Chapter 13.9, Problem 4AE

To determine

To make model to find the shortest connection between the vertices of pentagon.Also find the measures of the angles where the soap film meets.

Answer to Problem 4AE

The angle between any two tangent planes to the intersecting surfaces, at any point along the line of intersection of three surfaces, is ${120}^0$

Explanation of Solution

Concept used:

Soap films minimises the surface area thus leading to shortest connection.

Explanation: 

To make model

Requirements:

• A sheet of clear plastic
• 10 split pin paper fasteners
• A drinking straw cut into four 3-cm long pieces

Bend the plastic sheet without creasing it. Cut four small slits (to represent five vertices of square) through both the layers of the sheet. Insert paper fasteners through all ten slits. Slip two fasteners through each of five straws, so that model looks like the figure below. The halves of the plastic sheet should be parallel, and the straws perpendicular to them.

  

Dip the model in a soap solution and carefully lift it out. A system of vertical soap films between two sheets of plastic and joining the straws will be seen.

After taking out the model from soap solution we will obtain 3-D figure (octahedron).

By placing the protector on top of the model where soap films meet, measure the angles.

The angle between any two tangent planes to the intersecting surfaces, at any point along the line of intersection of three surfaces, is ${120}^0$