College-level math exploration in 3D at JMU 3SPACE

Also published at Ultimaker Education

At James Madison University, every student must satisfy a math requirement. For those students who are not majoring in subjects that require specific mathematics courses, one option is our general education MATH 103 course, “The Nature of Mathematics.” The goal of MATH 103 is to expose students to mathematics and mathematical thinking in a broad sense, with whatever topics the instructor chooses to cover.

Teaching math with 3D printing

This semester I taught MATH 103 at JMU using 3D printing and design as the basis for exploring fractals, infinity, and other mathematical curiosities. My overarching goal for the course was for students to “learn to learn” on their own, and to practice thinking like a mathematician: asking questions, getting stuck, starting all over again, and eventually telling a mathematical story or making a mathematical argument.

The students learned Tinkercad, OpenSCAD, and other 3D design tools to construct mathematical objects from scratch, and then investigated those objects and presented their work in written and oral form. They used WordPress as their main form of documentation, so you can actually see all of their projects at our class website.

Last semester I wrote about the MATH 103 students’ first projects with fractals in the Ultimaker Education article 3D printed fractals at JMU 3SPACE. That semester is now over and I can report on their other projects and explorations.

Projects from the Fourth Dimension

The main project for the course was to work in groups to investigate various topics chosen from Matt Parker’s excellent book Things to Make and Do in the Fourth Dimension.

Things to maker and do in the fourth dimension

The students chose the projects themselves and then pitched them to me for approval. My goal in approving projects was to help students identify and research the mathematics they would be explaining through the objects they created, and to push them to do more than they thought they could do mathematically.

For example, one group studied Prince Rupert’s Cube, the famous construction whereby a cube can pass through another cube of the same size (or, in fact, actually one of a smaller size!). As a warmup, they used Tinkercad to recreate a well-known form that can pass snugly through a triangular hole, a square hole, and a circular hole, depending on its orientation:

Three holes

In the Prince Rupert’s cube construction, a cube passes through another cube through a hexagonal cross-section. This can be difficult to represent in a physical print, because the cube being passed through has parts where it is connected only at single points. The students had the ingenious idea to make loops holding the pieces of the passed-through cube together. Here’s what their final design looked like, with a red cube passing through a green cube of the same side length:


Part of the mathematical story that this group of students chose was to calculate the largest red cube that could pass through a green cube of side length 1 unit. It turns out that the red cube can actually be about 6% larger than the green cube, and still pass through! Here’s some of the mathematics that the students wrote up to explain this:

Rupert Calculations

To read the entire story on Prince Rupert’s cube, check out the students’ final blog post on the course website.

Each group’s project was different; topics included hinged dissections, equitable cake cutting, knot theory, and polyhedra. Some groups had to learn Fusion 360 or OpenSCAD to complete their projects, some had to perform certain calculations or proofs, and others had to provide historical background, depending on their topics.

All groups were expected to design a 3D print from scratch, share their work on Thingiverse, and compose blog posts on the design and mathematics behind their projects. To see all of the students’ blog posts for this assignment, check out the Open Projects section of our class blog.

Blog list

Fractal Carpets

After having the students work on many different projects, it seemed like a good idea to have them switch gears and all do different versions of the same, more guided, project. Returning to our original study of fractals, the students were given some OpenSCAD code and asked to modify it to create their own “Fractal Carpets”. These fractals were basically 4×4 variants of the well-known Sierpinski Carpet. My goal was to have each student construct and 3D print their own unique Carpet fractals, and then calculate the surface area and dimension of their individual Fractal Carpets.

Here are some of the Fractal Carpets that the students created. They named these the “Space Invader”, “Tetris”, and “Centerless” Fractal Carpets, from left to right:


The figures above show the “third level” of each fractal. Students printed levels 1, 2, 3, and, if possible, level 4. However, their calculations and investigations always had to do with the infinite/final level of their fractal.

Below is the OpenSCAD code I gave to the students. The students for the most part had no prior experience with OpenSCAD, so we did a short introduction to coding before getting started on this project. Students only had to make simple modifications to the template code in order to make a unique Fractal Carpet.

