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As 3D printers come into homes and schools, you will be looking for tools that let you make your own designs.  Here are a few free titles that I think should be on your list:

Inkscape is primarily a drawing program for two-dimensional designs. It is an amazingly powerful tool that even automates the process of drawing complex objects like gears.  Drawings created in Inkscape can be saved in the SVG (scalable vector graphic) format so they look great at any magnification.  You can also export images in traditional graphics formats like PNG (portable network graphic) that looks great when used on websites, etc. But the real power of Inkscape as a 3D drawing tool comes when you install the support plug-in for 3D extrusion to an OpenSCAD file (described later)  that can be rendered and exported as an STL file for the printer to use.  The way this works is that you select the part of the drawing you want to extrude into a three-dimensional shape.  When you choose the extrusion option, you just indicate how many millimeters you want the extrusion to be, and a 3D file for OpenSCAD is generated automatically.  To get this shape to your printer, the next step is to open it in OpenSCAD, compile the image, and save it as an STL file.  STL (Stereolithography) files are the format your printer expects to see when it starts the process of getting your model ready to print. This sounds laborious, but it is easy to get the hang of it, and the whole process goes very quickly.

You may be wondering why I would mention a (primarily) two-dimensional drawing tool in the context of 3D printing.  The reason is that, while building 3D objects on the computer screen is likely a new task to students, they probably use two-dimensional art programs all the time.  Our goal is to build from this strength on the path to (later) creating designs with 3D drawing tools.

While I will largely use OpenSCAD as an extrusion tool for Inkscape, it is, in fact, a full 3D modeling program that builds models from text commands.  It has its own programming language that might be appropriate for high-schoolers to play with.  An advantage of building geometric models in OpenSCAD is that they can be “parameterized” – expressed in a way that lets one design make several related shapes by changing the values of a few variables.  For example, a propeller can be designed in OpenSCAD in a way that lets the end user change the number and size of the blades.  This is a real feature, and quite a few Thingiverse models include OpenSCAD files for just that reason.  Used in this way, students can tinker with existing models to create a custom part for their construction.  The final model is displayed on the screen to be sure it is what you want before saving it as an STL file.

Sketchup is a professional 3D modeling tool that is super for creating geometric structures from scratch (architectural designs, for example).  The free version (Sketchup Make) has all the features that students might need to build models of the parts they want to print.  If your model can be built from boxes, cylinders, and balls, it is a great tool.  It is not what I would choose for more organic shapes, though.  The Sketchup Extension Warehouse has a free plug-in that lets you export your finished part as an STL file directly.  My only caution about this tool is that it is not the best program for editing completed STL files.  They show up as a mass of dots and triangles, and I haven’t found a way to render the surfaces as nicely as you can from models made in Sketchup in the first place.  This is a shame, because older versions of Sketchup handled imported STL files much better.  The good news is that there are many other alternatives for you to use.

This program also lets you create projects from scratch using a library of geometric shapes.  My experience is that it is easier to align parts in 3DTin than it is in Tinkercad (another cloud-based design tool).  3DTin lets you download your drawing as an STL file ready to print!

Autodesk is one of the premiere publishers on computer-aided design software.  Their products are found in design firms and architects offices all over the world.

They decided to support the beginning 3D designer with a rich suite of tools that covers the gamut from parts designed from geometric pieces, to the more organic designs suitable for modeling living organisms.  In fact, Clark Barnett, a teacher in  the Conejo Valley Unified School District  in California does a project with his kids using one of the Autodesk applications on the iPad – 123D Creature.  With this tool, students design their own insects that could live in the ecosystem of their classroom.  Once printed, these “insects” are mounted in a display tray and students explain why their insect is likely to survive on its own in the classroom ecosystem.

While not geared specifically for “creature” creation, Autodesk has a wonderful free product called Meshmixer that is perfect for creating organic, rather than geometric shapes.

This tool lets you sculpt by hand as if you were working with clay.  Anyone who has worked with modeling clay will know how to use the tools in this program, and there is a great manual to show exactly how to get the most from this program.  Tools like this bring 3D printing into the life sciences classroom.

