Some subjects sit beautifully at the crossroads of several disciplines. Fibonacci, Da Vinci and the golden ratio are two of them.
What begins as a simple mathematical sequence can lead us into medieval history, Renaissance art, geometry, biology, botany, architecture, astronomy, nature study and even some fascinating questions about whether mathematics is something we discover in nature or something we use to describe what we observe.
It is also a wonderful subject for homeschooling because it lends itself so naturally to hands-on learning. Children can count, measure, draw, build, photograph, paint, investigate plants, study famous artworks and even design their own mathematical artwork.
And perhaps most importantly, this subject can grow with a child.
A five-year-old can investigate patterns in numbers and arrange objects in Fibonacci-like groups. A ten-year-old can calculate ratios and construct a golden rectangle. A teenager can investigate the mathematics behind the golden ratio, examine claims about its appearance in nature and consider whether some commonly repeated examples are actually mathematically justified.
That makes Fibonacci and the golden ratio an unusually rich topic for a mixed-age homeschool.
As this post is part of my Renaissance Unit Study, and I’ll link to all my Renaissance posts throughout. Click below for more about the Renaissance resources we have used
Fibonacci, Da Vinci and the Golden Ratio
Who Was Fibonacci?
Fibonacci was the nickname of Leonardo of Pisa, an Italian mathematician who lived around 1170–1250.
His best-known mathematical work in the Western world is associated with a book called Liber Abaci, published in 1202.
The book helped introduce European readers to mathematical methods that used the Hindu-Arabic numeral system. These numbers and calculation methods eventually became the system we use today.
This is important because Fibonacci’s famous rabbit problem was not originally intended as a mysterious explanation for patterns throughout nature.
Instead, the rabbit problem was a mathematical puzzle.
It asks us to imagine a pair of rabbits reproducing according to a particular set of rules. If each pair produces another pair after a certain amount of time, and the rabbits continue reproducing according to those rules, how many pairs will there be?
The resulting numbers form this sequence:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89…
The rule is wonderfully simple:
Add the previous two numbers together to find the next number.
For example:
1 + 1 = 2
1 + 2 = 3
2 + 3 = 5
3 + 5 = 8
5 + 8 = 13
And so on.
For younger children, you can introduce this as a growing number pattern.
For older children, it becomes much more interesting and is an excellent introduction to Fibonacci, Da Vinci and the golden ratio.
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Fibonacci, Da Vinci and the Golden Ratio Activity 1: Build the Fibonacci Sequence
Ages: 5–8
Gather counters, buttons, blocks, Lego pieces, stones or anything else that can be counted.
Create groups containing:
1
1
2
3
5
8
13
Ask your child to work out what comes next.
Rather than simply writing the numbers down, let them physically build the sequence.
For example:
- 1 block
- 1 block
- 2 blocks
- 3 blocks
- 5 blocks
- 8 blocks
Ask:
“What do you notice?”
Then ask:
“How did we get from one group to the next?”
This turns an abstract mathematical sequence into something children can actually see and touch.
Extension for older children
Ask them to continue the sequence to 20 or 30 terms.
Then ask them to investigate:
- What happens to the numbers?
- How quickly do they grow?
- What happens when you divide one Fibonacci number by the previous number?
This leads naturally to the golden ratio.
Click here to learn all about our project-based learning focusing on Leonardo Da Vinci
What Is the Golden Ratio?
The golden ratio is a special mathematical relationship.
It is approximately:
1.618033988…
It is traditionally represented by the Greek letter φ (phi).
If two quantities are in the golden ratio, the relationship can be written as:
a/b = (a+b)/a
where a is the larger quantity.
This can look intimidating at first, but the idea is surprisingly simple.
Imagine a line divided into a long section and a shorter section.
The golden ratio occurs when:
whole line ÷ long section = long section ÷ short section
In other words, the relationship between the larger and smaller parts is the same as the relationship between the whole and the larger part.
This is why the golden ratio is sometimes described as a particularly harmonious or self-repeating proportion.
The Fascinating Connection Between Fibonacci and Phi
Here is where our simple Fibonacci sequence becomes much more interesting.
Take two consecutive Fibonacci numbers and divide the larger by the smaller.
For example:
8 ÷ 5 = 1.6
13 ÷ 8 = 1.625
21 ÷ 13 ≈ 1.615
34 ÷ 21 ≈ 1.619
55 ÷ 34 ≈ 1.618
89 ÷ 55 ≈ 1.618
The numbers get closer and closer to the golden ratio.
This is one of the beautiful surprises of the Fibonacci sequence.
