Maths IA · Geometry and trigonometry · topic 28 of 40

Golden ratio claims: do they hold up? · IB Maths IA idea: guide, sample graph and step-by-step plan

very popular topic: make it YOURS

Some people see the golden ratio in everything. You're going to measure, and find out whether it is really there.

Difficulty●○○ Gentle
KitPhotos or prints, ruler, spreadsheet
TimeAbout 12 to 15 hours
FitsAA and AI, SL and HL
  • ✓ you like art or architecture
  • ✓ you want to test a claim, not repeat it
  • ✓ you can handle measurement, ratios and some statistics

The big idea

← test it, don't repeat it

The golden ratio, about 1.618, turns up in claims about paintings, buildings and faces. This is among the most common IA topics, so test the claim; do not repeat it. Choose a work you care about, decide what is measured and where the edges are, and ask whether the ratios are closer to the golden ratio than to other ratios (3 : 2, √2). Measurements have uncertainty, so a ratio is never exactly 1.618. Strong explorations decide in advance what would count as 'close enough' and test it with numbers.

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 →

Ask what 'close' means

Fix a tolerance in advance, using the uncertainty in your measurements.

→ Your conclusion becomes a test with a stated criterion, not an opinion.
Compare with rival ratios

Test the golden ratio against 3 : 2, 5 : 3, √2 or the ratio of a standard paper size.

→ You may find that a different ratio fits as well or better, and that is a perfectly good result.
Use a sample, not one picture

Measure several works by one artist, or buildings from one period.

→ A sample mean and a confidence interval for the ratio. Check the sample is not chosen to fit.
Test the mathematics behind it

Show where the number comes from (a ratio equal to the ratio of the rest), and how the Fibonacci numbers approach it.

→ A short proof from an equation, which makes the number meaningful rather than magical.
Measure inside the work

Check internal divisions: the horizon, a doorway, a window.

→ More measurements but also more decisions about where an edge is; you must justify each.
Model the measuring error

Estimate how far your ratio could be from the true one just from the ruler and the photo's angle.

→ Error propagation in a ratio, and a clear sense of how much a measurement can really tell you.

Your Maths IA research question: run the check

say what 'close enough' means

Write your own question first. Then make sure it ticks every box:

  • Names the works or buildings (or the sample) and says why you chose them.
  • Says exactly which lengths are measured, and where the edges are (frame, canvas or image).
  • Names the ratio claimed and at least one rival ratio, so there is something to test.
  • Says what would count as support for the claim, with a tolerance that follows from your uncertainty.
  • Is narrow enough for 12 to 20 pages and not a tour of all of art history.
  • Can be answered with numbers and a stated precision.

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The picture to draw

A rectangle with a square cut off. Draw yours on your own work, and mark exactly which edges you measure.

whsquarecut off a square: is the rest the same shape? w/h = 1.618…?which edges? frame, canvas, image?

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.

Title page, question and rationale~1 page
Your question; the work you care about and what the claim says.
Aim and plan~0.5 page
The ratios you will test and what would count as 'close enough'.
The mathematics of φ~2 pages
Where the golden ratio comes from (an equation and its solution), its link to Fibonacci, and a short proof of one property.
Measuring~2 pages
How you measured, from what (print or photo), the instrument resolution and where the edges are; a table of lengths with units.
Uncertainty in a ratio~2.5 pages
A derivation of the uncertainty in w/h, and a graph of how it depends on the size measured. At HL this is where the HL-level maths must show.
Testing the claim~3 pages
Ratios against φ and rival ratios, with a confidence interval or a test and a tolerance chosen in advance.
A wider sample~2 pages
Several works, with a comparison and a check of whether the sample was fair.
Reflection throughout~1.5 pages
What the measuring cannot tell you, how your edge choices matter, and what it means that a ratio is 'close'.
Conclusion~0.5 page
Answer with numbers, and say whether the aim was met.
Bibliography and appendixnot counted
Sources, images and extra tables.

What your Maths IA graph could look like

can your ruler tell? this tells you

Made-up data, to show you the form of a strong graph, not the answer. Yours will look different. example only

1020304050607080901000.00.51.01.52.02.53.03.54.0Width of the work, w / cm (ruler ± 0.2 cm)Uncertainty in the ratio w / h, in %your derived uncertainty in w / h (h = w / φ)the gap between 1.6 and φ: 1.1%invented spread of repeated measurementsbelow the dashed line, canyou tell 1.6 from φ at all?ILLUSTRATIVE · MADE-UP DATA
  • The curve comes from error propagation in a ratio: the axes say what is measured and in what units.
  • A dashed line marks a difference you want to detect, so the graph answers 'can this measurement tell?'.
  • Horizontal bars show the ruler uncertainty on the width; vertical bars show the spread of repeats.
  • The graph says that small objects cannot settle the question, which is useful to know before you start.
  • Another graph to consider: the ratio w / h for each of your works, with error bars, and a horizontal line at φ.

How close is 'close enough'?

rival ratios make it a real test

Rank the rival ratios by how far they are from φ. If your measurements cannot separate them, say so. This is arithmetic you can check yourself.

√2 (A-series paper), 1.41412.6%3 : 2 (1.5)7.3%5 : 3 (1.667)3%8 : 5 (1.6)1.1%can your data separate these?% difference from φ = 1.618…ILLUSTRATIVE · MADE-UP NUMBERS

Your game plan

Tick steps off as you go. We'll remember on this device.

0 of 11 done

  1. Pick a work you care aboutChoose a painting, building or object with a link to you. Write the claim exactly as you found it, with its source.
  2. Learn where φ comes fromFind the equation behind the golden ratio and solve it. Write the proof of one property in your own words.
  3. Predict and set a toleranceSay which ratio you expect and how close counts as 'close enough', before you measure. Explain the tolerance.
  4. Trial run (one afternoon)Measure one work twice, with two edge choices. How much does the ratio change? Is your photo square-on?
  5. Decide what and how to measureList the lengths, the instrument and its resolution, and your rule for where an edge is. Add rival ratios to test.
  6. Collect and processRecord raw lengths with units. Work out the ratios and their uncertainties. Show one worked example.
  7. Test the claimCompare the ratios with φ and the rival ratios using your tolerance, a confidence interval or a test. Check any conditions.
  8. Widen the sampleAdd more works. Is your sample fair, or did you choose the ones that fit?
  9. Reflect as you goWhat did your edge choices change? What does 'close' really mean for a human artist?
  10. Write, then cut12 to 20 pages. One worked calculation per type, then tables. Check notation, axis labels and units.
  11. Check before your teacher's draftDoes the conclusion say whether the claim held, with numbers and a stated precision?

Maths you'll need

Traps to avoid in this Maths IA

  • Starting from 'it is the golden ratio' and measuring until it fits. Test it with a rival ratio.
  • Choosing edges after seeing the numbers, so the ratio comes out right.
  • A photo taken at an angle, so the measured ratio is wrong before you start.
  • The same Fibonacci and golden spiral story as every other golden ratio IA (personal engagement).
  • Quoting ratios to four decimal places when the ruler cannot support more than two.
  • At HL, stopping at SL-level ratios and a mean: add a proof, a confidence interval or a test, and check its conditions.
  • A sample picked to fit the claim, with no way to say what a failure would look like.

More Maths IA ideas like this

Make sure it's yours

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.

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