Participate in nature’s patterns

Complex patterns in nature form from simple rules, repeated over time and space. A striking example of this is a process that yields beautiful space filling spirals, as in daisies, pine cones, dandelions, sunflowers, and pine apples. 

See our movie on how floral spirals form. You see these spiral patterns emerging on pine cones, pineapples, sunflowers, daisies, roses, and more. In another movie, we describe a computer model of spiral formation, and apply it to making spirals.

Each rose petal is placed at the golden angle (137.5 degrees) relative to the prior petal.  You can download the below instructions for doing this rose with paper, and then see the same rose petal project in simple Scratch code.

Notice how the petals each point in a new direction, never in the same direction. The petal tips thus fill gaps, fill the spaces, very efficiently. This space filling is a special property of the golden angle, which is based on an irrational number with no denominator. 

Remember, if you subscribed to Beautiful Discovery Box, you get a full free course on floral spirals

The same golden angle creates beautiful spirals in sunflowers, pine cones, and pine apples. Learn how to code these spirals in Scratch (simple drag and drop commands), NetLogo, or Python. 

Here’s our spiral model in NetLogo and also in Python. Here’s  our simplest spiral models in Scratch.  Our more interesting Scratch spiral model is colored to highlight the spirals.  This one uses moving florets. Using this Scratch model you can see how these kind of spirals appear in pine cones.

Below is an example of code for the spirals in Python.

Click “Code” below to see code. Click “Run”  to see the spirals output, then enter a number of colors.

Below is a sheet to download and color. It’s output from the above golden spiral models. 

Here’s a movie of the above golden angle spiraling in Scratch. Here are those golden spirals with skulls and hearts. Movie of spirals with 32 arms. In this Scratch model, the turn angle is 1/32 of a full 360 turn, so the spirals turn into 32 arms.

One beautiful thing about the golden angle is that it so perfectly fills up space in a growing spiral of florets (tiny flowers, future seeds, on a flowerhead). That’s because the underlying number is the golden ratio, an irrational number that cannot be expressed as a whole number fraction. If we could write it out completely, the number seems to have infinite decimal places  1.618033988749…. on and on without end. Another famous irrational number is pi, 3. 14159…. on and on. But pi is not quite so irrational as the golden ratio and when it spirals around in an angle based on pi, it doesn’t fill space as perfectly as does the golden ratio turn (golden angle), as we show here. But you need not understand the comparison of pi and the golden ratio to get a feel for the beauty of the golden angle.  

Here’s our movie that applies this model of golden angle spirals to a craft with pinecones based on our computer model of spirals in NetLogo and a similar spiraling model in Python. We see such golden spirals in sunflowers, daises, dandelions, pine cones, pineapples and more.

There is another way to make golden angle spirals which we show here. This version of the model helps show how the golden angle, approximately 137.5 degrees, fills space.

With the model you can connect the dots with lines, forming triangles, to help show how each golden angle turn (137.5 degrees) creates a point that fits in between previous points.

Using a “forever” loop, we can use Scratch to show the great variety of spirals that can be found in flowers.  

Armed with Scratch code, you can make 3D spheres of golden angle spirals, essentially a painted pine cone. A variant is a cone of spirals, which looks much like a Christmas tree decorated with lights, which you can rotate in different directions. 

Dry out a pine cone and it opens up. The seeds may drop out, as they did for us. Run water over a cone (like rain) and the cone closes up. Here are photos we took of our wet cone every few minutes, showing the cone closing. Looks more like a pineapple when closed.
Spirals on dandelion seed head

We found that the pine cones always have a number of  clockwise spirals and a number of counter-clockwise spirals that were Fibonacci numbers. Above are some pine cones we painted to make it easier to see the number of clockwise and counterclockwise spirals. These are Fibonacci sequence numbers.

This is also true of daisies and dandelions, the number of spirals are Fibonacci numbers.

See if pineapples have numbers that are from the Fibonacci sequence. Try painting a pineapple in the same way as you did for the pine cone. You can also pin colored paper pieces to the pineapples. We tried painting spirals going one way and pinning paper on the spirals going the other way. Use your imagination.

See our page on pine cone spirals. There we describe more about the science of golden angle spirals and offer a simple craft, a pine cone mobile (doesn’t require any knowledge of the science).  Our subscription box explains more about golden angle spirals, along with our coloring book, pine cones, paints, and some materials for making a pine cone mobile. There’s also a demonstration of instant origami spirals (see our video below).