This amazing tool is a great next step for Meshmixer users.  It was designed for sculptors (and would-be sculptors), instead of a blank screen you are presented with a round ball of “clay” that can be shaped into just about anything you want.  While not geared toward the creation of geometric objects, it is a perfect tool for building models of various creatures – both real and imagined.  Finished projects are exported as OBJ files that can be easily converted to STL files by Meshlab (see below).  Once you start working with this tool, hours happily go by as you build amazing things, all of which can be built on your 3D printer.  This software comes with good documentation and links to some video tutorials I highly recommend for anyone interested in this tool.

This program lets you build mathematical knots of all kinds.  While created for math geeks, knots are pretty to look at, and students can use this program to explore this branch of mathematics – a worthwhile activity in itself.  One great feature of this program is that it lets you export your finished knot as an OBJ file if you want to tweak it in Meshlab (see below).  You can also export your image as an STL file directly and send it to your printer software with no further work required.  Finished knots can be sent out for metal plating in case you want to make your own jewelry. (You probably have some service providers in your area that will do this inexpensively.)

Other tools:

Sometimes (as with Sculptris) your 3D images will be exported as OBJ files that need to be converted to STL files so they can be printed.  Meshlab does this job beautifully and even lets you adjust the mesh from which the model is defined to optimize it for printing.  This optimization process lets you clean up your model so it will print perfectly.

This is the plug-in you need to allow Inkscape to create extrusions for OpenSCAD.  All the instructions are provided in the web link shown above.

And there are more good programs coming out all the time, so keep your eyes open and let us know what you find (!

The Trinity Fractal

The story behind this discovery dates back to the 1970’s when I used to volunteer as a math resource specialist at a small school near my office.  One day, a teacher introduced me to a 10 year-old girl (we will call her “Ann”) who was (in the teacher’s  words) “bad at math.”  I found that to be  strange announcement since, in my view, Ann had never been exposed to math, but only to arithmetic.  So one day I invited my class to do an experiment.  

I brought my own bucket of pattern blocks to school, in which I added some regular pentagons I had made.  Piles of the same shapes were put on several desks, and students were asked to tile the surface with just one shape – and I made sure Ann was at the table with the regular pentagons.  After a few minutes, all the students had succeeded – excepting those at the table with the pentagons.  No matter how hard they tried, there was always going to be a gap.

Ann said, “Well, we can do it, but we are going to need a lot of grout.”  And, after looking at the other tables, she said, “This is a strange kind of math – 3 works, 4 works, and 6 works, but 5 doesn’t work.  Why is that?”

I was delighted to hear that question because this is the kind of question mathematicians ask themselves often.  Ann told me she wanted to experiment more with the pentagons, and I gave her a bunch to take home so she could report her findings the next week when we met again.  At this point I didn’t know what to expect, but it sure wasn’t what she showed up with.

The next week, she started off with the following pattern:

Starting pattern

Ann pointed out that this shape, while not a tiling pattern, looked like a pentagon if you “squished” your eyes a little bit.  “So,” she said, “suppose we start with this pattern, shrink it in size and build a new pattern with the same shape.”

First generation pattern

Then, she said, just keep repeating this process.  You will always need some grout, but the picture should be very pretty.

The next two generations of patterns are shown below:

Second generation

Third generation

As you can see, Ann was quite right.  Yes, you still need grout, and, the resulting pattern is quite pretty.

Basically, what Ann had discovered (and accurately described) is a fractal – a shape with a non-integer dimension.  I’ve told Ann’s story many times, but never before constructed the fractal patterns to show people.  I decided to call this the Trinity fractal, named after the school where I volunteered (Trinity Parish School in Menlo Park, California.)

I lost touch with Ann, but talked with her on her first day of college at UC Berkeley, where she was majoring in mathematics.

I’m glad I may have played a small role in helping her see the beauty in this subject, and have no doubt that she has gone on to do great things.

