3D printed fractals at JMU 3SPACE

  • University
  • Mathematics

Pioneer Professor Laura Taalman, (a.k.a. mathgrrl), reviews a multi-week study of fractals by general education math students in the JMU 3D printing classroom.

The James Madison University 3SPACE classroom kicked off the Fall 2017 semester with ten new 3D printers incorporated into our classroom stations: five Ultimaker 2+ printers, and five Ultimaker 3 printers.


This new equipment enabled us to increase our capacity to 24 students and to offer courses that require more challenging print jobs, which made it possible for us to offer two new 3-credit general education courses: MATH 103 - The Nature of Mathematics and ART 300E - 3D Printing and the Creative Community. In these courses, we’ll be exploring fractals, four-dimensional representations of objects, extreme remixes, and everything our students can dream up!

In this article, we’ll talk about how we implemented the first unit in the MATH 103 course: exploring fractals. For information on all of our JMU 3SPACE courses and workshops, see the 3SPACE website.

First fractal prints

The 3SPACE classroom is not part of our university Engineering or Design departments; we serve the general university community. This means that the students that come to us typically have no 3D printing or design experience. In the case of MATH 103, which satisfies the JMU general education math requirement for non-majors, our students also don’t know much math. In fact, most of them will freely volunteer that they don’t like math at all! So, we started slow, having students find fractals on Thingiverse to download and print.

The purpose of this assignment was to help students become familiar with fractals and fractal properties, and to train them to use the 3D printers. A secondary goal was to have students participate in the online 3D printing community; they use the community to find designs, and then they give back to the community by documenting their prints on our public class Wordpress blog, in the First Fractal section.

While students printed and documented their first fractals, we spent class time discussing exactly what makes something a fractal, and what fractal properties were illustrated by each of their printed fractals. Students also started reading Falconer’s book Fractals: A Very Short Introduction. We also watched some YouTube videos on fractals, including this excellent Numberphile video on the Dragon Curve:

Dragon Curve
Watch the video

Here are some of the great fractal models that our math students found to print: A Nautilus shell, a Pythagorous tree, and a Vicsek fractal cross.

Pythagorous tree
Vicsek fractal cross

Student-designed fractals

The next fractal assignment for MATH 103 was for students to design their own, brand-new fractals. We gave the students a brief introduction to Tinkercad and then let them struggle to produce something that they felt had fractal properties. We purposely didn’t give them much direction here, in the hopes that the students would really have to think hard about “what makes a fractal a fractal”.

The students came up with some very interesting things! Here are three of them. First, a simple Whirly Cross Fractal that gets smaller and smaller as you go to the inside:


Second, a beautiful self-intersecting Pyramid Spiral Fractal that twists in on itself (this one is courtesy of “Control-D” in Tinkercad):

self-intersecting Pyramid Spiral Fractal

And third, a Mandala Fractal where each ring is ⅘ of the size of the one outside it:

Mandala Fractal

Perimeter, area, and volume

Eventually, we had to buckle down and do some calculations. We discussed in class how various fractals might have infinite perimeter but finite area, or infinite surface area but finite volume. A good starting point for this topic is Randy Dobson’s video on the Koch snowflake:

Koch Snowflake
Watch the video

Students attempted to compute perimeter, area, or volume of their new fractal creations, but were permitted to fall back to doing computations on one of their earlier “First Fractals” if necessary. (The complexity and self-intersections of some of the students’ original creations sometimes made it too difficult to do these calculations.)

Here is some student work finding the area of Level 2 of the Pythagorean fractal pictured earlier:

fractal math

During class, we talked a bit about geometric series and what the behavior of these fractal measurements after infinitely many iterations. Since our MATH 103 students come from many different backgrounds and most of them have not had calculus before, the challenge as an instructor is to pick out just that one piece of math that students need, and to try to put it into context. Nearly all of the students’ fractal calculations ended up being related to geometric series, so we focused on understanding finite and infinite geometric series. Pretty much everything boiled down to understanding the following:

geo series


The final step in our fractal exploration was to think about fractal dimension. Fractal dimension can be difficult to calculate, but it is easy for certain types of self-similar fractals. We used the method explained in the first half of the excellent video Fractals Are Typically Not Self-Similar from 3Blue1Brown.

