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January EDU Project: Fractal Tree In The Forest


January EDU Project: Fractal Tree In The Forest

Welcome to the first EDU project of 2017 by our own Dr. David Thornburg! Winter is still in full effect, but hopefully this project will have you thinking about a verdant, green tree on a beautiful spring day. Without further adieu, here's Dr. Thornburg's Lesson...

Followers of this blog are no strangers to BlocksCAD, and this is the tool we'll be using today. The best way to access BlocksCAD is through its own site:


Good programming languages (like BlocksCAD) support recursive programming. Instead of looping a single set of commands, a recursive module uses replicas of itself as a part of the module definition. Once a module is called, the values of variables are local to that instance of the module. The result is the ability to create complex objects that can't be made with simple loop instructions.

In celebration of this capability, we'll create a famous recursive structure, a fractal tree. Fractals are complex structures in which any part of the structure is a replica of the structure as a whole. An entire branch of mathematics (chaos and complexity theory) is devoted to the exploration of these structures.

Consider a tree, for example, branches are similar to each other, but their size changes as smaller branches grow out of larger ones.

A similar pattern is found in ferns where the whole structure is similar to the smaller parts that make up the plant.

I became so engaged by these structures that I wrote a couple of books on the topic back in the 1980's: Discovering Apple Logo : An Invitation to the Art and Pattern of Nature, Addison Wesley, 1983.

The language I wrote about, Logo, supported recursion so it was a natural choice. Of course, in those days, the graphics were low resolution 2D images on an Apple II computer ― a far cry from what we can do with BlocksCAD and a 3D printer today.


As our tree is being built, the size and length of branches will change, just as they do with real trees. We start by creating a set of global variables used in our structure.

Next we define a recursive module called tree. This module uses three local variables ― the starting branch length and width and the depth (number of levels) of the tree. These three variables will have their local values changed during use of the module. All other variables will keep their global values.

There are two things to notice in this module. First, you'll see that we don't use the Centered option for the cylinder. This makes it easy to get all the branches connected. Second, the cylinder is tapered from its previous radius at the bottom to the new one at the top ― just like the branches of many real trees.
Once this module has been created, we can create our final instructions.

When this BlocksCAD program is run, it produces the following tree. I changed the rendering color by clicking on the Render window box with the color swatch and choosing green as the new color since it looks more tree-like. Of course the color of your printed object will be chosen by the filament you use.

Here's our printed tree.

This tree also looks like a stalk of broccoli. You should do some research to see why this is.


Now Taking Submissions For STEAMtrax PLUS


Now Taking Submissions For STEAMtrax PLUS

A letter from our Director of Education, David Thornburg, PHD.

To all teachers enthusiastic about 3D printing,

Have you ever thought about a really cool lesson on a subject you teach that involves 3D printing? 

Polar 3D is building the STEAMtrax PLUS Teacher Idea Center — a place where you can put your amazing curriculum ideas and lesson plans to share with the world!  Your lessons will be indexed by subject area and grade level with a focus on STEAM topics - although submissions in other areas are more than welcome as well.  While our focus is on the K-12 world, we'd love to see college level projects too!

Our goal is to make creating great curriculum as simple as possible for you.  We have a template for you to use for your lesson, to bring consistency to the offering.  Here's how the process works:


  1. Download and use the template here to build your lesson.
  2. Send it to Dr. Thornburg, our Director of Education.
  3. We will choose a professional mentor to help you tweak and publish your lesson if needed.
  4. When approved, your lesson will be posted along with printable files for others to use and comment on in the new Polar Cloud 2.0.




There is no cost associated with any of these activities, and your materials will be released under a Creative Commons Copyright.  We'll be watching to see how folks resonate with your ideas, and look forward to promoting & rewarding those with the most engaging content!

If you have any questions, please send them to Dr. David Thornburg.  We can't wait to see what great ideas you come up with!


David Thornburg, PHD

Director of Education, Polar 3D



Introducing: Educational Projects By David Thornburg


Introducing: Educational Projects By David Thornburg

Today we're starting something that we think you'll love. This is the first in a monthly series of projects in the STEAM (science, technology, engineering, arts, and mathematics) education areas written by David Thornburg, that we want you to try with your class! They involve the design and construction of projects that can be used to explore a variety of topics. 

Our emphasis is on the design process and we'll primarily focus on two excellent free design tools suitable for students at most grade levels: Tinkercad and BlocksCAD. These entry-level tools are cloud-based, and accounts are free. 

As you work with these projects, we want to hear from you! Especially if you have cool ideas for new projects, and ways you have extended these projects in your own schools. Have fun and enjoy David's first project: Sea Shell Spirals!



Sea shells are amazing objects to explore.  Young children enjoy finding them, exploring their colors and shapes.  They even hold large ones to their ears to “hear the ocean.”  This project explores the spiral shape of many shells, with special focus on shapes similar to that of the chambered nautilus― a cephalopod whose shell is an equiangular spiral.  In this project, you will build 3D-printable shapes of quarter circles with different sizes based on Fibonacci numbers (connecting this activity to math) that, when assembled, produce a spiral similar to that of the chambered nautilus.  These shapes will be made using Tinkercad.

Fibonacci numbers are generated from a simple calculation and show up quite commonly in nature.  Starting with the numbers 1 and 1, the next number in the series is the sum of the previous two numbers.  This gives us 1, 1, 2, 3, 5, 8, 13, 21, and so on.  Our project will use quarter circle pieces whose width is based on Fibonacci numbers.  We will start with two quarter circles 10 mm across, and add more from there.

If you'd like to download the entire lesson as PDF for usage in the classroom, use the link below!

