biorhythm - cymatics (3/3)

Third Dimension

In order to apply the Chladni patterns in spatial, architectural settings, we need a third dimension. For this we can simply map one of the parameters to the y-axis. Here I added y to the m-parameter, so that the pattern evolves up and forms a Chladni-volume:

#declare chladni = function(x, y, z, l, m, n) {
    cos(n * x * pi / l) * cos(m * z * pi / l) -
    cos(m * x * pi / l) * cos(n * z * pi / l)
}

#declare height = 5;
#declare bottom = 0;
#declare n = 5; 
#declare m = 4;
#declare l = 2;

isosurface {
    function {      
        abs(chladni(x, y, z, l, m+y, n)) -0.15
    }
    contained_by {
        box { <-l,bottom,-l>, <l,bottom+height,l> }     
    }       
    accuracy 0.001
    max_gradient 32    
}

Animating a thin slice along the y-axis illustrates clearly how the pattern unfolds in time and space:

Then I defined three zeroes of the function pouring out into the floors of the building.

Some visual experiments. The first one shows the complete amplitudes of the standing waves instead only the zeroes, with a second threshold inside in red:



Finally some excerpts from the early biorhythm-brochure from Nik based on the Chladni pattern:

biorhythm - cymatics (2/3)

Visualization of the Chladni figures

A simple formula for the amplitude of a standing wave for a point (x,y) on a quadratic surface with a support in the center exists :

cos(n * x * pi / l) * cos(m * z * pi / l) - 
cos(m * x * pi / l) * cos(n * z * pi / l)

with l is the length of the plate and m and n are integer factors for the various modes.
Locations where this formula evaluates (close) to zero are the areas where the sand settles and forms the Chladni pattern.

More information and background at the amazing website of Paul Bourke.

POVRay

For the visualization of the Chladni figures I used POVRay, a free 3D visualization software that achieves an amazing rendering quality by using raytracing with radiosity and photon-mapping and is capable of volume rendering.
Scenes in POVRay are  defined in a textual scene description language, similar to a programming language. POVRay is not limited to rendering polygons and surface meshes, but can also directly render various forms of mathematical formulas, like parametric and implicit surfaces.
Implicit surfaces or isosurfaces are defined by a equation f(x,y,z)=0 which means their surface is defined and rendered in space where their function evaluates to zero.
With isosurfaces we can directly render the Chladni figure in POVRay. However the node-lines are currently still infinitely thin, because only the zeroes of the function are rendered. In reality, the sand collects also on locations, where the vibrations are very soft. but not zero. Therefore I thickened the isosurface a bit by taking it's absolute value and subtracting a small constant, that controls the thickness of the nodal lines.
Here the first simple POVRay scene:

camera{ 
  orthographic
  up y 
  right x 
} 

background { rgb 1 }

#declare l = 0.4;                            
#declare n = 5; 
#declare m = 1;

isosurface {
  function {                   
    abs(cos(n*pi*x/l) * cos(m*pi*y/l) - cos(m*pi*x/l) * cos(n*pi*y/l)) -0.1
  }
  contained_by { 
    box { <-l,-l,0>, <l,l,0.1> }    
  } 
}

Now the parameters can be varied and animated. In the visualization we are not limited to integer values but can also use continous values. In the next example n=5 and m is animated between 1 and 5:

In the next post I'll show how to extend the figure into the third dimension...

biorhythm - cymatics (1/3)

Around 2005 I created a couple of visualizations and algorithmic design-patterns for my friend Nik Schweiger. Nik just founded his design label 3deluxe/biorhythm in Berlin and was working on a concept for the conversion of the Café Moskau into a multifunctional venue and recreation center.
His design subdivided the building into areas of contemplation, communication und conclusion.
The formal aesthetics, the ornamentation and the layout of the spatial organization should be deduced from the phases of the chladni figures.

Ernst Chladni invented a way to visualize the modes of vibration of a rigid plate. By drawing a violin bow over the edge of a sand-covered metal plate he made the plate resonate. At certain frequencies of resonance, the sand forms into spontaneous and symmetric patterns.



Today this is more commonly achieved with loudspeakers and a electronic frequency generators. The effect is not limited to sand but works similar with fluids or grain.

When the plate vibrates with certain frequencies, it begins to resonate - the waves running through the plate and their reflections from the edges interfere with each other - they amplify or soften each other and therefore create standing waves. Various areas of the plate vibrate in opposite directions and at the boundary lines between them, where there are no or only few vibrations, the sand particles settle and collect.


In the next post I will describe a simple method to visualize the various chladni figures in software...