<code>// mathgrrl 4x4 fractal carpet maker for MATH 103
// template cells
template = [
// parameters
level = 3;      // SET TO 1, 2, 3, or maybe even 4
thickness = 1;  // do not change
width = 100;    // do not change
// renders
// module for making the fractal
module fractal(n,cells,diam){
 if (n>0) {
 for (i = cells)
 if (n==0) {

Other than the code, the students were given the instructions below. The math instructions refer back to the surface area and dimension calculations that students first learned about in their initial fractal projects, earlier in the semester.

Design instructions

  • STEP 1: Comment in/out some number of boxes from the template
  • STEP 2: Render with n=1; re-choose boxes if necessary to make connected
  • STEP 3: Render with n=2; re-choose boxes if necessary to keep connected
  • STEP 4: Render with n=3 and n=4 (any higher will CRASH)

3D printing instructions

  • Record: Make a note of which cells you commented out to make your design
  • Export: Run “F6” and Save STL files for Levels 1, 2, 3
  • Print: Levels 1, 2, 3 (each should take 1-2 hrs with Racer X, no support, no adhesion)
  • Print: If possible, try Level 4! (may work for certain open/blocky designs? or not)

Math instructions

  • PROBLEM 1: Compute the surface area of the “k^th level” for k=1, 2, 3, 4
  • PROBLEM 2: Compute the surface area of the “infinity level” using geometric series
  • PROBLEM 3: Compute the fractal dimension of the “infinity level” using scale/copies equation

The students also 3D printed grids to help illustrate the self-similarities of their Fractal Carpets. For example, here is the Space Invaders Fractal Carpet with a grid to illustrate that the level 3 fractal is made up of smaller copies of the level 2 fractal:

carpet grid


One major goal of this MATH 103 class was to help students learn how to communicate mathematics. At the end of the semester the students put together a public showcase and presented their projects to various student and faculty visitors in the Department of Mathematics and Statistics. They showed off their fractal creations, constructed giant Dragon Curves, cut up some cake, and put together hands-on demonstrations of their mathematical explorations. Here are just some of the pictures from the event:

Showcase Carpet dragon

What’s next?

At the end of the semester, the JMU 3SPACE classroom finally packed up and said goodbye to its original location. We have now moved our gear into the main campus library in a much more high-visibility area, and we are excited for the future (and to have windows!). Here’s our equipment moved into the new space, just waiting to be set up for next semester…

Library Move In

Wish us luck and stay tuned for our future endeavors!



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3D printed fractals at JMU 3SPACE

Also published at Ultimaker Education

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.

JMU's Ultimakers

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:

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:

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:


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.

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.



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Creating a Community of 3D Educators at the Construct3D Conference

See the original post at Shapeways Magazine

These days 3D printing and digital fabrication are key elements of college and K-12 curricula. Students “learn through making” while developing crucial 21st-century skills in hardware, software, and 3D design. These efforts are happening at hundreds of institutions around the United States. The thing is, there’s often just one faculty or staff member leading the 3D effort, working alone. If you’re lucky, maybe you’re part of a small team.

What happens when you put hundreds of those leaders in the same place? We found out at the Construct3D Conference at Duke University last weekend!

Duke University campus tower

The mission of Construct3D is to bring together college and K-12 educators already active in 3D printing and give them a forum to share information and build a community. The conference, which grew out of Ultimaker’s Pioneer Program, was hosted this year by Duke University’s Innovation Co-Lab. Sponsors included Autodesk, ShopBot, FormLabs, LulzBot, 3DPrinterOS, Adobe, and, of course, Shapeways. These companies brought workshops, talks, and exhibits to Durham. And more than 250 educators from 150 different colleges, K-12 schools, libraries, and makerspaces showed up to share what we know and learn from our peers.

Ultimaker machines and prints

Ultimaker machines and prints // Photo: Aleph Objects, Inc.

LulzBot machines and prints // Photo: Aleph Objects, Inc.