I’m also happy to finally share her discovery with others in the hope that it encourages other teachers to move beyond arithmetic to see the beauty in real mathematics, as encouraged by the Common Core State Standards for mathematics.  Our workshops on CCSS Math can be scheduled by e-mailing me at  Also, our work with pentagon tiling has continued in a new way.  See to see what we’re doing in this area!

Before getting into the content of this blog, I want to ask the flame brigade to hold off until they get to the end of the message.  This blog is not anti-iOS, not anti-Android, not anti-tablet.  It is simply my view of how things seem to be turning out.  So here goes:

Schools around the world have diven into the deep end of the tablet pool, purchasing these devices by the thousands (or more) in the quest to bring powerful technology into the hands of students.  The reasoning behind tablets is that they are rugged, have amazing battery life, and provide access to various apps that may be of value in the classroom.

This last point has been a sticking issue for some.  I’ve argued for decades that the choice of a computer platform for kids needs to be driven by the software they will use, and this message has been lost on some districts who chose the platform first, and then tried to figure out how best to use it.  As with the Apple vs. Microsoft battles of the past, the fight quickly broke down into two camps – the iOS folks (iPads) and the Android enthusiasts.  While some have found ways to use these tools in remarkably powerful ways, the question arises: should we have been looking at tablets at all?

In a 2012 piece in THE Journal (, Therese Mageau argued that the race to buy iPads (for example) largely came without thinking about the deeper educational shifts implied by every child having his or her own connected device.  While she is correct, I’d like to take a different approach to the question – to ask if tablets were the right choice at all!

While the world was focused on iPads and the like, Google announced the Chromebook at their developer’s conference in May, 2011.  Like tablets, Chromebooks have long battery life (8 hours or more), virtually no boot time (8 seconds from a cold start), a low price (under $300) and the ability to run some applications (word processing, spreadsheet, presentation tools, watch videos, etc.) without Internet access – although this tool was designed to be used when you are online.

While some schools started to adopt Chromebooks, many did not, even though the Chromebook looks like a thin laptop with a full keyboard and high quality display.  But, once again, the question arises on the application front.

Google has done a wonderful job of helping developers create apps for the Chrome Web Store ( and amazingly powerful educational apps abound – a great many of which are free!  For example, you can get Geogebra, the Scratch programming language, even all fifty of our own Knights of Knowledge inquiry-starter videos that span grade levels and subject areas.  The list of educational apps is growing daily, along with the adoption of this tool as the one-to-one device of choice for many.

And this brings us back to tablets – or more particularly to the schools and districts who purchased so many of these devices for student use. Given what we now know, would different purchasing decisions be made?  As the pundits say, hindsight is always 20-20.

The fact is that the Chromebook emerged as a wild card in a field that never seems to stop and catch its breath.  Does this relegate tablets to the storage closets?  Of course not.  It merely suggests that we need to base our purchasing decisions on the best information we have at the time.  And, make no mistake about it, there will be something someday that eclipses the Chromebook.  This just reinforces the importance of ensuring that whatever purchase we make is based on the actually utility of the device to kids in support of their learning.  As long as we do that, we are on solid ground.

Earlier this month I got a Samsung Chromebook because many of the schools we work with have adopted them in their one-to-one programs.  I just admit that I had some misperceptions that kept me off the Chromebook wagon for over a year, but now I think I can use it as my primary road-tool for giving presentations, creating documents (such as this blog) and doing other things (but not all things) I used to use my laptop for.

My original thinking was that this could be a very cool device.  From a historical point, it all started in 1984 when Sun’s John Gage said: “The network is the computer.”  When he said this many computers (including the brand new Macintosh) came without a modem or ethernet port – so this was a very bold statement.  About a decade later, when the first graphical web browser (Mosaic) was released, I said: “The browser is the operating system” and, just two years ago the Chromebook hit the market.  This ultralight laptop replacement uses the Chrome browser interface for all applications.  The browser is built on a Linux base (just like iOS, MacOS, and Android) thus leaving Microsoft out in the cold.

