Students and busy bees

From children’s pictures to architect’s drawings, most of us imagine houses and buildings as being made up of squares and rectangles; not surprising given the majority of the buildings we see in the city around us are based on  right angles.  But we do have some fascinating examples of differently shaped buildings here in Oxford.  Some are circular, such as the Radcliffe Camera or the Sackler Library, some have curved sides such as the front of the Sheldonian.  But only one building in Oxford is hexagonal.

When this student accommodation was built in the North Quad of St John’s college in 1960 the architect made the unusual decision of using hexagonal rooms.  This design is rare in the built environment but is found more frequently in nature. [Where do we see hexagons in nature?]  The honey comb in beehives, which also serves as accommodation for the next generation, is formed from hundreds of hexagonal wax cells packed together.  So it’s not a surprise that this building has come to be known as the Beehive.

Making the most of your wall

So why should bees, or architects, use hexagons?  For bees, making the wax for the walls of the honeycomb is a very expensive business. (A single bee produces just 1/12th of a teaspoon of honey in their entire lifetime.  Bees in a hive need to consume 6-8 pounds of honey to produce 1 pound of wax, which means they collectively need to fly more than 6 times around the world to produce that amount of wax!)    So understandably they would want to use this expensive resource most efficiently, building the largest cells possible for a given amount of wax.

If you know how much wall you have to use, say a fixed number of bricks or a fixed amount of honey, how should you build your room so that it encloses the largest space possible?

You can explore how to increase the area enclosed by a fixed length of wall using a loop of string and asking volunteers to add a corner to the room, one corner at a time.

You can explore this using a loop of string for your fixed length of wall.  Starting with (an admittedly very useless) room with just two corners, each time you add in another corner (going from triangular, to square, to pentagonal, etc), you increase the area the string encloses.  If you carry on adding corners you shape becomes more and more like a circle, and it is a circle which encloses the most area for fixed perimeter.  (If you’ve got a mathematical bent why not try deducing the area of these shapes, and prove that it increases as the number of sides increases.  You can check you calculations for the areas with those listed on Wikipedia)

Making the most of your space

A circle is the most efficient shape for a room on its own – it encloses the largest possible space for a given length of wall.  Although we have some examples of circular buildings, including those here in Oxford, it isn’t a good choice for the shape of rooms within a building. [Why aren’t circular rooms common?] Circles don’t fit neatly together and wasted gaps of space would be left between circular rooms.

In order to make the most of your space you need a shape that tessellates or tiles the floor space of your building, just like the paving stones that neatly cover the courtyard near the Beehive.

There are only three regular shapes that can tile the plane – triangles, squares and hexagons.  So by choosing hexagons, the architects of the beehive, both this building and that of the bee, have chosen the most efficient shape for a room – the hexagonal walls enclose the largest possible areas while wasting no space between them.

Bees have known about the benefits of hexagons for millennia, and we have suspected that hexagons were the most efficient way to divide up a flat plane from at least 300AD when Pappus of Alexandria posed this as a question.  However this Honeycomb Conjecture was only proved mathematically just over a decade ago, by Thomas Hales in 1999.

 Making the most of your energy

Although the architects of the Beehive were aware of all this maths, we can be pretty sure that the bees haven’t learnt any geometry.  But nature, like mathematicians and architects, is keen on efficiency and will form shapes and arrangements that require the least amount of energy.  The bees actually start by making roughly circular cells as these are the most effecient use of their wax.  But as these pack together the walls bend to create a hexagonal arrangement.  You can see how this happens by tossing a handful of marbles in a curved wok – the spherical marbles naturally settle into a hexagonal pattern. 

Mathematically hexagonal rooms have many benefits, but they have some practical drawbacks too.  Any of the students who have lived in these rooms will tell you that it can be quite hard to fit normal, predominantly right-angled, furniture into these hexagonal rooms.  Some students apparently even went so far as to take to their beds with a saw to make them fit! Which is why most of us won’t be living in a beehive soon.

Demonstration

We've included questions (in italics in the description above) that you can ask the group to get them involved.

You can use a large loop of string (one about 6m long works welll) to explore how the area of a room (enclosed by a fixed length of wall) changes as you alter the shape.  Ask for two volunteers and ask them to pull the loop taut., giving you a room you with just two corners – it’s not a very useful room as it encloses no room at all.  The room becomes much more useful if you add a third corner, creating a triangular room which now encloses some space. Continue asking for new volunteers to add a corner to the room and note that the area enclosed continues to increase.

It is also useful to have a wok (or similarly curved dish) and a bag full of marbles.  If you empty the marbles into the dish they will settle into a hexagonal pattern, demonstrating that making the honeycomb arrangement uses the least energy.