Data-Driven Rhino/Grasshopper Workshop with Andres Gonzalez // Photo: Josh Ajima

Dale Dougherty, founder and CEO of Maker Media, kicked off Construct3D at a collaborative makerspace at Duke called The Foundry with a rousing call to action for making and sharing in education, and a vision for an educational climate where students can explore their passions. The following morning, Skylar Tibbits, founder and co-director of MIT’s Self-Assembly Lab, ran through a dizzying array of future-forward ideas including 4D printing, self-assembling structures, and auxetic materials. Lunchtime that day included Q&A sessions with Marius Kintel, lead developer of OpenSCAD, and Shapeways design evangelist Lauren Slowik, who announced the upcoming public 3D design curriculum that will grow out of her current New York Public Library course. The final day of the conference included a ShopBot keynote; a materials panel that brought together representatives of Proto-PastaStructur3DEssentium, and Reflow; and a top-secret peek into the future of Tinkercad and Fusion 360 with Autodesk’s Guillermo Melantoni.

Just one of the many things Tinkercad has in store for the future…

But the heart of the conference was more than 100 presentations by educators. For most of us, Construct3D was the first time we could give a talk about 3D printing and know that the audience was already familiar with 3D printing and its educational opportunities. This enabled every talk and workshop to dive deep into techniques, pedagogy, design walkthroughs, and extensive Q&A sessions, with topics ranging from quadcopters and multivariable calculus to photogrammetryart history, and social impact.

Over three days at Construct3D, we shared our successes (and failures!), got inspired with ideas to try out next fall, and most importantly, formed a community of educators that together will take 3D printing to the next level.

Didn’t get a chance to attend Construct3D this year? Stay tuned for the next one in Spring 2018 (location to be announced)!

In the meantime, you can find a complete rundown of talks and workshop at the Construct3D 2017 schedule, follow the hashtags #Construct3D and #Construct3D2017 on Twitter, and read recaps of each day of Construct3D at Karen Blumberg’s blog.

If you were at Construct3D and want to share what you experienced there, tell us about it in the comments; we’d love to hear about it!



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JMU 3SPACE: Building a 3D printing classroom

See the original post at Ultimaker Education

In the beautiful Shenandoah Valley of Virginia lies a state university with over 20,000 students and the country’s first college-level 3D printing classroom for general education. This is the first in a series of posts about how JMU 3SPACE got started, what we’ve done so far, and where we’re headed next.

Many universities have 3D printing facilities that serve individual departments or function as service bureaus or labs. In 2013, James Madison University built something brand new: a hands-on 3D printing classroom whose mission is to serve students of all majors and backgrounds. The classroom has eight stations, each with its own computer and 3D printer where students can do interactive work in groups. Over the last four years, the JMU 3SPACE classroom has supported 3D printing across the university curriculum by hosting general education classes, courses in mathematics and art, projects in history and biology, workshops for local K-12 school groups, faculty workshops, and even a 3D printing club. In this post, we’ll walk through how JMU 3SPACE went from ideas to equipment to curriculum, with the goal of providing advice for other schools that want to establish their own 3D printing classrooms.

during class
Students working during a GSCI class in the JMU 3SPACE 3D printing classroom

First steps: the JMU MakerLab

In early 2013, the Engineering Department, the Institute for Visual Studies, and a few other sites on the JMU campus already had 3D printing experience and equipment, but in the Department of Mathematics and Statistics, we were just getting started. During the spring semester of 2013 we used Engineering and IVS facilities to run an independent study course on 3D printing mathematical objects, and began the process of converting a small closet in our own department into the JMU MakerLab for Mathematics and Statistics, with three 3D printers and two computers. In the MakerLab, students and faculty could 3D print mathematical objects to support research projects and create course manipulatives.

The MakerLab was fun, exciting, and opened right up into the hallway where students walked to their calculus and statistics courses. As a result, a lot of those students asked to have access to the MakerLab. However, we didn’t have enough equipment to serve such a large population, and we couldn’t give lab keys to students that weren’t majors in our department. We also lacked the staff and resources to help teach that many students how to design for 3D printing, and we didn’t want to just help people print things they found on the internet. We started dreaming of having a whole classroom of 3D printers at JMU, with access for students of all majors and the ability to teach people how to get started with 3D printing and design.