It means that although Fibonacci’s original rabbit problem was not created as an explanation of the golden ratio, the sequence has a very close mathematical relationship with it.
Fibonacci, Da Vinci and the Golden Ratio Activity 2: Discover Phi for Yourself
Ages: 10–16
Give your child a list of Fibonacci numbers:
| Fibonacci number | Previous number | Ratio |
| 2 | 1 | 2 |
| 3 | 2 | 1.5 |
| 5 | 3 | 1.667 |
| 8 | 5 | 1.6 |
| 13 | 8 | 1.625 |
| 21 | 13 | 1.615 |
| 34 | 21 | 1.619 |
| 55 | 34 | 1.618 |
Ask them to calculate the ratios themselves.
Then ask:
“What number does the ratio seem to be approaching?”
This is a lovely example of mathematics that children can discover rather than being told.
Further investigation
Ask older students to research why the ratio approaches approximately 1.618.
They can explore the quadratic equation:
x² = x + 1
Rearranging:
x² − x − 1 = 0
Solving this equation produces:
x = (1 + √5) / 2
which is the golden ratio.
For younger children, there is absolutely no need to introduce the equation. Simply observing the pattern is enough.
Golden Rectangles
One of the most famous geometric constructions associated with the golden ratio is the golden rectangle.
A rectangle is called a golden rectangle when the ratio of its longer side to its shorter side is approximately:
1.618: 1
So, for example, if the short side measures 10 cm, the long side would measure approximately:
16.18 cm
A golden rectangle can be divided into a square and a smaller rectangle.
Remarkably, the smaller rectangle has the same proportions as the original rectangle.
This means that the process can continue.
Remove the square.
Another square can be removed.
Then another.
And another.
The remaining rectangles become progressively smaller while retaining the same basic proportion.
This is where our spiral begins to emerge.
Fibonacci, Da Vinci and the Golden Ratio Activity 3: Make a Golden Rectangle
Ages: 8–16
Give your child graph paper, a ruler, a pencil and a calculator.
For a simple construction, start with a rectangle measuring approximately:
10 cm × 16.2 cm
Ask your child to:
- Draw the rectangle.
- Divide off a 10 cm square.
- Look at the remaining rectangle.
- Continue dividing off squares.
- Mark the corners of the squares.
- Draw a smooth curve through the appropriate corners.
The resulting curve approximates a golden spiral.
Younger children can draw and colour the squares.
Older children can calculate the dimensions at every stage.
Fibonacci Squares and Golden Spirals
There is another wonderful way to visualise the connection.
Take squares whose side lengths correspond to Fibonacci numbers:
1, 1, 2, 3, 5, 8, 13…
Arrange them beside one another.
The squares fit together to create an expanding rectangular pattern.
If you draw a quarter-circle through each square, you create a familiar spiral-like shape.
It is important, however, to make a distinction here.
The spiral formed from Fibonacci squares approximates a true mathematical golden spiral. The two are closely related, but not identical.
This is a useful opportunity to teach older children that mathematics often requires precise language.
Fibonacci, Da Vinci and the Golden Ratio Activity 4: Build a Fibonacci Spiral
Ages: 5–16
This can become a fantastic whole-family project.
Younger children
Give them square pieces of coloured paper with side lengths based on Fibonacci numbers.
They can arrange them into a spiral-like pattern.
Middle ages
Ask children to measure and cut their own squares.
Older children
Have them calculate the side lengths and investigate the relationship between the Fibonacci spiral and the mathematical golden spiral.
Once the construction is complete, draw curved lines through the squares.
Then display the finished piece as mathematical artwork.
Leonardo da Vinci: The Artist Who Was Also a Scientist
The story becomes even more fascinating in the Renaissance.
Leonardo da Vinci was not simply a painter.
He was an artist, engineer, anatomist, inventor, observer and mathematician.
Leonardo filled notebooks with observations of the natural world.
He studied:
- human anatomy
- birds
- horses
- water
- plants
- geology
- machines
- perspective
- geometry
- light and shadow
For Leonardo, art and science were not necessarily separate subjects.
Careful observation was essential to both.
An artist needed to understand how the human body was constructed.
An engineer needed to understand forces and movement.
A painter needed to understand perspective, proportion, light and geometry.
This makes Leonardo an ideal person to study when exploring the connections between mathematics, science and art.
How Leonardo da Vinci Applied the Golden Ratio in Art
Leonardo illustrated a mathematical work called De Divina Proportione, written by Luca Pacioli and published in 1509.
The book explored the mathematical proportion now commonly called the golden ratio.
