Designing for Fashion with the Van der Pol Equation

Year: 2025 Authors: Ralf Jacobs; Loe Feijs

Core claim

The Van der Pol equation can serve as a generative source for fashion artifacts that merge mathematical visualization, historical tribute, and textile fabrication.

Topics

Van der Pol equation, limit cycles, fashion design, textile fabrication

Domains

differential equations, nonlinear oscillations, numerical methods, dynamical systems, fashion, embroidery, laser cutting, wallpaper motifs

Methods

phase plot visualization, Euler’s method, Python and Matlab, Brain Dynamics Toolbox

Media

vacuum tube oscillator, textiles, embroidery, laser engraving

Paper text

The text below is the locally extracted OCR/Markdown version of the paper. Raw PDF files remain local and are not published here.

Bridges 2025 Conference Proceedings

Designing for Fashion with the Van der Pol Equation

Ralf Jacobs ${}^1$ and Loe Feijs ${}^2$

${}^1$ Fashion Tech Farm, Eindhoven; ralfjacobs@gmail.com ${}^2$ TU/e and Laurentius.Lab, Sittard, The Netherlands; l.m.g.feijs@tue.nl

Abstract

This paper celebrates Balthasar Van der Pol’s equation, a significant contribution to mathematics developed in Eindhoven a century ago, by exploring its artistic potential. We delve into the equation’s history, rooted in early 20th-century radio technology, and its mathematical properties to generate visualizations. We translate the oscillations and limit cycles into fashionable outfits featuring embroidery and laser-cut designs inspired by solutions to the equation. The work is a tribute to Van der Pol by merging mathematics, technology, and fashion.

Background

One hundred years ago, Eindhoven was a small town with a growing manufacturing industry. The industry produced cigars, textiles, hats, soap and light bulbs. From light bulbs, it is a small step to vacuum tubes and Philips established a physics laboratory where our hero Van der Pol was given the task in 1922 to develop radio technology [1]. By that time, it was known how to create oscillations with a vacuum tube and a tuned circuit. Van der Pol considered the equation $d^{2}x/d{t}^2+{\omega }^{2}x=0$ , which was widely used to explain oscillators; it has solutions of any amplitude. This was in contrast with the circuits in the lab that gave oscillation of bounded amplitude. Without loss of generality, Van der Pol set ${\omega }^2$ to one. He knew that an extra term $kdx/dt$ would work like damping (or amplification if k < 0 ). He replaced $k$ by $-\mu (1-x^2)$ , which is non-linear because of the $x^2$ ( $\mu$ is a constant). Thus, the famous formula was born [5]:

Today, the laboratory is home to a restaurant named “NatLab”, where the equation is printed on the wall.

Playing with the Equation

The first author is fascinated with visualizing oscillations, often using a synthesizer to find interesting waveforms. In previous artwork he used cyanotype photosensitive paper to laser-project phase plots of Lissajous-like waves (ralfjacobs.net/laser_cyanotypes/).

Figure 1: Photograph of Van der Pol oscillations.

Working in Eindhoven, we got interested in the Van der Pol equation, its history and its potential for art – including fashion. We went back to the roots of the equation, including the historical and mathematical

Jacobs and Feijs

perspectives. Van der Pol had to cope with a messy radio world: ad-hoc trial and error, ignorance about the behavior of radio waves, and absence of standards and regulations (like the Titanic disaster, when not even the SOS code was standard). But he had received the tools of mathematics and systematic experimentation as he had worked with J.J. Thomson and Hendrik Lorentz [1].

We began building vacuum tube oscillators, the raison d’etre of Van der Pol’s equation from 100 years ago. Figure 1 is a photograph of the phase plot of one of our circuits. We deployed a 1930s “EL3” pentode (another NatLab invention, see frank.pocnet.net/sheets/046/e/EL3.pdf). Pentode means “five electrodes”, although the third grid is internally connected. We drew this diagram with the symbols as used in the 1960s by the American Radio Relay League. The circuit produces limit cycles, but without interaction, these are

boring. So, we added a capacitor-resistor pair to the grid circuit (pin 6 in Figure 2) to choke the oscillation - which then slowly starts again. In Van der Pol’s days, a positive feedback circuit was called “regeneration”, quenching and restarting was called “super-regeneration”. The capacitor-resistor trick is known as Flewelling’s circuit.

The oscillation always starts at the middle hot spot, then spirals out. The spurious high-frequency oscillations which can be seen in Figure 1 are interesting details. An interactive circuit will be packaged as artwork for the Bridges exhibition. Besides the circuit, we designed fashionable outfits. To get the phase diagram in a digital form to feed our embroidery machine, we turned to Python and Matlab.

Figure 2: Vacuum tube oscillator circuit.

