Math of Soap Bubbles and Honeycombs

Have you ever blown a soap bubble and wondered why the bubble is spherical? Or admired a bee honeycomb and wondered why the honeycomb forms a hexagonal tiling? We will explore these shapes of nature and the mathematical principles behind their formation.

Blowing soap bubbles can be fun, entertaining, and fascinating for all ages. Have you ever wondered why soap bubbles form the way they do? Here is a simple soap bubble blown from a circular wand:

Why is the soap bubble spherical instead of another shape, such as a long and skinny ellipsoid with the same radius as the circular wand? Try using the following soap formula to blow your own bubbles.

Ingredients:

• 6 parts water
• 1 part dishwashing liquid

• For longer lasting bubbles, add 1/3 part glycerin or corn syrup.

  Experiment with different types of wands, such as pipe cleaners, metal coat hangers, or flexible wire bent into shapes such as polygons, spirals, or stacked circles.
  

What happens when you bend your wire into the shape of a circle and dip it into the soapy water? The result is a film of soap stretching across the circle. Soap films adopt a shape that minimizes elastic energy due to surface area while still spanning the outline of the wand. (In the physical world, gravity will also play a role, but we will ignore these effects for the time being.)

When you blow a soap bubble, some amount of the air you blow becomes trapped within the bubble. Then the soap bubble or cluster of bubbles naturally tries to minimize surface area for the volume(s) they enclose due to surface tension.

Minimal surface principle: Soap bubbles try to assume the shape of least surface area possible containing a given volume.

Mathematically, the question of what shape the soap bubble will form is a minimization problem: the surface area seeks to be as small as possible under a constraint (the volume is constant and the boundary spans a given countour). This is known as Plateau's problem. Let's compare the surface areas of few different shapes with the same volume.

To generalize this problem, consider the surface areas of different regular polyhedra with volume 1 cm\(^3\):

\[ \begin{array}[ccc] \mbox{Shape} & \mbox{Number of sides} & \mbox{Surface Area}\\ \hline\\ \mbox{Octahedron} & 8 & 5.72 \mbox{ cm}^2\\ \mbox{Dodecahedron} & 12 & 5.32 \mbox{ cm}^2\\ \mbox{Icosahedron} & 20 & 5.15 \mbox{ cm}^2\\ \mbox{Sphere} & - & 4.84 \mbox{ cm}^2 \end{array}\]

Observe that the surface area decreases as the number of sides increases for regular polyhedra. Is the sphere the shape with the minimum surface area over all shapes with volume 1 cm\(^3\)? While this problem was formulated by Archimedes, it was not proven until 1884 by Hermann Schwarz in the following theorem.

Isoperimetric theorem for three dimensions: The shape with the minimum surface area for a given volume is the sphere.

This theorem shows that the sphere is indeed the three dimensional shape minimizing surface area over all possible shapes. Since soap bubbles try to minimize surface area (in the absence of other physical forces, such as gravity), this explains why soap bubbles form spheres instead of other shapes.

What happens when clusters of bubbles form together? What shapes do they take? Similar to single soap bubbles, clusters of bubbles find the minimal surface area shape that encloses multiple regions of volumes. Let's first consider what happens in the two dimensional plane.

The first observation is that for fixed areas \(a_1\) and \(a_2\), two circles that are pushed together to share a common wall separating the areas \(a_1\) and \(a_2\) has smaller perimeter than two disjoint circles. How far should these circles be pushed together? Does this shape give the minimum perimeter over all possible shapes enclosing areas \(a_1\) and \(a_2\)? These questions were resolved by the following theorem, proved in 1993.

Double Bubble in the Plane: In 1993, Alfaro, Brock, Foisy, Hodges, and Zimba showed the configuration of shapes with the minimum perimeter enclosing two equal areas is achieved for two intersecting circles separated by a line such that the arcs all meet at 120 degree angles:

The red dots indicate the angles meeting at 120 degrees.

Triple Bubble in Plane: In 2002, Wichiramala showed the configuration of shapes with the minimum perimeter enclosing three equal areas is achieved for three intersecting circles such that the arcs all meet at 120 degree angles:

The problem of finding the optimal configurations with minimum perimeter for four or more bubbles in the plane is currently still open. The conjectured configuration of shapes follows the patterns above with all intersecting circles meeting at 120 degree angles:

In general, soap bubbles always meet in groups of threes at equal angles of 120 degrees. Showing configurations of four or bubbles are optimal is still an open problem -- perhaps you will be the one to make progress on these conjectures!