The illustrations in Leonardo’s geometric work demonstrate his extraordinary ability to combine mathematical precision with artistic observation.
This period also saw a growing interest in proportion, perspective and geometry among Renaissance artists.
How The Vitruvian Man Embodies the Golden Ratio
Leonardo da Vinci’s Vitruvian Man is a fascinating example of the Renaissance interest in using mathematics to understand human proportion.
The figure is drawn within both a circle and a square, illustrating relationships among different parts of the human body.
Leonardo carefully recorded measurements such as the relationship between the head, arms, legs and overall height, showing how geometry could be used to study the structure of the human form.
While the drawing is often associated with the golden ratio, don’t assume every proportion in the Vitruvian Man was deliberately based on φ; its stronger, better-documented connection is to the proportional ideas described by the Roman architect Vitruvius.
How the face of the Mona Lisa aligns with the Golden Ratio
Some modern analyses of Leonardo da Vinci’s Mona Lisa suggest that parts of her face fit proportions close to the golden ratio, or Divine Proportion. For example, researchers sometimes compare the relationships between the face’s length and width, the positions of the eyes, nose, and mouth, and other facial measurements. These comparisons can make an interesting mathematical exercise, but they do not prove that Leonardo deliberately constructed the face using the golden ratio. This makes the Mona Lisa a useful example of the difference between noticing a mathematical pattern and having evidence that the artist intentionally used it.
Understanding the Golden Spiral: From Math to Masterpiece
The golden spiral offers a beautiful way to turn the mathematics of Fibonacci numbers into something we can see. Start by arranging squares with side lengths based on consecutive Fibonacci numbers (1, 1, 2, 3, 5, 8, 13 and so on). As the squares join, they form a series of increasingly larger rectangles. Drawing a quarter-circle through each square creates a spiral-like curve that closely approximates a mathematical golden spiral. A true golden spiral is a logarithmic spiral whose size increases by a constant factor as it turns, while the Fibonacci-square construction is an approximation. This distinction reminds us that mathematics can produce both exact relationships and visual models that closely represent them.
This offers an intriguing way to view the work of Leonardo da Vinci. In paintings such as the Adoration of the Magi, Leonardo carefully arranged figures, gestures, lines and areas of light and shadow to create movement through the composition and guide the viewer’s attention. Modern viewers can sometimes find spiral-like or golden-ratio patterns within these arrangements, and overlaying a golden spiral can be an interesting way for children to investigate how their eyes move around a painting.
However, we should be careful not to claim that the Adoration of the Magi was deliberately constructed along an exact logarithmic golden spiral. The painting is unfinished, and there is no simple proof that Leonardo used the golden spiral as a compositional template. Instead, it offers a wonderful opportunity to explore how geometry, proportion, perspective and artistic composition can work together and to ask whether a pattern we discover was intentionally designed or is simply something we can see in the finished artwork.
An Important Caution When Studying Leonardo and the Golden Ratio.
The golden ratio is frequently claimed to have been deliberately used throughout Leonardo’s paintings, including the Mona Lisa and The Last Supper. Some analyses identify proportions that are close to the golden ratio. Still, historians and mathematicians do not agree that every modern golden-ratio overlay proves that Leonardo deliberately constructed those paintings using phi.
That uncertainty actually makes this an excellent homeschool discussion.
Rather than simply telling children, ‘Leonardo used the golden ratio everywhere,’ we can ask, ‘What evidence would we need to prove that?’
That is a much more interesting question.
It introduces children to an important part of scientific and historical thinking:
A pattern appearing in something does not automatically prove that the pattern was deliberately put there.
Fibonacci, Da Vinci and the Golden Ratio Activity 5: Become a Renaissance Art Detective
Ages: 8–16
Choose a famous Renaissance painting.
Give your child a ruler and a reproduction of the painting.
Ask them to investigate:
- Where is the horizon?
- Where are the main figures?
- Where are the important objects?
- How is the painting divided?
- Where does your eye travel first?
- Can you identify rectangles?
- Can you find repeated proportions?
- Are there strong horizontal or vertical lines?
- Does the composition appear symmetrical?
- Where does it deliberately break symmetry?
Then let them create their own geometric overlay on tracing paper.
The important question
Ask:
“Does finding a golden rectangle prove that the artist used the golden ratio?”
The answer is no.
This is a wonderful opportunity to teach children the difference between:
observation → hypothesis → evidence → conclusion
Fibonacci, Da Vinci and the Golden Ratio Activity 6: Draw Like Leonardo
Ages: 5–16
Leonardo was famous for careful observation.
