What it shows
Circles packed in a hexagon pattern leave less gap than circles in a square grid. The pen stand shows the hexagonal packing, and drawings on graph paper let 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 cans and pipes are stacked this way. In a large pattern, the share of the area covered by circles is about 90.7 percent in the hexagonal packing and 78.5 percent in the square packing. Honeybees use hexagonal cells for a related reason: hexagons fit together with no gaps and use the least wax for their walls.
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.
Step 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.
On graph paper use compasses to draw circles of radius 2 cm touching in a square grid, and mark the 4 cm by 4 cm square cell around one circle (area 16 cm²).
Draw circles of the same size touching in the hexagonal pattern, and mark the regular hexagon around one circle that is 2 cm from its centre to each side (area about 13.9 cm²).
Count the squares of empty space left in each cell; expect about 3.4 cm² in the square cell and about 1.3 cm² in the hexagon cell.
Work out the percentage of each cell covered by its circle (circle area about 12.6 cm²) and compare the two percentages.
Write which pattern wastes less space and which one the pen stand uses.
What you should see
The circle covers about 78.5 percent of its cell in the square pattern and about 90.7 percent in the hexagonal pattern, so the hexagonal arrangement wastes less space, and the pen stand holds pens in seven neat compartments.
Viva questions
A teacher or a judge will ask how it works. Try to answer each question before you open it, then compare.
01Why do bees use hexagons?
Hexagons tile with no gaps and use the least wax.
02Which circle packing is denser?
The hexagonal packing.
03How many circles touch one central circle in hexagonal packing?
Six.
04Why 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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