What it shows
Circles packed in a hexagon pattern leave less gap than circles in a square grid. The pen stand lets you compare the two packings by measuring the wasted space.
The science
Seven equal circles can fit with one in the centre and six around it, in a hexagon. The wasted space between circles is smaller in this arrangement than in a square layout, which is why honeybees build hexagonal cells and cans are stacked this way. The area fraction filled is about 90.7 percent in the hexagonal pattern and 78.5 percent in the square pattern.
Materials
Cost: free. Many items come from home or from waste; spare parts such as LEDs, motors and wire are in the store.
- Seven identical small tins or cardboard tubes (free)
- Strong glue or tape
- Coloured paper
- A ruler
- A sheet of graph paper
- Compasses
Safety
For this project: An adult must check the cut edges of tins and cover them with tape. Wear gloves when handling sharp edges.
Never use mains electricity: power models with cells, batteries or a low-voltage adapter. Sharp and hot tools, flames and lab chemicals need an adult or your teacher beside you. Keep magnets, button cells and small parts away from young children, and wash your hands after making.
All the safety rulesStep by step
8 steps. Do them in order, and test each part before you go on to the next. Write what you notice in a notebook as you go.
Collect seven tins of the same size and make sure all edges are smooth and covered.
Arrange six around one and glue them together at the touching points.
Cover the sides with coloured paper to make a pen stand.
Draw the same seven circles on graph paper in the hexagonal pattern and measure the area of the enclosing hexagon.
Draw seven circles in a square grid pattern and measure that enclosing area.
Count the squares of empty space in each drawing.
Compare the two percentages of area used.
Write which pattern wastes less space.
What you should see
The hexagonal arrangement wastes less space than the square one, and the pen stand holds seven pens neatly.
Viva questions
A teacher or a judge will ask how it works. Try to answer each question before you open it, then compare.
1Why do bees use hexagons?
Hexagons tile with no gaps and use the least wax.
2Which circle packing is denser?
The hexagonal packing.
3How many circles touch one central circle in hexagonal packing?
Six.
4Why cover the edges of the tins?
To avoid cuts.
Take it further
Test with marbles in a box: pack them in a square then hexagonal way and see how many fit.
Change one thing, measure what happens, and add a table or chart of your readings: that is what turns a model into your own investigation.
Free ebooks for this project
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