Take the children outside with sketchbooks.
Choose one natural object:
- a leaf
- flower
- feather
- pine cone
- stone
- shell
- branch
Ask them to draw what they actually see rather than what they think the object “should” look like.
Younger children
Concentrate on:
- shape
- lines
- texture
- colour
Older children
Add:
- measurement
- symmetry
- proportion
- scale
- scientific labels
- cross-sections
- observations
Encourage children to write notes alongside their drawings.
This combines art, biology, observation and scientific recording.
Fibonacci in Nature
This is the part of the subject that children find most exciting.
Fibonacci numbers can appear in some patterns in nature, particularly in the arrangement of leaves, petals and seeds.
One reason is that plants often need to arrange leaves or seeds efficiently to access light and space.
The arrangement of plant structures around a growing stem can sometimes be described using mathematical models involving Fibonacci numbers and the golden angle.
The golden angle is approximately:
137.5°
It is related to the golden ratio.
When repeated around a central point, an angle close to 137.5° can produce arrangements in which new structures are positioned to reduce overlap.
This is particularly interesting when looking at plants.
But again, precision matters.
Not every plant contains Fibonacci numbers.
Not every spiral in nature is a golden spiral.
And not every beautiful pattern is evidence that nature is “following the golden ratio.”
Nature is much more complicated than that.
That complexity makes it even more interesting to investigate.
Fibonacci, Da Vinci and the Golden Ratio Activity 7: Hunt for Fibonacci in the Garden
Ages: 5–16
Go outside with notebooks.
Look for:
- daisies
- sunflowers
- pine cones
- pineapples
- artichokes
- leaves
- flower petals
- seed heads
Count what you find.
For example:
How many petals does the flower have?
How many visible spiral directions can you find?
And how many leaves appear around a section of the stem?
Record the results.
Older children can create an observation table and compare different plants.
Younger children can count, draw and photograph their discoveries.
The most important part is not finding the “right” Fibonacci number.
It is learning to observe and record evidence.
Fibonacci, Da Vinci and the Golden Ratio Activity 8: Make a Sunflower Investigation
Ages: 7–16
If you have access to a sunflower head, examine the seed arrangement.
Look closely at the spirals.
Can you find spirals running:
- clockwise?
- anticlockwise?
Count them.
Then compare your numbers.
Older students can investigate whether the counts correspond to neighbouring Fibonacci numbers.
They can also discuss why spiral arrangements might be advantageous for packing seeds efficiently.
Science extension
Ask:
“If plants are trying to fit many seeds into a limited space, what arrangement would make efficient use of that space?”
This turns a simple nature walk into an introduction to:
- packing
- geometry
- optimisation
- plant growth
- spatial organisation
Fibonacci, Da Vinci and the Golden Ratio Activity 9: Fibonacci Pine Cone Investigation
Ages: 5–12
Find a pine cone.
Look at the scales.
Can your child trace spirals around the cone with their finger?
Try using a pencil to mark one spiral.
Then mark another spiral travelling in the opposite direction.
Count them.
You can make a simple recording sheet:
Object:
Number of spirals one way:
Number of spirals the other way:
Are the numbers related?
Younger children can draw the pine cone.
Older children can compare several pine cones and graph their results.
Fibonacci, Da Vinci and the Golden Ratio Activity 10: Create Fibonacci Art
Ages: 5–16
Now bring everything together.
Give each child:
- paper
- ruler
- pencil
- compass
- coloured pencils or paint
Begin with Fibonacci squares.
Construct the spiral.
Then turn the mathematical construction into artwork.
Children might:
- colour each square differently
- create an ocean inside the spiral
- turn the spiral into a galaxy
- transform it into a plant
- make an animal from the geometry
- incorporate leaves and flowers
- create an abstract composition
The important thing is that the underlying structure is mathematical.
The finished piece becomes a visual representation of mathematics.
Fibonacci, Da Vinci and the Golden Ratio Activity 11: Make a Fibonacci Collage
Ages: 5–16
Cut squares from paper using Fibonacci dimensions.
For younger children, keep the numbers simple:
1, 1, 2, 3, 5
For older children:
1, 1, 2, 3, 5, 8, 13, 21
Arrange the squares into a Fibonacci pattern.
Then create a collage around it.
You could make:
- a Fibonacci butterfly
- a Fibonacci flower
- a Fibonacci galaxy
- a Fibonacci tree
- an abstract geometric picture
Ask older children to label the mathematical dimensions.
Fibonacci, Da Vinci and the Golden Ratio Activity 12: Measure Yourself Like Leonardo
Ages: 8–16
Leonardo’s famous Vitruvian Man provides an excellent opportunity to explore proportion.