Fractals are typically not self-similar
Watch the video

With this method, students identify a linear scaling factor that shrinks the self-similar fractal onto an exact copy of itself, and then count how many of those smaller copies it takes to make up the entire fractal.  Then they use the formula (1/scaling)^D = 1/(number of copies) and use logarithms to solve for D. Here is an example of a student calculation, for the dimension of the Vicsek fractal pictured earlier:


What’s next?

We’re still just six or seven weeks into the semester, so we have lots of time to explore new mathematical objects with 3D printing. We just started our second unit, where students choose interesting topics from Matt Parker’s book Things To Make and Do in the Fourth Dimension and then design and print 3D models that illustrate those topics. Students have chosen topics ranging from Prince Rupert’s Cube and the Reuleaux Tetrahedron to Trefoil Knots and the Borromean Rings. You can see all of their projects-in-progress at the Open Projects category on our class blog.

Read more education blogs

  • College of the Desert 3D printing club

    Feel the Action: Learn about how a college is changing lives in ways we can touch!

    A 3D printing club emerges at College of the Dessert to help make ideas accessible to all students

    Testing Quadcopter

    How the tools we use influence the designs we make

    Michael Delaney writes about his iterative process of integrating electronics and 3D printing in his programmable quadcopter project.

    Ultimaker Met

    Bringing 3D printing to the Metropolitan Museum of Art

    Part of the mission from Ultimaker North America's Community team is to help educators get up and running with 3D printing and to help them find meaningful ways to integrate the technology into their curriculum while celebrating their successes. When we w

  • Smithsonian Learning Lab

    3D printing as part of the Smithsonian Learning Lab

    Pioneer Christopher Sweeney shares the work he did with 3D printing and the Smithsonian Learning Lab to create collections of educational resources, organized and structured for teaching and learning.

    Harvard calculus courses use 3D printed models to engage students

    Harvard calculus courses use 3D printed models to engage students

    What is the best way to teach concepts like surface area and volume relating to mathematical objects in a three-dimensional world? The curricular design team at Harvard University thinks the...

    Mathematica Tea Light Holder

    3D Design in Mathematica: Tea Light Holders

    If it’s happening in Ultimaker’s world, you can find out about it here. 3D printing stories about inspiring moments, original 3D printed projects and much much more.

  • Code your 3D designs with Tinkercad’s new Codeblocks app

    Code your 3D designs with Tinkercad’s new Codeblocks app

    Tinkercad released Codeblocks last June, and now that it's out of Beta, Pioneer Rob Morrill tells us how he uses it.


    Modifying an STL with Tinkercad

    When Ultimaker Pioneer Alex Larson contacted me about the Ultibot-D project, I was super excited. As a teacher and parent, I am a huge cheerleader for risk. I printed the base models at ⅕ scale and gave it to my students to explore

    Ultimaker at Camp

    Math Camp: Having fun doing stuff

    Pioneer Dr Toni Szymanski writes about summer fun at camp with 4th and 5th graders, math, and 3D printing.

  • 3D printing for glass


    Guest blogger Astrida Valigorsky writes about combining the old and the new at Timothy Belliveau's GlitchCraft class where students combined 3D printing and glass blowing.


    Using Basecamp to manage your classroom/makerspace projects and print queue

    Pioneer Andrew Woodbridge uses Basecamp to organize his students' projects, and he explains how you can too.

    3D printing in math and chemistry

    A unique 3D printing collaboration between mathematics and chemistry faculty

    Passing it on. After integrating 3D printing into her own math courses, Kristen Schreck helps spread 3D printing across disciplines at Saint Xavier University.

  • World monument project

    Living world monuments assignment

    Pioneer Joanne Barrett shares about a middle school project that combines 3D printing, Augmented Reality, History, and Art.

    parts in cura (1)

    My reintroduction to Netfabb

    A recent NetFabb workshop convinced the writer that they can't live without this application. See why it's time to take another look at Netfabb.

    hero wrenchs

    Wrench Engineering

    Inspired by NASA printed in space wrench, Pioneer Rob Morrill gave his fifth graders a design challenge to design their own real-world tool.

