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Fibonacci MobilesAuthor(s): Alison Frane and Susan GoldstineReviewed work(s):Source: Math Horizons, Vol. 16, No. 2 (Nov. 2008), pp. 24-25Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/25678789 .

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"We can view this process as a formofmobile arithmetic.To

take the sumof two smallermobiles,we hang themfrom ither

end of a bar tomake a new, largermobile."

Fibonacci MobilesAlison Frane and Susan Goldstine

Looney abs and SaintMary'sCollegeofMaryland

Mobiles, the kinetic sculptures originated byAlexan

der Calder in the 1930s, comprise an art form in

which aesthetics, physics and geometrymeet. As it

happens, theyalso provide an excellent method formodelingtheFibonacci sequence.

When you construct amobile, you build itfrom thebottom

up. Inmost cases, you begin with two hanging elements and

suspend each fromone end of a bar. Once the elements are in

place, you find the balance point of the bar and mark itwith a

loop, notch, or hole. The entire bar is now a unit to be hungfroma subsequent bar, and themobile isbuilt up recursivelyfromhanging elements and bars.

We can view thisprocess as a formofmobile arithmetic.To

take thesum of two smallermobiles, we hang themfromeither

end of a bar tomake a new, largermobile. Using this defini

tion,Fibonacci mobiles arise from the same formula as the

Fibonacci sequence: each bar is the sum of theprevious two

bars. The beginning of thisprocess is shown inFigure 1,which

begins with two hanging elements, a disc and a loop.

o +?=

<5?)

?+ 6&=

<?<5<?)

Figure I.The FibonacciMobile Recursion

By repeating thisprocess untilwe had a total of 21 hanging

elements, 13 loops and 8 discs, the authors assembled the

mobile pictured inFigure 2, which currently hangs in the

Mathematics andComputer Science Department at St. Mary's

College ofMaryland. Naturally, the recursive rule underlyingitsconstruction results inotherFibonacci-related patterns.For

instance, themobile contains eight 2-element bars, five 3-ele

ment bars, three 5-element bars, two 8-element bars, one 13

element bar, and one 21 -element bar.

For optimal spacing of themobile, the lengthof each bar is

thegolden ratio times the lengthof theprevious bar.The paper

_._

. ;"i'? ..liri3?

foO <0oio? &

0?? I0?oj

Figure 2. SteelWire andWatercolor PaperMobile byAlison Frane

and Susan Goldstine.

iscut so thatthedisks and loops have exactly the same

weight.This means that ifwe assume that theweight of thewires has

a negligible effecton theposition of thebalance points (and

surprisingly, this is empirically the case), the balance pointsdivide thebars into theFibonacci ratios 1 : 1,2: 1,3:2,5:3,

and so on. Since these ratios converge to thegolden ratio, as

themobile grows larger,the final balancing point gets closer

and closer todividing the topbar into thegolden ratio.

The Fibonacci recursion can be applied tomany styles of

mobile. Ifyouwant to tryyour hand at a Fibonacci mobile but

don't have access towire and wire-bending tools, youmight

attempt a dowel-and-thread mobile like the one inFigure 3.

The use of threadmakes iteasier to experiment with depth,and in themobile pictured here, the beads hang one or two

units down, the2-bead bars hang threeunits down, the3-bead

bars hang five units down, the 5-bead bars hang eight units

down, and so forth.

An interesting question spawned by these projects is

whether and how frequently thisparticular Fibonacci patternmakes its way into artistic mobiles. In perusing

www.calder.org/SETS/work/work.html, the cata

logofAlexander Calder's work, we find thatwhile Calder did

not use the strictFibonacci recurrence described here, he did

24 NOVEMBER2008

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