Have children investigate human proportions.
Measure:
- height
- arm span
- hand length
- foot length
- head height
- distance from floor to navel
- distance from shoulder to fingertips
Record the measurements.
Then calculate ratios.
For example:
arm span ÷ height
Ask:
“Are these the same?”
“Are they approximately the same?”
“How much variation is there between different people?”
This introduces children to:
- ratios
- measurement
- averages
- variation
- human biology
- scale
It also demonstrates an important scientific principle:
Real humans vary.
Mathematical models can describe patterns without every individual matching the model.
Fibonacci, Da Vinci and the Golden Ratio Activity 13: Design Your Own Golden Rectangle
Ages: 10–16
Challenge your child to design something using golden rectangles.
Possible projects include:
- a bookmark
- a picture frame
- a card
- a poster
- a book cover
- a small painting
- a room layout
- a logo
- a phone wallpaper
First, calculate the dimensions.
For example:
If the short side is 8 cm, calculate the long side:
8 × 1.618 ≈ 12.94 cm
Then construct the rectangle.
Once the mathematical construction is complete, let the child design something beautiful inside it.
This is an excellent example of mathematics as a design tool rather than simply a worksheet exercise.
Fibonacci, Da Vinci and the Golden Ratio Activity 14: The Golden Ratio Photography Challenge
Ages: 8–16
Take photographs using a camera, phone or tablet.
Challenge children to compose photographs using different layouts.
Take one photograph with:
- the subject in the centre
- the subject using the rule of thirds
- the subject positioned using a golden-ratio-inspired composition
Compare the photographs.
Ask:
“Which composition catches your eye first?”
But resist telling children that one arrangement is automatically more beautiful.
Instead ask:
“What did you notice?”
This turns the activity into an investigation of visual composition rather than a lesson in what children are supposed to find attractive.
A Wonderful Question to Discuss
One of my favourite aspects of this topic is that it allows us to ask children a question that has nothing to do with memorising Fibonacci numbers.
Ask:
Did we discover the pattern, or did we create the pattern?
This question has several layers.
Fibonacci numbers are a mathematical sequence that humans define through a particular rule.
The patterns we observe in plants are physical phenomena.
The golden ratio is a mathematical relationship.
A spiral in a plant is a biological structure.
Sometimes mathematics provides an extraordinarily useful way of describing what we see.
But that does not necessarily mean that nature is consciously using mathematics.
This distinction can lead to fascinating discussions about mathematics, science and philosophy.
Fibonacci, Science and the Problem of Overclaiming
This is also a wonderful subject for teaching children how to evaluate information.
The internet contains many claims that the golden ratio can be found everywhere:
- the human body
- famous paintings
- architecture
- music
- galaxies
- hurricanes
- shells
- DNA
- financial markets
- and almost everywhere else imaginable.
Some mathematical relationships are genuinely present in natural systems.
Others are approximate.
Some examples are disputed.
And some are simply the result of drawing a golden-ratio rectangle over an image until something appears to line up.
This doesn’t make the golden ratio less interesting.
In fact, it makes it more interesting.
A child studying this topic can learn to ask:
How was the measurement made?
What exactly was measured?
How close does it need to be to phi?
Was the pattern predicted or noticed afterwards?
Could another ratio fit equally well?
Is there evidence that the artist or designer intentionally used the ratio?
These are sophisticated questions for older children, and they provide an excellent introduction to scientific reasoning.
The Most Important Lesson
Perhaps the greatest lesson in this entire study isn’t actually the Fibonacci sequence.
It isn’t even the golden ratio.
It is learning to look carefully.
Leonardo da Vinci spent enormous amounts of time observing.
Fibonacci noticed patterns in numbers.
Scientists observe patterns in nature.
Mathematicians describe relationships.
Artists look at proportion, balance, shape and movement.
And children can do all of these things.
The beauty of homeschooling is that we don’t have to put each subject into its own little box.
A single pine cone can become a science lesson.
Then it becomes a mathematics lesson when we count its spirals.
It becomes an art lesson when we draw it and a writing lesson when we describe what we observed.
It becomes a history lesson when we compare our observations with those made by naturalists and artists hundreds of years ago.
And it becomes a lesson in critical thinking when we ask whether the pattern we found really means what we think it means.
That is the part of Fibonacci, Leonardo and the golden ratio that I find most exciting.
Mathematics gives us a language for patterns.
Science teaches us to test those patterns against evidence.
Art teaches us to see them.
And children get to explore all three at once.
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