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.
The Mathematics and Magic of Bubbles: A Guide
This guide explores the fascinating connection between bubbles and mathematical principles, providing instructions for demonstrations and bubble solution recipes. Bubbles are real-world examples of minimal surfaces—shapes that enclose a given volume with the smallest possible surface area.
Safety and Cleanup Precautions
- Slippery Surfaces: Exercise caution, as spilled bubble solution creates a slip hazard.
- Location: Perform bubble activities outdoors or on an easily cleaned indoor surface. Avoid carpets.
- Metal Equipment: To prevent corrosion, metal bubble frames should be cleaned with water, then a vinegar wipe, followed by another water rinse immediately after use.
- Storage: Store unused bubble mix in an airtight container.
The Golden Rule and Minimal Surface Theory
The fundamental principle governing bubble behaviour is simple: Bubble film will always try to make the smallest possible surface area.
- Sphere Formation: A sphere has the smallest surface area for its volume of any solid shape. This is why a single, free-floating bubble always forms a perfect sphere.
- Minimal Surface: A bubble is a minimal surface—a surface with the smallest possible area under certain boundary conditions (like being stretched over a frame or containing a volume of air).
- Minimal Surface Theory: This is a whole field of mathematics dedicated to the complex, minimal shapes bubbles and films form.
Bubble Mix Formulas and Preparation
The success of bubble demonstrations relies on a quality solution. Additives like glycerol (glycerine) or corn syrup/Karo syrup are crucial as they prevent the film from drying out, making the bubbles last longer.
Recommended Recipe (Questacon Maths Squad)
This recipe is recommended for its reliability and use of common ingredients:
| Part | Ingredient | Notes |
|---|---|---|
| 3 | Dishwashing Liquid (e.g., Dawn, Joy) | Anti-grease agents work best. |
| 7 | Hot Water | Use soft water; hard or well water may require distilled water. |
| 1 | Glycerol (or Sugar) | Glycerol is available from chemists/drug stores. |
Other Popular Recipes
| Name | Dish Soap | Water/Liquid | Additive (Glycerine/Syrup) | Notes |
|---|---|---|---|---|
| Basic Bubbles | 2 Tbsp | 1 Cup | – | Simple and easy. |
| Magic Bubbles | 2 Tbsp | 9 oz | 1 Tbsp Glycerine | Known to make bigger, longer-lasting bubbles. |
| Sweet Bubbles | 2 Tbsp | 1 Cup | 1 Tbsp Corn Syrup | Avoid if bees/wasps are a concern. |
| Ultra Bubbles | 1 Part Ultra Detergent | 15 Parts | 0.25 Parts (Optional) | Proportions are key here. |
Mixing and Conditioning Tips
- Mixing: Mix ingredients very thoroughly. A flat stick, paint stirrer attachment, or electric mixer can be used.
- Age: Bubble mix improves with age. Prepare solutions the day before use for the best results.
- Froth: Allow the froth on top of the mix to settle before using.
- Fine-Tuning: If bubbles pop easily, try adjusting the mix by adding more water, then more detergent, stirring thoroughly after each adjustment.
Environmental Factors
Bubble performance is sensitive to weather and air quality:
- Humidity: Bubbles burst when they dry out. The more humid the environment, the longer bubbles last.
- Best Conditions: Bubbles work best at night, in the shade, or while it’s raining/damp.
- Worst Conditions: Air conditioners dry the air, causing bubbles to burst faster.
Bubble Wands and Tools
Wands can be created from many household items.
| Material | Construction | Notes |
|---|---|---|
| Coat Hanger & String | Cut two 6-inch straws and one 24-inch piece of cotton string. Thread the string through the straws and tie a knot to form a loop/rectangle. | Creates large, giant bubbles. |
| Plastic Lid | Cut the center out of a yogurt/margarine lid and attach it to a chopstick. | Simple, basic wand. |
| Twig/Yarn | Find a flexible twig. Wrap the twig tightly with yarn and bend the tip to meet the starting point, forming a circle. | A natural, customizable option. |
| Other Tools | Fly swatters (make lots of small bubbles), plastic cookie cutters (for fun shapes), and strawberry baskets (for millions of tiny bubbles). | Great for toddlers and high-volume bubble making. |
Demonstrations of Minimal Surface Theory
These activities illustrate how bubble films minimize their surface area and obey specific physical rules.
Demonstration 1: The Bubble Trampoline
- Dip a simple round ring (at least 10 cm wide) into the bubble mix to form a flat film.
- Hold the ring horizontally and move it up and down.
- Observation: The film stretches and bounces like a trampoline, demonstrating the elasticity and tension of the bubble film.
Demonstration 2: Woolly Jumper
- Tie a piece of loose wool or thread across the middle of the ring.
- Dip the ring to form a bubble film on both sides of the thread.
- Use a clean, dry finger to burst the film on one side of the thread.
- Observation: The thread will be violently pulled taut towards the remaining film. The tension in the remaining film is always trying to shrink it to the smallest possible area, pulling the thread with it.
Demonstration 3: Shapes (Why Spheres?)
- Gather models of simple geometric shapes (cube, pyramid, etc.).
- Blow several free-floating bubbles.
- Observation: Note that the bubbles always form spheres. Explain that a sphere requires the least amount of surface film to contain the volume of air, proving the “Golden Rule.”
Demonstration 4: Cubical Bubble
- Dip a cube-shaped bubble frame entirely into the mix to form multiple films.
- Dip again, immersing only one face, and gently burst the surrounding films until one large bubble is trapped inside the center.
- Observation: The resulting bubble will appear cubical. Burst the surrounding films to reveal that the “cube” was actually a stretched sphere constrained by the minimal surfaces around it.
Rules for Intersecting Films (Plateau’s Laws):
When multiple bubbles or films meet, they obey strict minimal surface rules:
- Three films meeting at an edge will always form an angle of 120°.
- Four edges and six films may meet at a single point (vertex), forming an angle of approximately 109.47°.
- The apparent “cube” bulges because its internal corners and edges must follow these mathematically derived angles, not the 90° angles of a true cube.
Further Activities
- Explore other frame shapes: Try frames shaped like a pyramid or a helix.
- Hypercycloid: Dip two circles held close to each other to produce a hypercycloid (a shape resembling a cylinder that narrows in the middle).
- Hand Bubbles: Dip hands in the solution, lock thumbs together, and slowly spread them to form a circular opening. Blow gently to create massive bubbles without a wand.