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 and more. In another movie, we describe a computer model of spiral formation, and apply it to a spirals coloring book.

Learn how to code the golden angle spiral model in either Scratch (simple drag and drop commands), NetLogo (very flexible and thorough for studying emergence), or Python (an advanced language used in all industries). We created the model in all three programs.

Here’s our spiral model in NetLogo (the most flexible model) and also in Python. Here’s  our simplest spiral models in Scratch. Below are screenshots of the code in Scratch.  Here’s a more interesting Scratch spiral model that’s colored to highlight the spirals.  And here’s one that uses moving florets to make the spirals. Using this Scratch model you can see how these kind of spirals appear in pine cones.

Below is a coloring sheet you can download and color. This is output from the above models of golden spirals. 

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.

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.

Try our Patterning Spirals: Coloring Book with Two Meditations and Illustrated “Science of Floral Spiraling.” We also have a Kindle version, which is not a coloring book, but includes many color images of spirals, the illustrated science of floral spiraling, and two meditations. The meditations promote awareness of symmetrical patterns in nature.
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. We explain why in our coloring book and in our subscription box. 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 comes with colorful cards that explain more about golden angle spirals, along with our coloring book, pine cones, paints, and some materials for making a pine cone mobile, and a paper sunflower plant with leaves.
The subscription box on spirals includes the origami leaves, several printed images of spirals, and pipe cleaners that you can use to connect the spiral disk and leaves . Cut out your choice of spiral and attach it to a pipe cleaner with tape or glue.
If the paper circle needs more support you can attach it to two pipe cleaners twisted together. Attach a paper leaf to a pipe cleaner. Repeat for a total of two or three leaves. Twist all the pipe cleaners together to make a single stem that holds the flower and leaves together on one stem.

Spread out the wire ends at the bottom  to make a base. You can place the wire stems in a small flower pot, or cover with paper. This is just a fun craft for adults to complete with young children. Yet it visually emphasizes that both the leaves and spiral florets on the flower head have patterns that optimize space. The spirals on the flower head tightly packs the florets, which later turn into seeds. The folds on the leaf allow the leaf to tightly fold.

All the above is the theme of one of our popular boxes that comes with your subscription to Beautiful Discovery. The box includes our golden ratio spirals science booklets, spirals coloring sheets, real pine cones, computer models of the spiral formation, crafts and much more.