Award for Excellence

Chladni Towers
Martin Skrodzki, Ulrich Reitebuch, Konrad PolthierMartin Skrodzki, Ulrich Reitebuch, Konrad Polthier “Chladni Towers” 2017, Material: RGD720, Size: 101.6W x 101.6D x 103.4H (mm)In 1802, Ernst Florens Friedrich Chladni published his book “Acoustics” containing visualizations of vibration modes in 2D. We
generalize the well-known concept to 3D. Via computer simulations, we solve a corresponding physical system and thereby obtain anisosurface in a cubical model. This isosurface visualizes the elimination of sound waves and their reflections, showing points in threespace where different frequencies vanish. Using a 3D printer, these vanishing points can be physically realized. Application are given
in instrument design or architecture of music halls. With transparent filament, constructors or architects can print problematic regions
directly into their prototypes for awareness.

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Award for Excellence

Keywords: Vibration Modes, Acoustics, Chladni Figures

Figure 1: Left: Portrait of Ernst Florens Friedrich Chladni by H. Adlard, 19th Century. Center: Illustration of the violin bow experiment, takenfrom [3]. Right: Table of Chladnis figures from Chladni’s book “Acoustics”, 1802.

Margaret Watts-Hughes, Henry Holbrook, or Hans Jenny

1 Introduction

have further developed the Chladni experiment [1]. Despite

In 1802, Ernst Florens Friedrich Chladni (November 30th,1756 – April 3rd, 1827, see portrait in Figure 1) published
his book "Acoustics". The book describes amongst otherthings an experiment by which different modes of vibration
can be visualized. Namely, sand is distributed over a thinmetal plate. A violin bow is then struck alongside the plate,causing it to oscillate, see Figure 1 for an illustration. Chladni
found that the sand grains form different patterns
corresponding to the varying vibration modes of the plate.
His book contains a table of patterns he was able to create in
his experiments, see Figure 1.

Figure 2: Application of Chladni Figures in building musical instruments.

Since its first description, several other scientists like15

Award for Excellence

their elegance and artistic value, Chladni figures have

Applications of our method can be found for example inapplications in the construction of musical instruments asArchitecture, where the printed model gives an impression ofillustrated in Figure 2.

the acoustics of a given room.

We generalize Chladni’s concept to the third dimension. In
terms of a corresponding physical experiment, this consists

2 Physical and Mathematical Backgroundof distributing light particles in some viscose fluid. For ex-

For simplicity, we generalize Chladni’s patterns from theample, speakers are then used to stir the fluid and the parti-square-shaped plate to a cubical model. In the following, ourcles form stable patterns, just as in the two-dimensional case.

cube will be assumed to occupy space [-1,1]³. Starting from

However, we do not build a physical experiment, but

the physical formulation of a damped oscillation of a string,

simulate the outcome of it, to print it via a 3D-printer.

we combine the solutions to the wave equation to obtain a16

Award for Excellence

formulation of a one-dimensional Chladni pattern as the zero

Given the first rough visualizations, more elaborate pictures

set of an expression on the string. From this formulation, welike the title image are created with the PovRay Raytracer

can obtain a three-dimensional pattern as the zero-set of thesoftware [6]. Finally, from these, a model is chosen for 3D

following expression:printing.

A · sin(μ · π · x) · sin(ν · π · y) · sin(w · π · z)
+B · sin(μ · π · x) · sin(ν · π · y) · sin(w · π · z)

Table 1: Exact parameter values of the “ChladniTowers” Model+C · sin(μ · π · x) · sin(ν · π · y) · sin(w · π · z)
+D · sin(μ · π · x) · sin(ν · π · y) · sin(w · π · z)

Objects
Integral Parameters
Real Parameters+E · sin(μ · π · x) · sin(ν · π · y) · sin(w · π · z)
+F · sin(μ · π · x) · sin(ν · π · y) · sin(w · π · z).
Note that the formulation permits two different types ofCaption
u=1, v=2, w=4
A=0.5, B=0.0, C=0.5,
D=8.0, E=0.5, F=8.0

figures: Those with sine (as shown) and those with cosine
formulations. These correspond to different boundary

Referencesconditions on the wave equation. For further details on the[1] Findeiß. F., Die Entdeckung der Chladnischen Klangfig-

mathematical and physical background, as well as otheruren. Ursprünge und Weiterentwicklung im historischenvisualizations, see [2].

Kontext (in German). Leuphana Universität Lüneburg,

The particular model generated for the Asian Digital Model-Germany, 2015.

ing Contest and shown as title image has been named[2] Skrodzki, M., Reitebuch, U., and Polthier, K., Chladni“Chladni Towers”. Its exact parameters are given in Table 1Figures Revisited: A Peek Into The Third Dimension.

and it corresponds to Dirichlet boundary conditions on theProceedings of Bridges 2016: Mathematics, Music, Art,wave equation.

Architecture, Education, Culture. pp. 481-484, 2016.
[3] Stone, W.H., Elementary Lessons on Sound, Macmillan

3 Software & System& Company, 1879.

Our approach consists of plotting the zero level set of the[4] Lorensen, W.E. and Harvey, E.C., Marching cubes: Aexpression given above in a cubical bounding box. Thishigh resolution 3D surface construction algorithm. ACM

corresponds to visualizing the isosurface given by theSiggraph Computer Graphics, Vol. 21, No. 4, pp. 163-expression. To do so, we utilize the marching cubes

196, 1987.

algorithm [4]. The algorithm has been implemented in the[5] Polthier, K. et al, JavaView, www.javaview.de

JavaView Framework [5]. Within this framework, we[6] Persistence of Vision Pty. Ltd. Raytracer (version 3.7),

visualize the zero set for different prescribed values of

http://www.povray.org

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