Proposing a dedicated classroom for 3D printing

To build a new technology classroom you need two things: funding and space. The necessary funding isn’t much different than what you would need to install a computer classroom since 3D printers are somewhat comparable in cost to computers. We thought of our 3D printing classroom as a computer lab where half of the computers were replaced by 3D printers. The facilities we needed to install were also the same as what you’d need for a computer classroom: multiple wired internet connections, many grounded electrical outlets, security camera, swipe card door access, and a projection system for the instructor.

For us, it was securing the space itself that was the most difficult. University space is always at a premium, and we were asking to set aside an entire classroom 24 hours a day, 7 days a week, for 3D printing courses and workshops. Having a dedicated space for 3D printing was key to our mission, so that we could keep 3D printers set up and accessible all the time for hands-on activities, ensure the safety of the equipment, allow people to run prints that extended beyond class times, control access to the space, and run outreach programs from the classroom whenever possible.

If you’re trying to set up a 3D printing space at your own institution, make it easy for administrators to see exactly what you need and what value it will bring to your campus. Three things helped us immensely when making our case to the university for creating the JMU 3SPACE classroom. First, we had a detailed budget with price quotes, including some educational discounts. The budget for our initial classroom setup, including all equipment but not including facilities costs, was about $35,000. This sounded like a lot to us, but to the university this was actually a fairly reasonable amount to pay for an entire room of equipment. Second, we created a small 3D model and printed out multiple copies to give to administrators that might not have seen 3D printed objects before. The model is based on the JMU logo and can be used to hold business cards.  We call it the “JMU Cube”.

JMU Cube
The “JMU Cube”, a 3D printed business-card holder

Third, we created a one-page inspirational handout that included concept diagrams of our computer-printer stations, a list of sample classes that one could imagine being taught in the space someday, a very brief copy of the budget, a list of some of the other universities that had already started 3D printing facilities of some kind, and text and references to help show the utility and relevance of building a new 3D printing classroom. This handout enabled us to put a piece of paper into any administrator’s hand to quickly represent the project. It also gave administrators a way to pass along information about our pitch without having to put together such a summary themselves.

3 space pitch flier

The inspirational flier we created for pitching JMU 3SPACE to the university

In addition, we had a claim to fame that helped in our particular case: we knew that if we moved quickly, we could be the first general education 3D printing classroom in the country. Universities like to find ways to break new ground, and JMU 3SPACE gave us a way to establish James Madison University as a leader in the popular trend of accessible digital fabrication.

Setting up the classroom: equipment and tools

I’m not sure which of the above actually helped our pitch the most, but we were successful! Campus facilities identified an unused science classroom in a transitional building, and the university agreed to pay for converting the space for our needs. Here’s a before and after of the classroom:

Science classroom
A science classroom ripe for conversion to a 3D printing classroom!
New Space
The JMU 3SPACE classroom on opening day, with eight student computer-printer stations, one teacher station, and even some new chairs

For the major equipment, we decided to go with Mac desktops and Afinia 480 printers, which at the time were the smallest reliable printers we could find. We started out using ABS filament and then switched over to more classroom-friendly PLA a couple of years later, once that was possible on the Afinias. Over the years we’ve leveled up to some fancier equipment, including a couple of MakerBot Replicator 2’s, an Ultimaker 2 Extended+, a CubeX Trio, and a MakerBot Digitizer. Next semester we’ll be flipping to a fleet of Ultimakers, but more on that in another post!

original station
One of the eight original 3D printing stations in the JMU 3SPACE classroom

Each station includes a computer, a 3D printer, and a box of post-processing tools, with a 3D printed Sharp Tool Holder to keep everything in place. We also designed and 3D printed a special fan grate to keep fingers and pencils out of the 3D printer fans.

A word of caution: the sharp platform-clearing spatulas and the curved X-acto knives have each done their share of damage on student and faculty hands. Nobody has ever been hurt by one of the 3D printers in our classroom, but lots of us have received scrapes and cuts from the sharp tools. We’ve since replaced them with tools that are less effective, but far less dangerous, such as this Cricut spatula.

