Bubbles can be great fun. They also behave according to mathematical rules. Here we look at some of the rules and why they occur. We also look at how bubbles behave as a result.
A word of warning: bubbles can be very messy. Do not do bubble activities on carpet. Either use them outside or on some easily cleaned indoor surface. In one of Questacon’s other outreach programs, the Shell Questacon Science Circus, bubble activities are done on a large piece of carpet which is occasionally blasted out with a high pressure hose.
Last, but not least, please remember slippery surfaces can be a safety hazard.
A sphere has the smallest area for its volume of any solid shape.
Bubbles always try to make the smallest possible surface.
For these activities you will need some bubble mix, several bubble frames and some wool or thread. For the demonstrations mentioned below, the frames needed are a simple round ring (at lease 10 cm wide) and a cubical frame. You may also want some models of various simple geometric shapes.
Whether it is a lone bubble or a group of bubbles joined together, bubbles get their shape by following one simple rule:
Bubbles are a real-world example of what mathematicians call a minimal surface. A minimal surface is one that has the smallest area possible, while meeting certain conditions. For example, the smallest surface that contains a certain amount of air. There is a whole field of mathematics called Minimal Surface Theory that deals with the sort of shapes that bubbles form. It also produces lots of fascinating and bizarre images that can be found by searching for “Minimal Surfaces” on the internet.
Take the ring and dip it into the bubble mix, so a film forms. Now hold the ring horizontally and move it up and down, without forming bubbles. The film will bounce up and down, like a trampoline (if you are doing this for an audience you may want to use a bit of mime and introduce a “performing flea”).
Bubble film is elastic. When you form a film on the ring, it has to stretch out over the gap in the ring. When you bounce the ring you stretch the bubble film even more. This elasticity is important, it lets the film stretch to make a bubble.
Use the ring again, but this time tie a piece of thread or wool across the middle of the ring. Don’t make it taunt, it should be reasonably loose. Make another bubble film. Use a finger to burst the film on one side of the thread. The thread will jump over towards the other side, until it is held taunt. It should be possible to hold it vertically, with the bubble film up the top so it is supporting the weight of the thread.
Just like a stretched rubber band, a stretched bubble film is under tension. When you burst one side of the bubble, the tension in the other side pulls on the thread.
This sort of thing happens with bubbles in general. The tension in the bubble film will always try to shrink the film to make the smallest surface possible. Sometimes there may be restrictions, like having to contain air (a bubble) or being stretched over a bubble frame.
Show various different shapes, such as a cube, pyramid and other geometric shapes. Explain that they have more surface area than a sphere with the same volume. Blow several bubbles. Bubbles form spheres because spheres have the smallest possible surface area for their volume.
The table below compares several different shapes and their surface areas to a sphere.
| Shape | Number of sides | Volume (units3) | Surface Area (units2) | Area/(Area of sphere) | (Area of sphere)/ Area |
|---|---|---|---|---|---|
| Tetrahedron | 4 | 1 | 7.21 | 1.49 | 0.67 |
| Cube | 6 | 1 | 6 | 1.24 | 0.81 |
| Octahedron | 8 | 1 | 5.72 | 1.18 | 0.85 |
| Dodecahedron | 2 | 1 | 5.32 | 1.10 | 0.91 |
| Icosahedron | 20 | 1 | 5.15 | 1.06 | 0.94 |
| Sphere | infinite | 1 | 4.84 | 1 | 1 |
Use a cube shaped bubble frame. Dip the frame completely into the bubble mix. This makes several bubble films forming a square in the middle. Dip again, but only immerse one side of the frame. A near-cubical bubble will form in the centre of the frame, surrounded by films (it may be necessary to burst a few bubbles near the centre to get just one bubble).
Burst the films around the bubble one at a time. The 'cube' will be revealed to be a sphere stretched out by the films around it.
Although a single bubble will always try to form a sphere, other shapes can be made by having several bubbles and films touching. When bubbles and bubble films meet, they always obey certain rules:
These rules are all consequences of the “Golden Rule of Bubbles”. From these rules, it can be shown that the bubble was not a true cube, as the edges and faces did not meet in right angles. It bulged slightly.
There are lots of different bubble mixture recipes available on the internet, but most of them are written by Americans and use American detergents. This recipe is used by the Tenix Questacon Maths Squad using ingredients available in Australia.
The glycerol helps the bubbles to last longer, by preventing them from drying out. Glycerol is available from chemists. If you cannot get any glycerol, use sugar instead.
Mix the ingredients very thoroughly. The Maths Squad's batches are mixed for around half an hour. If you are making a large amount, try an electric drill with a paint-stirrer attachment, or even a bent piece of coathanger wire. Otherwise a flat stick works fine.
There are a few things that may affect how well your bubble mix works.
Bubble mix gets better with age. With time, some water may evaporate from the bubble mix and it may need more water added.
Humidity can seriously affect your bubbles. Bubbles burst when they dry out, so the more humid it is the longer they last. Bubbles work best at night, in the shade and while it is raining. Air conditioners can dry the air which bursts bubbles faster.
The quality of the tap water in your area may also affect your bubble mixture. Hard water doesn't work as well as soft water. You may need to try distilled water instead of tap water.
You may need to “fine-tune” your bubble mix, either by adding more detergent or more water. Experiment a little. Remember to stir thoroughly.
Froth on the top can interfere with making bubbles.
Cleanup
Vinegar can be very effective for cleaning up bubble mixture. If you are using metal bubble frames, they will need to be wiped down with water, then with vinegar, then with water again. Otherwise they will corrode. The bubble mix can be kept in an airtight container.
Try other shapes.
Two circles held near each other can produce a shape called a catenoid, like a cylinder that gets narrow near the middle.
