands artists could not invent in the course of a year; and while it works with such unexampled rapidity, it works also with a corresponding beauty and precision (David Brewster).’


figure 14: a katoptric device by Kircher, 1646

The symmetrical patterns visible through the kaleidoscope are related to the above sound figures and ornaments. They have the same type of symmetry around a central point. In the actual kaleidoscope this symmetry is always based on triangles. Precursors of Brewster, however, used different arrangements of mirrors which generated different forms of symmetry. These precursors worked predominantly in the seventeenth century. From this period a number of descriptions survive of ‘katoptric devices’ that were meant to be used as scientific intruments and as philosophical toys for the nobility. A beautiful demonstration of reflection in four directions is the picture by Z.Traber from Vienna (1675)(fig. 1). Many of these katoptric devices are described in the ‘Ars magna Lucis et Umbrae’ by Athanasius Kircher from 1646. Amongst others he describes a system with two hinged mirrors. Because of these hinges the angle between the mirrors can be varied, and with it the number of reflections (fig. 14). If a sculpture of dragon’s head is placed in this device, it appears to have 4, 5 or 6 heads if the angle is 90, 72 or 60 degrees respectively. This is a hardware version of the mathematical principles discussed above.


figure 15: theatrical mirror by Hero of Alexandria (2nd century b.c.)

These arrangements of mirrors ultimately date back to the manuscript ‘Katoptric Magic or the Miraculous Representation of Objects by Mirrors’ by Hero of Alexandria (2nd century b.c.) In this book he describes many amazing magical machines, which had much more than a mere technical significance. He also descibes theatrical mirrors (fig. 15). These are sets of mirrors arranged in a circle and which form the impression of a big mass of people because of the multiple reflections. His designs were later descibed by Leonardo da Vinci (fig. 16) and Gianbatista della Porta (fig. 17). Theatrical mirrors were still used on a limited scale in the nineteenth century (fig. 18).


figure 16: theatrical mirror by DaVinci, 1488


figure 17: theatrical mirror by DellaPorta, 1589


H.W.Franke, G.Jaeger, “Apparative Kunst”, DuMont, Koeln, 1973
H.Robin, “The Scientific Image”, H.N.Abrams, New York, 1992
N.Wade, “Brewster and Wheatstone on Vision”, Academic Press, London, 1983
H.Jenny, “Cymatics”, Basilius Press, 1967
J.S.Gonzalez, “Puertra”, (website, not online anymore..)
M.C.Ghyka, “Essai sur le Rhythme”, Gallimard, Paris, 1938
J.Baltrusaitis, “Le Miroir”, du Seuil, Paris, 1978
L.Mannoni, “Le Grand Art de la Lumiere et de l’Ombre”, Nathan, Paris, 1995


figure 18: nineteenth century theatrical mirror

copyright 1998, Joost Rekveld.