Looking at the conjectured optimal configuration for four or more bubbles in the plane leads to the following question: what happens if we consider more and more bubbles? You might notice the pattern that the conjectured optimal configuration for more bubbles begins to resemble a hexagonal tiling of the plane. If these configurations are indeed optimal, this leads to the question: do we observe hexagonal tilings in nature?

Two examples of hexagonal tilings in nature are

• 1) soap film patterns formed between two glass plates
• 2) bees honeycombs, which are made of wax and are created by many bees working simultaneously in different parts of the honeycomb.

In the 19th century, Charles Darwin observed that honeycombs were engineering feats "absolutely perfect in economising labour and wax."

For a single bee storing a fixed amount of honey, we have seen that the shape with the smallest perimeter to store the fixed amount of honey is the circle. However, if we pack circles side-by-side in the plane, then these circles will leave gaps rather than filling the entire space. In other words, if several bees work in parallel, they would not be minimizing the sum of perimeters for the total amount of honey being stored. Bees have instead found a solution that minimizes the collective perimeter for the total amount of honey.

The Ancient Greeks knew that perimeter enclosing a fixed area of a hexagon is less than the perimeter of a square or triangle of the same area. However, there is no reason that the cells must all have equal side lengths, or why the cells do not have curved sides rather than straight sides.

The Honeycomb Conjecture remained unproved for centuries until Thomas C. Hales gave a proof in 1999.

Honeycomb Theorem (Hales): The hexagonal grid gives the best way to divide a surface into regions of equal area with the smallest total perimeter.

If the tiling has curved sides, then the side that bulges out will minimize the cell perimeter, while the side that bulges in will hurt the perimeter. Hales proved that the advantage of bulging out is less than the disadvantage of bulging in. In other words, no single cell can do better than a hexagon if it pays a penalty for having more than six sides or if it curves outwards. This gave a complete proof that polygon with straight sides work better than curved sides and a hexagonal tiling is the best of all configurations. Note that in a hexagonal tiling, the meeting points are always formed by triples of lines meeting at 120 degree angles. See below for our challenge #why120degrees

More complicated forms occur when multiple bubbles are joined together in space. The simplest example is the double bubble, and beautiful configurations can form when three or more bubbles are joined together.

Is this configuration give the minimal surface area to enclose and separate two different volumes? Of all possible configurations, how can we prove that it is the best? This question remained an open problem for a long time and sparked a great deal of interesting research in the area of math called Geometric Measure Theory. In 2000, Morgan, Hutchings, Ritoré, and Ros resolved this Double Bubble Conjecture for arbitrary double bubbles.

Double Bubble Theorem: The configuration enclosing two fixed volumes with the minimum possible surface area is formed by two intersecting spheres meeting at angles of 120 degrees on a common circle.

If the two bubbles in the double bubble enclose the same volume, the common circle where the intersecting spheres meet is a flat circle. If the two bubbles enclose different volumes, then the smaller bubble has a higher internal pressure and will bulge into the larger bubble. For bubbles of any size, the meeting points are always formed by triples of bubbles meeting at 120 degree angles. This 120 degree rule always holds, even for complex bubble collections such as foam. This again raises the question #why120degrees?

The Triple Bubble Problem is still wide open and the conjectured optimal configuration follows the pattern above, with intersecting spheres meeting at angles of 120 degrees.

Soap films between two parallel circular rings forms the shape of a catenoid, with equation \(r = cosh(z)\) in cylindrical coordinates.

The catenoid is a minimal surface whose points are all saddle points, which means that at every point, the bend of the surface upward in one direction is matched with the bend of the surface downward in the perpendicular direction.

As the rings are pulled further apart, the neck will narrow until the critical separation is reached, and then the catenoid will pop into two circular rings of film across the two parallel rings. Try this activity for yourself and experiment to find the critical separation point!

For many great problems in mathematics, the quest for a solution often leads to further questions. We have studied the minimization principle for soap bubbles and honeycombs, but another question arose: why are meeting points of minimal surfaces formed by triples of lines meeting at 120 degree angles? Do you have any intuition for why this occurs? We encourage you to ask your friends, your family, your teachers, and discuss with fellow Brilliant members. To entice further discussion, try making a cube frame out of wire and dip it into a soap solution. Do you get this shape? How many 120 degree angles are there? #why120degrees?

We will study this 120 degree angle principle in a future post.