Mathematics from "Soaps"Author(s): Brian PooleSource: Mathematics in School, Vol. 21, No. 1 (Jan., 1992), pp. 39-41Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30216435 .

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by Brian Poole, St. Luke's High School, Exeter

Investigational work in Mathematics is taking a very promi- nent position in the classroom, and rightly so. In particular, to meet the requirements of Attainment Targets 1 and 9 (AT1 and AT9) in the National Curriculum. This article attempts to use the fascinating topic of soap films to allow students to use practical methods to help investigations. Many other ATs are met as well as AT1 and AT9 of course.

Soap film has the property that in its equilibrium pos-

ition, it will always "pull" itself to the minimum surface area.

Consider the simple question: what is the shortest dis- tance between two points? By construction of the apparatus shown, immersing into a soap solution, and gently allowing the soap film to stabilise, one can see that it is, of course, a straight line. This experiment is effectively a 2-D problem when viewed from A.

STIFF* WIRE

,SOAP FILM

CLEAR PLASTIC

VIEW FROM 'A'

Fig. 1 Shortest distance between two points.

Mathematics in School, January 1992 39

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P O O

P o

R

VIEW FROM 'A'

Fig. 2 Minimum network for three points.

Supposing the question was then posed, "If three points are now considered, then ...?" (AT9.2c, AT9.8b), a predic- tion could be made before the next experiment (AT9.3c). If the student has an idea, a trial and improvement method could be applied to test this out and hence find the minimum network (AT9.6a, AT9.7a). However, if no idea is forthcoming, the soap solution can again be used to obtain the shape.

Possible questions then arise:

If A PQR is equilateral, what symmetries are there? (AT11.3a, AT 11.4b, AT 11.5a) What are angles x, y, z? (AT10.5b) What are the lengths of PO, QO, RO, if PQ = QR = RP = a, say? (AT10.8a) What is the total length of the network? Other arrangements and number of "points" can be

considered if two plastic plates, with a triangular lattice of holes, are used with further wires. Although an exact square cannot be formed, the dimensions shown are very close to a 7 by 7 cm square.

Fig. 3 Triangular lattice of holes in plastic

Four points forming a square give an interesting situation for study. From the soap film experiment, Fig. 4 shows the shape that is formed.

Let PQ = QR = 2a and XY = 2x

Using PYTHAGORAS (AT10.7a),

PX = /[a2 +(a-x)2] Total length of network, 1 = 4PX + XY

1 = 4,/ [a2+(a-x)2] 2x (1) Various approaches can be used to find the value of x

which gives the MINIMUM network, to different accu- racies.

1. Scale drawing with trial and improvement (ATS.Sa). 2. Calculation of the formula (1) with trial and improve-

ment, or starting with an actual number for "a" and varying x (AT6.5a, AT6.8c).

3. Graph of L = f(x) to find x for minimum L.

40 Mathematics in School, January 1992

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4. Computer program using iterative process. (FOR X = 0 TO a STEP DX).

5. Differential Calculus - A Level students.

If I= 4/[a2 (-x2]

+ 2x it can be shown that

dl 4(a - x) -2-

dx /[a2 + (a- x)2]

dl For max or min,d = 0

This leads to 3x2 6ax + 2a2 = 0 This leads to 3X2 - 6ax + 2a 2 = 0

It is therefore apparent, that the three films meet at 1200

A study of soap films in 3-D is fascinating and can be undertaken by immersing shapes made from plastic straws and joiners.

The algebraic Mathematics becomes quite involved, but the geometric properties of 2-D and 3-D shapes can be studied with much enjoyment. See below for two such examples.

AT10.4a,b, 10.5b, 10.6a,c appear in abundance with many discussion points arising - triangles, quadrilaterals, iso- metric drawing, plans and elevations, areas, to name a few.

A

P O

X

;CY P 2a Q r

0

X

2a 2x

S R

s

VIEW FROM 'A'

Fig. 4 Minimum network for four points

Solving: x = a+ 3

a i From a practical point of view .'. x = a- a

3

From Fig. 4, tan 80 a _ a

= a

- x a -,/ a-- a--

~3= 60 => 0 = 60'

This topic gives students the opportunity to investigate in an "open-ended" manner by setting themselves prob- lems that become apparent as experimenting takes place. The style of recording and presenting their findings enables AT9 statements to be met to high levels.

Note It is recommended that when making soap solution, not too much detergent is used, or bubbles form that spoil results.

The equilibrium positions formed are not always the minimum network arrangements. (See "Patterns in Nature" by Peter S. Stevens - Peregrine Books.)

Mathematics in School, January 1992 41

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