Sharp tools for post-processing prints, and a custom 3D printed holder to keep them in place

If you’re setting up your own 3D printing classroom then something to keep in mind is that all tools are consumables. That is, they break and wear out, and you’ll need to replace them someday. Make sure to include these costs in your future yearly budgets so that you can refresh your tools when needed. You’re also going to run out of filament! Here’s how much filament we went through in a single year:

One year of empty filament spools from JMU 3SPACE

A great first print: Quarter Traps

We kick off most of our classes with a 3D printer demonstration and safety training that involves printing “Quarter Traps”. Students insert a quarter halfway through the printing process and it becomes trapped within the model.

quarter traps
Quarter Traps – a great first print for students!

Coin traps are a great illustration of the process of additive manufacturing, and give the students something to take home after the first day and show to their friends and other teachers. While the models are printing we talk about printer safety and show the students how the design is constructed in Tinkercad.  We also use coin traps as a starting activity in our outreach workshops to local K-12 schools and clubs.

Students from Mercer County Middle School
Students from Mercer County Middle School showing off their 3D printed Quarter Traps

Students get really, really excited on the first day of class while watching the Quarter Traps print, and often take pictures and videos of the process with their phones. On a personal note, after teaching so much calculus I was really not prepared for the amount of joy and excitement involved with teaching 3D printing.

first day of GSCI 104
Students on the first day of GSCI 104 class, watching their first 3D print
students posing with their 3D printed Quarter Traps
Students posing with their 3D printed Quarter Traps. Notice that students come to us from every corner of the university to learn about 3D printing!

3D design, failure, and iteration

Of course, after having fun printing a pre-designed model, the next step is for students to design something themselves and print it. For software, we use anything that is free for students, which now includes pretty much everything from Autodesk. Our favorite classroom design tools include Tinkercad for getting started, 123D Design and Fusion 360 for going to the next level, OpenSCAD for exploring code-based design, and Mathematica for the math geeks. All of those are free except for Mathematica, for which we have some university licenses. We also routinely use Inkscape, MeshLab, Meshmixer, TopMod, Blender, Morphi, 123D Catch, Sketchup, KnotPlot, and Sculptris, depending on what projects people are working on. Most of the time our students come to us with no prior design experience, so we start them off with Tinkercad.  After attempting to print their first 3D design, their initial joy turns to… something more realistic.

much sadness
Students experiencing their first of many failures with 3D design, right before they pick themselves up and try again

Failure is perhaps the most important thing that students experience in our 3D printing classes. When their designs don’t work, the students don’t give up. They problem solve, discuss with classmates, and iterate their designs. The skill of failing – and then getting back up again – is naturally developed when studying 3D design, but not always natural in other subject areas. One of our main pedagogical goals is to empower students to learn to create and make, to learn how to learn on their own, and to learn how to fail and iterate productively.

By the end of the semester, these students – who come from every major on campus and have no previous design experience – make incredible things that they never imagined were “make-able” before. The picture below shows some of the final projects from our very first GSCI 104 class, and includes a working Ferris Wheel with hanging cars and a crank, a fully functional modular ukulele made to look like a flower, frames and signs, a mechanical Jansen leg assembly, geological models, custom drawers, and a mathematical Seifert surface for Borromean rings.

3 space projects
Some of the amazing 3D models that students make after just one semester

Next steps

Since JMU 3SPACE opened in 2013, we’ve hosted lots of 3D printing classes, faculty workshops, outreach programs, and 3D printing projects in math, art, history, and biology. This spring we’ll be moving from our converted science classroom into the first floor of the main library on campus, and improving our 3D printing technology to a set up with Ultimakers that will enable us to make finer prints, take advantage of variable nozzle sizes, and experiment with multi-material printing. Our first batch of Ultimakers are already being tested at one of the JMU Library Makerspaces; in the picture below you can see some future high school teachers in a JMU Education masters degree class printing out 3D designs from their technology lesson plans.

Ultimaker Lab
JMU graduate students testing out four of JMU 3SPACE’s new Ultimaker printers in the Library

JMU 3SPACE is now run by a team of four directors: Laura Taalman and Rebecca Field from the Department of Mathematics and Statistics, Jamie Calcagno-Roach from Innovation Services, and Daniel Robinson from the Institute for Visual Studies. In future posts we’ll all share more of our curriculum, projects, and progress, so stay tuned!



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