380 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 41, NO. 2, FEBRUARY 2013

Generalized Leonardo da Vinci Rule for theDischarges, Sliding on Electrolyte Surfaces

Alexander E. Dubinov, Igor L. L’vov, Sergey A. Sadovoy, Leonid A. Senilov, and Dmitry V. Vyalykh

Abstract—Photo images of branching discharges and sliding onelectrolyte surfaces are presented. Based on the analysis of theseimages, the diametrical index for the generalized Leonardo daVinci rule is calculated. It turns out that the value of the diametricindex is close to 0.7. We attempt to explain such a low index value.

Index Terms—Branching surface discharge electrolyte, diamet-rical index of Leonardo da Vinci, photo image.

I. INTRODUCTION

S TUDYING the structure of trees, the great naturalistLeonardo da Vinci articulated a rule in one of his note-

books [1], according to which the sum of squared thicknessesof all the branches of a tree at any height is equal to to thesquared thickness of its trunk, i.e.,

D2 =

N∑

j=1

d2j . (1)

What led Leonardo to this rule is illustrated on his geniussketch Fig. 1. Then, Leonardo mentions that rule (1) is applica-ble to any multibranch flow-carrying system [1], such as a riverwith its tributaries or vascular and bronchopulmonary systemsof humans. Rule (1) stems from the law, which states that thetransverse area of channels remains constant to guarantee theconstancy of incompressible flux carried through the system.As to its physical sense, the rule is similar to the first Kirchhofflaw in the circuit theory.

The power α = 2 in (1) is known as Leonardo’s index ora diametric index [3]. Recently, it has become clear that, forcertain objects, having fractal geometry, the power index in (1)can be different (α �= 2). The structure of these objects satisfiesthe generalized Leonardo rule, i.e.,

Dα =

N∑

j=1

dαj . (2)

For the aforementioned vascular system, for instance, α ≈2.7, for the bronchopulmonary system α ≈ 3 [3] and for thebranch and root systems of trees 1.8 < α < 2.6 [4], [5]. In the

Manuscript received August 6, 2012; revised October 31, 2012; acceptedDecember 12, 2012. Date of publication January 11, 2013; date of currentversion February 6, 2013.

The authors are with the Russian Federal Nuclear Center, All-RussianResearch Institute of Experimental Physics, Sarov 607188, Russia (e-mail:dubinov-ae@yandex.ru).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TPS.2012.2234765

Fig. 1. Sketch made by Leonardo da Vinci, illustrating rule (1) and the methodfor analyzing the images (from [2]).

book [3], Mandelbrot wrote that the diametrical index α oftencoincides with the fractal dimension Δ of an object. Althoughsome specific systems behave differently, and the inequalityα < Δ is observed. Above all, the higher α is, the higher isthe throughput capacity of the flux-carrying system.

Lightnings and streamer discharges, having a multibranchstructure, can be also attributed to flux-carrying systems. Thefractal dimension of streamer discharges are extensively studiedby many scientists [6]–[15]. Their works conclude that, forthe fractal dimension, which is calculated from the dischargebranched skeleton and neglecting the branch finite thickness,and for surface discharges, the inequality 1.3 < Δ < 1.9 isusually true, whereas the inequality 2.0 < Δ < 2.5, as a rule,takes place for volumetric discharges.

At the same time, the mentioned authors never cite thepapers, in which Leonardo’s index α for the multibranch dis-charges is measured and/or calculated, taking the thickness ofbranches into consideration. These papers let us estimate thecapacity of the discharge channel. The aim of this paper is tofind the values of index α for flat streamer discharges on thesurface of liquid electrolytes.

II. TAKING PHOTO IMAGES OF THE DISCHARGE

The experiments are performed on the setup similar to thesetup described in [7]. Our setup is schematically depictedin Fig. 2.

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DUBINOV et al.: LEONARDO DA VINCI RULE FOR THE DISCHARGES ON SURFACES 381

Fig. 2. Schema of the experimental setup. 1: grounded ring electrode; 2: pinhigh-voltage electrode; 3: dielectric vessel; 4: discharger.

