PURPOSE:

This explainer will present information to acquaint visitors with how a mathematical question is addressed in a natural phenomenon
 

OBJECTIVES:
 

1. The visitor will build minimum spanning trees and see that 120 degree angles are present.
2. The visitor will observe examples of the regular behavior of soap films.
3. The visitor will relate the use of minimum spanning trees to city/town and grid planning and to their own lives.
4. The visitor will relate the forms in natural phenomenon to geometry.

To be realistic, all of these activities can't be done by one explainer. The "soap sandwich" activities using soap solutions, plex devices and and overhead projector will keep one person occupied. But prepared to use the other activities as background if questions come up and you have the opportunity. Ideally, if two explainers worked together, all could be done. In ordinary circumstances, the "soap sandwich" activity can stand alone.
 

MATERIALS

area - a measure of a surface (Caution! This is not an easy concept for middle school students)
circuit ­ a path on a graph (network) that begins and ends at the same vertex. With a Hamiltonian circuit, all vertices are traveled. With an Euler circuit, all edges are traveled.
Cuisenaire rods - a commercial name, known by elementary school teachers,
for a standard set of rods made by the Cuisenaire Company. Each length
of rod, 1 cm to 10 cm, is color coded. As a mathematics manipulative
learning aid, the Cuisenaire Rods have multiple uses. They are most
traditionally used to teach properties of addition, multiplication and fractions. We use them to teach about Fibonacci Numbers, symmetry and minimum spanning trees.
graph (network) - A graph is a set of points (called vertices) and a set of lines (called edges) joining these vertices; a finite set of dots and connecting links. The dots are called vertices and the links are called edges. Each edge must connect two different vertices. Two vertices are adjacent if there is an edge joining them.
path ­ a sequence of vertices in a graph (network), each one adjacent to the next one; a path along a network which covers all vertices and edges (each edge only once) An Euler circuit is a path that ends in the same place that it began, but an Euler path can end on a different vertex.
perimeter­ the outer boundary of a two dimensional figure or the length of that boundary.
regular hexagon - a hexagon with 120 degree internal angels at every vertex
"soap sandwich" device - two pieces of plexiglass
held at an interval by bolts. When this device is
placed in a soap solution, the soap film forms lines
to connect the bolts
in the most energy efficient way.
spanning tree - a network of lines of minimum length
to connect 3 or more locations
¹ subgraph ­ a subset of vertices and connections belonging to the original graph (network)
² tree ­ any graph (network) which is connected and has no circuits. This means that there is one and only one path joining any two vertices

The shape of soap films has to do with two problems: minimizing distance and maximizing space.
Delta Airlines and the Pricing Policy Steiner point in red

This is a real­life problem that arose in 1960. At that time Delta Airlines claims that Bell telephone overcharged them for the private lines between their main terminals in Chicago, New York and Atlanta. The mathematicians at Bell Labs researched Delta's claims and found that Delta was correct. A fourth point, called a Steiner point (after the nineteenth century mathematician Jacob Steiner) could be located so that the shortest route between the three cities is a minimum. This route could be the basis for the phone line charges.
Today Bell Labs is not really concerned with the location of Steiner points, but in the length of the network that results if Steiner points are used. The cost of the service is based on the total length of the network.
­ by Yvonne Greenleaf and Jeanette McGillicuddy

See the attached notes for more from Greenleaf and McGillicuddy's workshop.

See the Maryland Science Center's soap film explainer for related information

Periodic Tiling Manual interactive

Costa Sculpture

Costa Models Manual Interactives

Soap Films Two Mechanical Interactives

PROCEDURES:
Choose a position on the exhibit floor that is near an electrical outlet. After plugging in the overhead projector, be sure that the electrical cord does not pose the threat of tripping visitors.
 

MATERIALS:

• overhead projector
• loop of chain
• laminated piece of plain graph paper
• washable marker
• clips on suction cups to hold up paper
• regular hexagon shaped block

So that the visitor can observe that the regular hexagon has 120 degree angles and that, of all the shapes, the circle covers the largest area of a surface, and the hexagon is the next largest:

Take a loop of chain, lay it on the overhead projector (to create a shape such as a triangle, a square, a rectangle, a circle and a regular hexagon) and project it onto a piece of graph paper. Point out that the chain represents the perimeter of the figure and that it does not change.