Computer Solutions

Today, it is easy to generate approximate solutions of the differential equation by computer, using numerical methods, the easiest of which is Euler’s method. Van der Pol had to do it by manual calculations and graphical methods [1]. We introduce an extra variable $y$ for $dx/dt$ . The program of Figure 3 (a) is a translation of the equation for $\mu =1$ , initial values for $x,y$ set to 0.01. When $t$ is incremented in tiny steps $\Delta t$ , $x$ and $y$ spiral outward, then enter a limiting cycle. For lower $\mu$ , spiraling takes longer, and the limit-cycle resembles a circle. Different initial values yield different inner spirals, yet the limit behaviors are the same. We could use a random generator to get a variety of initial values (as the electronic noise does, as shown in Figure 1).

(a)

(b) Figure 3: Python program: (a) code, (b) output.

Designing for Fashion with the Van der Pol Equation

Inspired by the richness of the oscilloscope images of our vacuum tube (Figure 1), we were not satisfied with plots like Figure 2 (b) and turned to the Brain Dynamics Toolbox (bdtoolbox.org) setting $\mu =1.5$ , with random initial values and perturbations. Now the limit cycle itself has variations, which is great for embroidery and laser cutting – it avoids the line going over itself and visually it is more interesting.

Printing and Embroidery

We took a stylized limit cycle for $\mu =1.5$ as a starting point for the motif of Figure 4 (a). The limit cycles overlap, and we added a 3D effect, so they appear interlocked, like chainmail. If we neglect the colors (and the tiny grey 3D edges), it is a wallpaper which has rotation centres of order two ( $180^{\circ }$ ). It has wallpaper group $p2$ (orbifold notation 2222). Taking the three colors into account, it still has $p2$ , yet with larger horizontal translations. The $180^{\circ }$ originates from the Van der Pol equation, which is invariant under the simultaneous substitution $x\mapsto -x$ , $y\mapsto -y$ (but not under $x\mapsto y$ , $y\mapsto -x$ ; i.e. no $90^{\circ }$ symmetry).

We took a more complex phase plot from the Brain Dynamics Toolbox and turned it into embroidery (Figure 4 b). The embroidery work is limited by the hoop size of our embroidery machine, so we also added laser engraving to the jacket. The wallpaper is for the trousers, while the embroidery and laser engraving are for one of the jackets. The completed fashion outfits are shown in Figure 5.

(a)

(b) Figure 4: Textile designs: (a) wallpaper motif, (b) embroidery.

Historic Remarks

The birth of the famous equation was not a sudden event – it was “in the air”. Poincaré had found limit cycles in celestial mechanics and in a singing arc device in 1908 [3]. Van der Pol himself, a prolific writer, made steps towards the equation’s final form in successive papers. In his 1920 paper [4] he still had the ${\omega }^2$ and circuit parameters such as $R,L$ , and $C$ , but he already drew a down-open parabola, the precursor of the term $\mu (1-x^2)$ . With only one parameter $\mu$ , left he brought the matter to its essence [5].

Balthasar Van der Pol had an eye for fashion and was interested in textiles. On all public photos he is extremely well-dressed. He can be seen in the office wearing a black suit, a neck tie or bow tie, and cufflinks. There is a famous NatLab photo of Giles Holst and Van der Pol with a vacuum tube in his hands where he even wears a pocket square [1]. He designed a textile pattern, representing Gaussian prime numbers, which was produced by linen manufacturer Van Dissel [2].

On June 1, 1927, the Dutch queen and princess Juliana visited Eindhoven and for the first time addressed citizens of The Netherlands all over the world through a radio transmitter with a crystal oscillator and powerful vacuum tubes. On the very same day, Van der Pol was knighted for his contributions to radio technology. We are proud to add our small tribute from Fashion tech Farm Eindhoven, one km from NatLab.

Jacobs and Feijs

Figure 5: Ralf and Loe with an oscilloscope and the Van der Pol vacuum tube circuit. Photographer Robin Kühn, stylist Suze Van der Vleuten.

Acknowledgements

We thank Marina Toeters, Sophie Van Kooten, Sion Choi, and Jesse Van der Jagt of Fashion Tech Farm for their great help in realization, and Rong-Hao for reviewing the manuscript. Marthe Bouwens helped with the print design and Remco Mulders scripted the csv to svg conversion for the embroidery.

References

[1] H. Bremmer. “The Scientific Work of Balthasar Van der Pol.” Philips Technical Review, vol. 22, no. 2, 1960, pp. 36-52. Online: https://archive.org/details/philips-technical-review-vol-22 [2] J. Bukowski and S. De Zoete. “Van der Pol’s Tablecloth.” Math Horizons, 26:1 (2018), 14-17. [3] H. Poincaré. “Sur la télégraphie sans fil.” Lumière Electrique, vol. 4, 1908, pp. 259-266, 291-297, 323-327, 355-359, and 387-393. [4] B. Van der Pol. “A Theory of the Amplitude of Free and Forced Triode Vibrations.” Radio Review (London), vol. 1, 1920, pp. 701–710 and 754–762. [5] B. Van der Pol. “On Relaxation-Oscillations.” The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, vol. 2, no. 11, 1926, pp. 978–992.