SEA SHELL SPIRALS : Step-By-Step Design

1. Using Tinkercad, drag a cylinder to the workplane and drag the ruler tool onto this plane as well.  The reason for using the ruler is that it lets us enter numbers for sizes as text rather than having to drag a corner of a shape by hand which sometimes is hard to control accurately.

2. Make the cylinder 20 mm in diameter by entering these numbers in the text field associated with each side on the x-y plane.


3.  Change the height of the cylinder to 5 mm and drag a box onto the workplane.

4.  Set the box dimensions to 25 mm on a side, with a height of 7 mm.

5.  Use the black triangle at the top to lower the box beneath the workplane by 1 mm.

6.  Using the arrow keys, move the box so the left edge is at the center of the cylinder.  If you don't get it the first time, you can fix it later.


7.  Copy and paste the box and use the arrow keys so the upper edge is at the center of the cylinder.


8.  Select both boxes and turn them into holes by clicking on the hole button in the Inspector window near the top right of the workplane window.


9.  Select all three objects and choose the Group button at the top of the screen.  This will leave you with just the visible quarter circle.


10.  If the dimensions are not exactly 10 mm by 10 mm, Ungroup the object, move the boxes to the right place, and Group everything again.

11.  Change the piece height to 3 mm.  This completes your basic piece, and everything else is made from copies of this.


12. Copy and paste your first shape, move it to the right and click on the rotate symbol (the red curved arrow shown below) and drag the piece by 90º.

13.  Use the arrow keys to move the piece to the right, leaving a small gap with the first piece.


14.  Copy and paste this piece and move it down a bit.

15.  Change the dimensions to 20 mm, rotate the piece 90º, and move it to clear the pieces you already made.


16.  Repeat this process with a size of 30 mm.


17. Repeat this process with a size of 50 mm.


18. Repeat this process with a size of 80 mm.


19.  You now have enough pieces to fill your build plate.  Change your view so you are looking at the model from the top and adjust any pieces that are too close together.


20.  Export your model as an STL file for printing.

Printing & Assembling The Model

Print a set of shapes in any color you wish and put them together into a spiral.  To my eye, this pattern is very pleasing to the eye.  As you can see, using Fibonacci numbers makes everything fit perfectly!


Things To Do & Notice

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     Chambered Nautilus Shell Section Source: Wikipedia

Chambered Nautilus Shell Section Source: Wikipedia

Assemble the shapes as shown in the photograph.  What would the dimension of the next shape be?  If you want, design and build the next shape in the sequence.  It will be so large that only one of them fits on the build plate.

Compare your shape with that of the chambered nautilus shell cross-section shown below.  In what way is this shape similar to that of the Fibonacci spiral you created?  In what way is it different? 

Fibonacci numbers are quite common in nature, but they don't seem to exactly match up for the spiral in the chambered nautilus shell.  Why do you think that is?

What about garden snails?  Are their shells similar to what we made?  If you have access to a shell collection, see how many different examples of spirals you can find.

Spirals of the kind we just made are called equiangular spirals.  For further learning, you can explore the math of these spirals with some help from Wikipedia.


Google's top education expert predicts what schools will look like in 50 years


Google's top education expert predicts what schools will look like in 50 years

Google's Lighthouse Dome Classroom - Credit: Google For Education Roadshow

Google's Lighthouse Dome Classroom - Credit: Google For Education Roadshow

We love this article by Business Insider that talks with Jonathan Rochelle, head of product management for Google Apps for Education. It affirms the same feelings we have here at Polar 3D, the next 50 years of education are going to be exciting, cutting-edge and collaborative. Some quotes we particularly like —

“…schools are poised to become highly collaborative spaces, thanks to the advent of virtual and augmented reality. Instead of needing to meet in the same physical space, kids could work on long-term projects remotely and interact through online platforms.”

This mindset is exactly why we’ve worked hard on the Polar Cloud, building collaboration into its core - allowing teachers and students to share & manage their 3D objects and 3D printers together. The same can be said for our STEAMtrax curriculum, which directly encourages 3D printing & collaboration to solve STEAM based problems as a team. 

“But Rochelle knows technology can't transform education on its own — it takes a smart application of cutting-edge products to help kids learn. As the world gets more technologically advanced, it's partly up to teachers to make sure kids feel comfortable using the latest products effectively.”

We also couldn’t agree more with this, and that’s why we are fully committed to training & promoting the teachers of the future to better use our products like STEAMtrax and Polar3D Printers through our professional development  & support offerings. Even direct features, such as Google Classroom Integration and Chromebook + Mobile compatibility, make using the Polar Cloud with lesson plans easy and comfortable — while still being on the cutting edge of technology. With a future as exciting as this, we can't wait to get back to school!


Polar 3D and PIE Partner to Develop New, Comprehensive 3D Printing Curriculum


Polar 3D and PIE Partner to Develop New, Comprehensive 3D Printing Curriculum


"There are very few 3D printer manufacturers who aren’t involved in education to some degree. It’s a refrain we hear again and again: teaching kids about 3D printing is critical, the earlier the better, because it’s the future of every industry. While many companies have included educational outreach in their business plans, others seem to have been formed for the express purpose of putting 3D printers in classrooms. Polar 3D is one of those companies. The Cincinnati-based corporation launched their first 3D printer only a little over a year ago, and they made it clear from the very beginning that their primary goal was to get students interested in 3D printing. "So far, they’ve been aggressively pursuing that goal through partnerships with leading educational and youth-oriented organizations like the Boys & Girls Clubs of America, with whom they teamed up last year to build new 3D printing labs at several club locations. Now Polar 3D is collaborating with Partnership for Innovation in Education (PIE) to develop new 3D printing curriculum for students in Cincinnati-area schools."

Read the complete story on 3D Print Magazine.