The number of enclosed squares gives an estimate of the area. After you have modeled this once, let the visitor try various shapes and record the results. (using a washable marker to trace the shadow and then writing the number of counted squares on the drawn shape makes a good record and can be referred to by other visitors.) 

MATERIALS:

• pulling string device to find the shortest system of lines connecting three points (Steiner's soda pop warehouse and the soda shoppes)
• regular hexagon -shaped block

(Use the pulling string device) to pull the lines into different configurations.

Which shows the minimum sum of all these routes? Count the black and white segments on the string. Use a hexagon cut out to show that the angle formed at the center vertex is always 120 degrees.

ACTIVITY C
 

MATERIALS:

• laminated charts for suggested ways to connect four points (four houses in the 'hood)
• device for laying Cuisenaire rods end to end to compare lengths
• overhead projector

To show that the minimal distance connecting three or more points is a tree with angles of 120 degrees at the vertices:

Show four laminated graphs (below) showing some of the possible ways to connect four houses(the number of all possible is infinite).

Demonstrate how Cuisenaire rods can be laid end to end to connect the four houses.

Which of these diagrams show the minimum distance? The solution can be arrived at by even young children by lining up the Cuisenaire rods in the corresponding wooden tray lines to see which line is shortest.

Use a hexagon cut out to show that the angle on the graph is always 120 degrees.

MATERIALS:

• water­tight transparent tray
• extension cord and cord cover for floor.
• Steiner point devices ("soap sandwiches")
• soap film solution made of multiples of this recipe:
• 1 c. liquid dishwashing liquid
• 2 1/2 tsp. glycerine
• 9 c. water
• (This recipe works best if made 24 hours in advance)
• tub for soap solution
• graphic of Delta Airlines problem or transparent map of Baltimore (or your museum's city)
• rags for wiping up spills.
• art straws

To show the visitor that natural phenomenon such as soap bubbles have 120 degree angles as away to form structural material efficiently.:

Refer to the Delta Airlines Problem map that fits a soap sandwich device (At this time, I don't have one made) to find how the lines could be configured.
One that I do have can be presented this way:
 

Present other soap sandwiches and have the visitor predict how the tree will form before dipping it and placing it on the overhead projector. Use drinking straws to blow the soap in different ways.

Frederick J. Almgren, Jr., and Jean E. Taylor; "The Geometry of Soap films and Soap Bubbles" Scientific American, July 1976

Greenleaf, Yvonne and Jeanette McGillicuddy of the Mathematics/Computer Science Department at River College, Nashua, New Hampshire"Trees and Soap Bubbles: Applications in Graph Theory" a workshop given at the NCTM (National Council of Teachers of Mathematics) 73rd annual meeting, April 6­9, 1995, Boston Massachusetts

Hildebrandt, Stefan and Anthony Tromba; Mathematics and Optimal Form, Scientific American Books, 1985

Isenbuerg, Cyril; The Science of Soap Films and Soap Bubbles, Tieto Ltd., no date given

Katz, David A.; Chemistry in the Toy Store, workshop handout, 1990

Noddy, Tom; Tom Noddy's Bubble Magic, Running Press, 1988

O'Neill, Catherine; "Blowing Bubbles", The Washington Post, August 1, 1989

Stevens, Peter S.; Patterns in Nature, Little, Brown & Co., 1952
This book is out of print but is referred to quite often. I have copied pages for this folder.-CB

Boys, C.V.; Soap Bubbles, Dover, 1959
The actual date has to be much earlier that this. It's a classic that is always referred to.

Johnsonbaugh, Richard; Discrete Mathematics, 3rd Edition, (Sections 7.3 - 7.4) MacMillan, 1993

Kolata, Gona; "Solutions to Old Puzzle. How Short a Shortcut," NY Times, October 30, 1990
 

COPYRIGHT: 1995 Maryland Science Center

AUTHORS:
Cathy Brady, Math Specialist

Maryland Science Center

		These diagrams can be found below.

^1,2 Greenleaf, Yvonne and Jeanette McGillicuddy of the Mathematics/Computer Science Department at River College, Nashua, New Hampshire"Trees and Soap Bubbles: Applications in Graph Theory" a workshop given at the NCTM (National Council of Teachers of Mathematics) 73rd annual meeting, April 6­9, 1995, Boston Massachusetts