Fibonacci Spiral – 30 STEM Links a Week

SketchUp is a free and fun program for 3D modeling. You can use SketchUp to design just about anything, from furniture to a dream bedroom to an entire city.

There is a downloadable version called SketchUp Make, which you can get here. But there is also a web-based version which works right in your Internet browser. This version is called my.sketchup, and to use it just go to www.my.sketchup.com.

The Fibonacci spiral is the basis for a lot of patterns you’ll find in nature. See how these are all similar?

In this project, we’ll create the spiral itself. Then, in the next issue, we’ll use the spiral to create some interesting flower patterns.

When you open SketchUp, you’re in this view, with Josh standing on the ground. He isn’t needed in this model, so press E for the eraser, and click on any of Josh’s edges.

To create the spiral, we want to start in Plan view (also known as top view or bird’s eye view). Click the Views icon on the right side.

In the Views window, click the Top view icon.

Now we’re looking straight down at the “ground.”

Press R for the Rectangle tool and draw a square – be sure to click when you see the “Square” popup.

Then press C to active the Circle tool. For best results, this circle should be as smooth as possible – SketchUp circles are actually polygons. Before clicking anywhere, look in the Sides field at the lower right corner; the default number of sides for a circle is 24. To make a smoother circle, type 60, which appears in the Sides field, and press Enter. (You never have
to click in this field; just type and the number will appear.)

Now create the circle by clicking Point 1 for the center, and Point 2 for a point along the circumference.

Activate the Eraser and erase the entire circle except for the quarter-circle inside the square.

This little object is the building block we’ll use to create the spiral. Because it will be repeated, and because we want it to act as a single object, it should be made into a component. Press the Spacebar for the Select tool, and drag a selection window around the whole square.

Right-click on any selected face and choose Make Component. Assign a name or accept the default name, and click Create. The component is highlighted as a single object, which means it’s selected – leave it selected.

This component will be copied multiple times, and each time the copy will be made with the Rotate tool (press Q). Tap the Ctrl key (PC) or the Option key (Mac) to add a plus sign to the cursor, which means a copy will be made.

The Rotate tool requires three clicks. The first click defines the center of rotation – click Point 1 shown below.

The second click defines the start of the rotation (zero angle) – click Point 2. Then click Point 3 to complete the rotation. You’ve just made a 90-degree rotated copy. (In the Angle field, you’ll see 90 listed.)

Now the copied component is selected, which is good since this is the one that will be rotate-copied next.

Rotate should still be the active tool, and press Ctrl / Option again to make another copy. This time click Points 1, 2, and 3 to create the next copy.

The new copy is now selected, and it needs to be enlarged to join the other two components. So while it is selected, press the S key to activate the Scale tool. You’ll see a set of eight green drag handles around the component.

Click the top right drag handle, and move it until that corner meets the top left corner of the other two components. The entire component is now twice as large as before, which you can see in the Scale field. Again, leave this component selected; this is the one that will be copied next.

Use Rotate and the three points shown below to make the next copy.

Then use Scale on the new component to make it large enough to reach the previous ones. The scale value this time is 1.5.

The scale values will vary the first few times, but will eventually all be ~1.62.

Why this number?

1.62 is the ratio of sides of a golden rectangle; this ratio equals (1+ square root of 5) / 2.

Keep going with the same steps, each time rotate-copying and scaling the newest component. When rotating, the protractor should always go to a corner of one of the original two components, on the side where the new component will go.

The scale value will eventually approach 1.62, after the 4th or 5th copy. Keep going until you have as many copies as you like – nine or ten is a good number.

We now have the Fibonacci spiral, made up of the quarter-circles inside each component. To reduce what we have now to just those arcs (no faces), right-click on any component and choose Edit Component. The component you’re editing appears in a dotted-line box, while everything else in the model appears faded in the background.

Use the Eraser to erase all four edges of the square, leaving only the arc. This change is made to each component.

When finished, right-click in blank space outside the component, and choose Close Component. Now you’re left with just the spiral.

For the next part of this project (coming in the next issue), it’s not very useful to have each part of this spiral locked inside a component. So press Ctrl + A (PC) or Cmd + A (Mac) to select all of the components, then right-click on any of them and choose Explode.

Use the Save button at the top left corner, and stay tuned for the next part!

What we’ll be doing in Part 2 is making rotated copies of this spiral, and using these copies to make Fibonacci flowers.

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THE FIBONACCI SEQUENCE, SPIRALS AND THE GOLDEN MEAN

What is the Fibonacci Spiral?

Golden Ratio Coloring Book

Golden Ratio in Nature

Understanding the Fibonacci Spiral

Why does the Fibonacci spiral appear in nature?

Fibonacci sequence in music