  • goblet project

    Integrating 3D printing and The Goblet Project

    Pioneer Chris Hanusa shares one way he integrated 3D printing into his Integral Calculus class

    Finland’s mobile libraries

    3D printers in the public library: Finland ahead of the curve

    US Professor Joshua Pearce, sponsored by Fulbright Finland for Research, shares his first impressions of Finnish libraries.


    Lessons in letting go - releasing student energy speeds creation and use of our art and innovation lab

    Pioneer John Nordell enlisted his students to put the Art and Innovation Lab together. The result was one of the best experiences he has had as an educator.

  • toni teaching fusion

    Teacher training for 3D printing

    Pioneer Dr. Toni Szymanski thought that 3D printing could engage students in math classes. To test this out, she had to learn all about 3D printing first.

    ceramic stamps

    Digital fabrication informs ceramics decorative process

    Pioneer Young Kim blends his classical arts background with digital fabrication to create a ceramic project that incorporates 3D printing.


    Modeling rocks at iCREATE

    iCREATE's Tiffany Huang shared an iCREATE success story about Jackie Zheng.

  • Empathy map

    Elementary-College Engineering Design partnership

    Pioneer Matthew Wigdahl writes about how his fifth graders and local undergraduate engineering students learn from each other.


    Introducing Ultimaker’s Core Lessons Set for STEAM Educators

    Designed by the Ultimaker North America Community Team, our new Core Lessons: STEAM Set is a resource for educators who need inspiration and ideas when they bring 3D printing into their classrooms.


    Empower one MakerGirl, empower the world

    how the group traveled 10,000 miles around the country to help bring 3D printing workshops to over 1,000 young girls. In this week's post we catch up with the organization that is still working to create a new generation of confident, creative women leade

  • Georgia Connections Academy

    Building a mobile maker space: part 2—up and rolling

    So what exactly does a Mobile MakerSpace look like? That's the question I had to ask myself once I began the project of bringing 3D printing technology to virtual students. When we started this project, we knew we needed something that would be easy to

    Building a mobile makerspace: part 1— getting started

    Building a mobile makerspace: part 1— getting started

    Pioneer Wendy Aracich is putting together and implementing a mobile MakerSpace for her virtual school of 4000 students spread across the state of Georgia.


    Tactile Problem/Solution Bank Community Project

    3D modeling and printing should be accessible of every educator so that they may offer their students tactile means to understand spatial concepts.

  • Design Engine Box

    Design Engine community project

    We want to challenge educators and students to help evolve the Design Engine game. We want to see how you're using or modifying the game with your students, and we want to incorporate your ideas into the next edition.

    coca cola upcycle

    Upcycling community project

    Upcycling challenges students and educators to use their creativity and 3D printing skills to breathe new life into a few familiar objects.


    Construct3D to Kamehameha Ed Tech Conference

    Last year Pioneer Greg Kent traveled from Hawaii to North Carolina to attend Construct3D 2017. We thought we'd share his reflections with you now since we recently announced Construct3D 2018

  • Plastic Ocean by Kevin Krejci

    Ocean Plastic Community Project

    The Ultimaker Community Team will be launching a series of interdisciplinary projects over the next few months that challenge students to research, explore, design, and 3D print. Ocean Plastic is the first project of this series.


    3D printed fractals at JMU 3SPACE

    Pioneer Professor Laura Taalman, (a.k.a. mathgrrl), reviews a multi-week study of fractals by general education math students in the JMU 3D printing classroom.


    Using drones and 3D printing to develop design thinking during a summer robotics camp

    Pioneer Yuriy Drubinskiy writes about his experience leading a summer program and how creating drones with 3D printing brings form, structure, and design together.

  • The Starter Pack launch event at Digital Harbor Foundation

    Introducing the Ultimaker Design Engine Starter Pack

    Presenting the Ultimaker Design Engine Starter Pack: a game created to provoke, inspire, and entertain students, educators, 3D designers, artists, and engineers of all experience levels!

    first puzzle cube and package to be made on a 3D printer entirely out of PLA

    Davidson Desktop Doohickeys: Puzzle cubes

    Pioneer Adam Davidson writes about a project in his curriculum that is a rite of passage for his high school's engineering program students. He explains at how it started and what caused it to change

    soft robot mold

    3D printing with UMaine Bioengineering students

    If it’s happening in Ultimaker’s world, you can find out about it here. 3D printing stories about inspiring moments, original 3D printed projects and much much more.