Maths IA · Geometry and trigonometry · topic 26 of 40
Honeycomb: why do bees use hexagons? · IB Maths IA idea: guide, sample graph and step-by-step plan
a classic story: make your angle personalEveryone says bees are brilliant mathematicians. You're going to test the claim and find out what is actually true.
- ✓ you like shapes and patterns
- ✓ you enjoy turning a story into an exact argument
- ✓ you can handle trigonometry and some algebra
The big idea
← the maths must go beyond the shapes!Only three regular polygons tile a flat surface: triangles, squares and hexagons. For a fixed cell area, which needs the least wall? Derive the area of a regular polygon from its perimeter, compare the three, and then ask what happens as the number of sides keeps growing. The shapes are easy, so the maths has to be deeper than the shapes: derive, justify and generalise rather than quoting the result. Good explorations prove why only three regular polygons tile and push on to a case you choose.
Make it yours: pick one twist
Copying a classic won't show your choices. Pick one of these (or invent your own) and your IA becomes yours.
Swipe for more twists →
Split a regular n-sided polygon into triangles and get its area from the perimeter using trigonometry.
→ One general formula covers every case, so your comparison is an argument, not three separate calculations.See what happens to the area for a fixed perimeter as n increases without limit.
→ A limit appears (the circle). Series expansions make the limit exact and show how fast you approach it.Work out which regular polygons can meet at a point with no gap.
→ A short, rigorous angle-sum argument you can state and justify in your own words.Compare irregular hexagons, or cells with curved walls.
→ Optimisation with more than one variable; a good chance to explain what you assume and what you cannot prove.Measure cells in a photo of real honeycomb, with a scale in the picture.
→ Your own data and uncertainty in the measurement, and a real test of how close nature is to the model.Think about the shape of the cell's base, or about packing spheres.
→ A harder geometry problem; keep it to one idea you can fully explain.Your Maths IA research question: run the check
fix what is held constant, in the RQWrite your own question first. Then make sure it ticks every box:
- Says exactly what is being optimised: least wall for a fixed area, or most area for a fixed perimeter.
- Says which shapes are allowed (regular only? any polygon? curved walls?) and why.
- Names the method that will carry the maths, such as trigonometry, a limit or a derivative.
- Can be answered with a derivation and a justification, not only by comparing three numbers.
- Is small enough for 12 to 20 pages with the working shown.
- Does not assume the answer: bees might be using the best shape or they might not.
The picture to draw
Three regular shapes with the same perimeter. Draw yours with your own side lengths and show where each formula comes from.
How your Maths IA could be laid out
The usual order for a topic like this, what each part needs to show, and roughly how much of your 12 to 20 pages it takes. The words are all yours.
Your question; what made you doubt or like the 'bees are mathematicians' story.
What you will compare and the maths you will use.
Which regular polygons tile and why, shown with an angle argument in your own words.
A derivation of the area of a regular n-sided polygon, with a clear diagram and every symbol defined.
A table and a graph of area (or wall length) against n, with the tiling shapes marked.
The limit as n grows, or an irregular case. At HL this is where series, a derivative-based argument or a proof must show.
A photo, a scale, your measurements and how close the cells are to the model, with uncertainty stated.
What the model leaves out (wax thickness, the bees' real behaviour), and what each result would mean for a claim about bees.
Answer the question and say whether the aim was met.
Every source for the maths and for the biology.
What your Maths IA graph could look like
formula AND measurementsMade-up data, to show you the form of a strong graph, not the answer. Yours will look different. example only
- The axis names the fixed quantity (perimeter) so the reader knows what is being held constant.
- A formula curve and measured points together: the comparison is the point.
- The circle is drawn as a limit line, so the reader sees what the curve approaches.
- Shapes that tile are marked, linking the graph to your argument.
- Another graph to consider: wall length needed for a fixed cell area, plotted against n.
Your game plan
don't skip the paper-shape trial!Tick steps off as you go. We'll remember on this device.
0 of 11 done
- Read and doubtFind where the 'bees and hexagons' claim comes from. Write down what exactly is being claimed, and what would show it true or false.
- PredictBefore any calculation, say which shape you think wins and by roughly how much.
- Trial run (one evening)Cut out a triangle, square and hexagon from the same length of string or paper strip. Compare the areas by measuring. Does the order match your prediction?
- Prove what tilesWork out which regular polygons fill a corner exactly. Write the argument so a classmate could follow it.
- Derive the area formulaSplit the polygon into identical triangles. Get area from perimeter and n. Check it against n = 3, 4 and 6.
- Compare and look for a patternTabulate and graph area for many n. What do you notice as n grows?
- Choose your own deeper questionPick one: the limit, an irregular shape, or real honeycomb. Say why you chose it.
- Do the deeper mathsJustify each step. At HL, ask where a proof, a series or a derivative can carry the argument, rather than a table of numbers.
- Test against realityMeasure real cells from a photo with a scale. State the uncertainty, and compare with the model by a number.
- Reflect as you goWhat does the model leave out? What would change if the wax had thickness?
- Write, then cutShow one worked example of each calculation type, then use tables for the rest. Check notation and axis labels.
Maths you'll need
- Trigonometry (AA HL) →
- Maclaurin series (AA HL) →
- Differentiation (AA HL) →
- Maths IA criteria, in plain words →
Traps to avoid in this Maths IA
- Stopping at 'the hexagon wins' with three numbers. That is prior-level maths and a very common idea.
- Quoting the area formula without deriving it, so criterion E shows no understanding.
- At HL, relying on SL-level trigonometry only: add a limit, series or proof argument so the maths has real sophistication.
- Claiming bees 'calculate' the best shape. Say what the maths shows and what it cannot show.
- A photo of comb without a scale, so measurements have no units or uncertainty.
- Copying the standard honeycomb treatment from a website with only the numbers changed (personal engagement).
- Mixing up perimeter and area as the thing held fixed.
More Maths IA ideas like this
- Building height from angles: how accurate is it?●○○
- Golden ratio claims: do they hold up?●○○
- Shortest route: how good is a quick heuristic?●●○
This guide shows how a topic like this is usually approached. It is not an IA. Your question, choices, data, analysis and words must be your own: change something, tell your teacher where the idea came from, and follow your school's rules on outside help.