Fig. 3. Oscillogram of the discharge current.

The basic element of the setup is a dielectric vessel of176-mm diameter. The high-voltage pin electrode is fixed alongthe axis of the chamber. The grounded electrode ring is locatedclose to the side surfaces of the vessel. The vessel is filledwith an electrolytic solution (e.g., water solutions of NaCl,CuSO4, KMnO4, and NaHCO3), so that the electrode is dipped40 mm deep into the solution. It turns out that the high-voltage electrode stays 1–3 mm above the liquid level (themeans for precise adjustment are used). The electrodes areconnected to capacity storage with 0.94 mF of capacity viaa noncontrollable discharger with the rated operating voltageof 4.5 kV. The capacity storage is connected to the chargingsource.

After the capacitors are fully charged and the voltage reachesits operating value, a pulse of streamer discharge with abranched structure appears on the surface of the solution. Fig. 3shows a typical oscillogram of the discharge current.

Photo images of the discharge were taken by a Sony DSC-H50 camera installed in the standby mode above in the shadedenvironment. A lot of pictures of the discharge were obtainedat positive and negative polarity of the voltage of the high-voltage electrode. The best pictures were taken at the followingparameters: distance between the lens and the pin electrode is100 mm; focal length is 16 mm; resolution is 72 points per inch;dimensions of the picture is 3456 × 2592 points; exposure timeis the discharge time within a 5-s open shutter; and color depthis 24.

Figs. 4 and 5 present the examples of the discharge obtainedfor a 2% solution of NaHCO3. Using these pictures, the width

Fig. 4. Photo image of the discharge at the negative voltage polarity of thehigh-voltage electrode.

Fig. 5. Photo image of the discharge at the positive voltage polarity of thehigh-voltage electrode.

Fig. 6. Photo images of the discharge at the negative voltage polarity of thehigh-voltage electrode. (a) At almost a vertical view. (b) At an angular view.

of the discharge channels at any cross section can be calculated.Unfortunately, the images do not let us exactly measure how thechannel-width changes in a vertical direction, when we movealong the channel. However, we have made few attempts totake pictures angularly. Look at vertical and angular picturesin Fig. 6. It is obvious that streamers have the same thicknesseson the pictures. Hence, we may assume that cross sections ofstreamers are circular with good accuracy.

382 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 41, NO. 2, FEBRUARY 2013

Fig. 7. S(r) diagrams at different α values (line 1 corresponds to α = 0.2;line 2 corresponds to α = 0.7; line 3 corresponds to α = 1; line 4 correspondsto α = 1.5; line 5 corresponds to α = 2; the blue line is the closest to thehorizontal one). (a) Negative polarity voltage for the pin electrode. (b) Positivepolarity voltage for the pin electrode.

III. PROCESSING THE PHOTO IMAGES AND CALCULATING

LEONARDO DA VINCI’S DIAMETRIC INDICES

Our approach to analyzing the photo images correspondsto Fig. 1. At first, linear transverse dimensions dj for all thevisible plasma channels are measured for each value of radiusr, counted from the axis of the vessel. Then, the sum

S(r) =

N∑

j=1

dαj (3)

is calculated with a value α selected from interval 0, . . . ,2. Thevalue of α at which S(r) diagram is least dependent on r is thetarget value, which satisfying the generalized rule of Leonardoda Vinci (2).

Fig. 6(a) and (b) shows S(r) graphs at several α values andcorrespond to the discharges on the surface of a 2% solution ofNaHCO3. It can be seen, that at high values of α, the diagramstend to grow and at low values of α, they tend to decrease. Ithas been found that S(r) is least dependent on r at α ≈ 0.7.This value of α is the same for all the discharges on the entiresurfaces listed above the electrolyte solutions, both at positiveand negative voltage polarity of the pin electrode.

