Drawing an elephant with four complex parameters

Jürgen Mayer

Max Planck Institute of Molecular Cell Biology and Genetics, Pfotenhauerstr . 108, 01307 Dr esden,

Germany

Khaled Khairy

Eur opean Molecular Biology Laboratory, Meyerhofstraße. 1, 691 17 Heidelberg, Germany

Jonathon Howard

Max Planck Institute of Molecular Cell Biology and Genetics, Pfotenhauerstr . 108, 01307 Dr esden,

Germany

! Received 20 August 2008; accepted 5 October 2009 "

W e deﬁne four complex numbers representing the parameters needed to specify an elephantine

shape. The real and imaginary parts of these complex numbers are the coefﬁcients of a Fourier

coordinate expansion, a powerful tool for reducing the data required to deﬁne shapes. ©

2010

American Association of Physics T eachers.

# DOI: 10.1 1 19/1.3254017 $

A turning point in Freeman Dyson’ s life occurred during a

meeting in the Spring of 1953 when Enrico Fermi criticized

the complexity of Dyson’ s model by quoting Johnny von

Neumann:

1

“W ith four parameters I can ﬁt an elephant, and

with ﬁve I can make him wiggle his trunk.” Since then it has

become a well-known saying among physicists, but nobody

has successfully implemented it.

T o parametrize an elephant, we note that its perimeter can

be described as a set of points ! x ! t " , y ! t "" , where t is a pa-

rameter that can be interpreted as the elapsed time while

going along the path of the contour . If the speed is uniform,

t becomes the arc length. W e expand x and y separately

2

as a

Fourier series

x ! t " =

%

k =0

!

! A

k

x

cos ! kt " + B

k

x

sin ! kt "" , ! 1 "

y ! t " =

%

k =0

!

! A

k

y

cos ! kt " + B

k

y

sin ! kt "" , ! 2 "

where A

k

x

, B

k

x

, A

k

y

, and B

k

y

are the expansion coefﬁcients. The

lower indices k apply to the k th term in the expansion, and

the upper indices denote the x or y expansion, respectively .

Using this expansion of the x and y coordinates, we can

analyze shapes by tracing the boundary and calculating the

coefﬁcients in the expansions ! using standard methods from

Fourier analysis " . By truncating the expansion, the shape is

smoothed. T runcation leads to a huge reduction in the infor-

mation necessary to express a certain shape compared to a

pixelated image, for example. Székely et al.

3

used this ap-

proach to segment magnetic resonance imaging data. A simi-

lar approach was used to analyze the shapes of red blood

cells,

4

with a spherical harmonics expansion serving as a 3D

generalization of the Fourier coordinate expansion.

The coefﬁcients represent the best ﬁt to the given shape in

the following sense. The k = 0 component corresponds to the

center of mass of the perimeter . The k = 1 component corre-

sponds to the best ﬁt ellipse. The higher order components

trace out elliptical corrections analogous to Ptolemy’ s

epicycles.

5

V isualization of the corresponding ellipses can be

found at Ref. 6 .

W e now use this tool to ﬁt an elephant with four param-

eters. W ei

7

tried this task in 1975 using a least-squares Fou-

rier sine series but required about 30 terms. By analyzing the

picture in Fig. 1 ! a " and eliminating components with ampli-

tudes less than 10% of the maximum amplitude, we obtained

an approximate spectrum. The remaining amplitudes were

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Fig. 1. ! a " Outline of an elephant. ! b " Three snapshots of the wiggling trunk.

648 648 Am. J. Phys. 78 ! 6 " , June 2010 http://aapt.org/ajp © 2010 American Association of Physics T eachers

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