Fig. 7(a) and (b) presents the results for two images. Similarmeasurements for other images showed identical results. Errorbars are mediated by slightly diffused boundaries of streamerson enlarged photo images. The total number of the obtainedimages for this discharge is several dozens, and this allowsreliably speaking about uniformity of the obtained dischargesat one and the same polarity even for different electrolyte solu-tions. For calculations, we selected the best and characteristicimages.

IV. DISCUSSION

The benchmark of this paper is the following. If α = 2,then it is guaranteed the constancy of an incompressible flux.If α < 2, then there are a flux leakage or/and the flux iscompressed.

Two important questions arise after the diametric index ofLeonardo da Vinci α = 0.7 is calculated.

1) What is the reason why the discharges have such a lowvalue of α?

2) What follows from the fact that α = 0.7?

Here, we briefly present our considerations and try to explainthe phenomenon.

First, the contraction of the total area inevitably causes theincreased local plasma velocities and pressures in channels.

Second, the growth of the plasma velocities (i.e., plasmaacceleration) is accompanied by the quasi-stationary electricfields occurring along the channels.

Third, the aforementioned processes should take place alongwith the plasma radiation shift into the short-wave (blue) areaof spectrum (it can be noticed in Fig. 4) and along withX-ray radiation. It should be added that the X-ray radiation instreamer multichannel high-voltage discharges was recorded,for instance, in [16].

Those three conclusions may be explained by the decreasingof the sum of all the thicknesses of the branches and bycontinuity equation for plasma.

Other types of discharges such as lightnings, streamer mi-crowave discharges may be tested on the generalized rule ofLeonardo da Vinci.

REFERENCES

[1] J. P. Richter, The Notebooks of Leonardo da Vinci (1452−1519).New York: Dover, 1970, compiled and edited from original manuscripts.

[2] R. Aratsu (1998, Dec.). Leonardo was wise: Trees conserve cross-sectional are despite vessel structure. J. Young Investig. [Online].1(1), p. 1. Available: http://www.jyi.org/volumes/volume1/issue1/articles/aratsu.html

[3] B. B. Mandelbrot, The Fractal Geometry of Nature. New York: Freeman,1983.

[4] C. Eloy, “Leonardo’s rule, self-similarity, and wind-induced stressesin trees,” Phys. Rev. Lett., vol. 107, no. 25, pp. 258101-1–258101-5,Dec. 2011.

[5] A. M. Oppelt, W. Kurth, and D. L. Godbold, “Topology, scaling relationsand Leonardo’s rule in root systems from African tree species,” TreePhysiol., vol. 21, no. 2/3, pp. 117–128, Feb. 2001.

[6] V. K. Balkhanov, Y. B. Bashkuev, V. I. Kozlov, and V. A. Mullayarov,“Spatial characteristics of radiation from lightning discharges,” Techn.Phys., vol. 54, no. 1, pp. 151–154, Jan. 2009.

[7] V. P. Belosheev, “Self-consistent development and fractal structure ofleader discharges along a water surface,” Techn. Phys., vol. 44, no. 4,pp. 381–386, Apr. 1999.

[8] T. Ficker, “Electrostatic discharges and multifractal analysis of theirLichtenberg figures,” J. Phys. D, Appl. Phys., vol. 32, no. 3, pp. 219–226,Feb. 1999.

[9] M. Murat, “2-D dielectric breakdown between parallel lines,” inFractals in Physics, L. Pietronero and E. Tosatti, Eds. Amsterdam,The Netherlands: Elsevier, 1986, p. 169.

[10] L. Niemeyer, L. Pietronero, and H. J. Wiesmann, “Fractal dimension ofdielectric breakdown,” Phys. Rev. Lett., vol. 52, no. 12, pp. 1033–1036,Mar. 1984.

[11] J. Saòudo, J. B. Gómez, F. Castaòo, and A. F. Pacheco, “Fractal dimensionof lightning discharge,” Nonl. Process. Geophys., vol. 2, no. 2, pp. 101–106, 1995.

[12] S. Satpathy, “Dielectric breakdown in three dimensions,” in Fractals inPhysics, L. Pietronero and E. Tosatti, Eds. Amsterdam, The Netherlands:Elsevier, 1986, p. 173.

[13] H. Takayasu, “Pattern formation of dendritic fractals in fracture and elec-tric breakdown,” in Fractals in Physics, L. Pietronero and E. Tosatti, Eds.Amsterdam, The Netherlands: Elsevier, 1986, p. 181.

[14] A. A. Tren’kin, “Fractal spatial structure of lightning discharge and itsrelation to the structures of high-voltage discharges of other types,” Techn.Phys. Lett., vol. 36, no. 4, pp. 299–301, Apr. 2010.

[15] H. J. Weismann and L. Pietronero, “Properties of Laplacian fractals fordielectric breakdown in 2and 3 dimensions,” in Fractals in Physics,L. Pietronero and E. Tosatti, Eds. Amsterdam, The Netherlands:Elsevier, 1986, p. 151.

[16] A. G. Rep’ev, P. B. Repin, and V. S. Pokrovskii, “Microstructure of thecurrent channel of an atmospheric-pressure diffuse discharge in a rod-plane air gap,” Techn. Phys., vol. 52, p. 52, 2007.

DUBINOV et al.: LEONARDO DA VINCI RULE FOR THE DISCHARGES ON SURFACES 383

Alexander E. Dubinov was born in Arzamas-16,Russia, in 1958. He received the M.S. degree (withhonors) from Moscow Engineering Physics Institute(MEPhI), Moscow, Russia, in 1988 and the Ph.D.and Doctor degrees in physics and mathematics fromthe Russian Federal Nuclear Center—All-RussianScientific and Research Institute of ExperimentalPhysics (RFNC-VNIIEF), Sarov, Russia, in 1997 and2004, respectively.

Since 1984, he has been with RFNC-VNIIEF,where he has been the Deputy Director of the Sci-

entific and Technical Center of High-Energy Density Physics and DirectedRadiation Fluxes. He is a Professor of the Chair of “Experimental Physics”with the Sarov Institute of Physics and Technology, National Research NuclearUniversity “MEPhI,” Sarov. He is the author of three books, more than 150articles, and 80 inventions. His scientific interests are plasma physics, high-power microwave electronics, physics of nonlinear waves, and gas dischargephysics.

Igor L. L’vov was born in 1976. He received theM.S. degree in applied mathematics and physics fromMoscow Engineering Physics Institute, Moscow,Russia, in 1999.

Since 1999, he has been a Research Fellow withthe Russian Federal Nuclear Center, All-RussiaScientific and Research Institute of ExperimentalPhysics, Sarov, Russia. His scientific interest isphysics of gas discharge.

Sergey A. Sadovoy was born in 1973. He re-ceived the M.S. degree in applied mathematics andphysics from Moscow Engineering Physics Institute,Moscow, Russia, in 1997.

Since 1997, he has been a Lab Leader with theRussian Federal Nuclear Center, All-Russia Scien-tific and Research Institute of Experimental Physics,Sarov, Russia. His scientific interests are physics ofgas discharge and plasma physics.

Leonid A. Senilov was born in 1987. He re-ceived the M.S. degree in applied mathematics andphysics from Moscow Engineering Physics Institute,Moscow, Russia, in 2010.

Since 2010, he has been a Research Fellowwith the Russian Federal Nuclear Center, All-RussiaScientific and Research Institute of ExperimentalPhysics, Sarov, Russia. His scientific interests areplasma physics and gas discharge physics.

Dmitry V. Vyalykh was born in 1981. He re-ceived the M.S. degree in applied mathematics andphysics from Moscow Engineering Physics Institute,Moscow, Russia, in 2003.

Since 2003, he has been a Research Fellowwith the Russian Federal Nuclear Center, All-RussiaScientific and Research Institute of ExperimentalPhysics, Sarov, Russia. His scientific interest is gasdischarges.