Because of the browser interface, I assumed (incorrectly) that all applications needed to be used when you were online.  This is not true.  Once you register with your Gmail account, you are able to create documents (such as this one) along with slideshows even if you have no internet connection at the time.  Once you go online, all your new documents and edits get synchronized to the cloud automatically.  This a great for kids who may only have good internet access from school.  They can still work on projects at home even though they are disconnected.

The automatic update feature applies to more than documents.  Applications reside in the cloud (unless you are running local versions on the Chromebook) so upgrades are automatic.  The Chrome operating system is virus proof.  If you completely mess up your system (hard to do), you can do a fresh restart and everything you were doing gets automatically put back in place as soon as you log in.  This means that if your Chromebook gets run over by a truck, you can turn on a fresh one, log in, and keeps working as if nothing happened.  Start-time (from cold start) is about eight seconds.  If you have the Chromebook sleeping, it wakes up immediately.

There are a few changes that need to be made.  Some of the applications (e.g., Geogebra) do not have all the features of the laptop version, and there seems to be a bug in the current release of ChromeOS that makes it hard to rename files in GoogleDrive.

Of course GoogleDocs is the home for word processing and other traditional mainstream applications.  Finished documents can be exported to a wide range of formats (.docx, for example) so you can share your work with others in the format they prefer.

As I continue to use this new device, I will post more insights on this blog.  In the meantime, if you want a reliable device with long battery life (I get over 8 hours) for the bulk of what you do that is web-based (and local to your machine when it is nowhere near the Internet), this can be a very good $250 investment.


My previous blog explored the eight elements of the Common Core math standards:

1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.

In this blog, we’ll look at another great project that brings 3D printing into the math classroom.

Anyone who has played with pattern blocks knows that many tiling patterns can be made with all the shapes in the set. But, while pattern blocks have polygons with three, four and six sides, the don’t have any regular pentagons. One reason for this is that you can’t tile a flat surface with regular (equal sides and equal angles) pentagons by themselves. Any attempt will fail and, as one of my math students said decades ago, “you need a lot of grout.”

But this can be fixed by adding one piece – a parallelogram that exactly fits the gap left when you try to build a tiling pattern with pentagons. And this is where 3D printing comes in. Using this powerful device and some free software, you can design you own set of tiles with two shapes: regular pentagons and special parallelograms. The software I recommend is Inkscape – a free two dimensional drawing program with a special plug-in that allows your shapes to be extruded into a 3D tile. I used a thickness of 2.5 mm, but you can use anything you want.

One reason for having students build their own tiles is that the process of designing the tiles incorporates several of the standards mentioned at the start. In particular, standards 1, 5, 6 and 7 seem relevant. And you haven’t even made the tiles yet!

Once the tiles are made, the real fun begins. To make the patterns interesting, students may want to have different colors for the pentagons and the parallelograms. A good set of tiles might have 50 of each shape, although most patterns will likely use more pentagons than parallelograms.

To get started, students might see if they can build a pattern that had rotational symmetry. One such pattern looks like this:

pentagonal pattern

While this pattern is pretty to look at, it also contains a lot of math. For example, what (if any) is the relationship between the number of pentagons and parallelograms in the pattern? What different patterns with this symmetry can be built with the same blocks? What patterns emerge in the number of tiles needed to expand this design with additional “rings” of polygons? The list goes on, and these explorations easily incorporporate all eight of the CCSS math standards.

While this pattern has rotational symmetry, can you build a pattern that has translational symmetry? Consider the various symmetries associated with this next pattern, for example:

pent tiles 2This pattern is completely different from the first, yet it is made with the same two kinds of tiles. This raises some more questions: How many different tiling patterns can you find? Is it possible to build a tiling pattern that has neither rotational nor translational symmetry?

Hours can be spent tinkering with these tiles, and all the while students are developing their skills at mathematical thinking.

If you want to experiment with these tiles in your classroom or school, send me an email at dthornburg (at) and I’ll gladly send you a pdf file with several educational projects that use 3D printers, each of which comes with step-by-step directions.

I’d say more, but the tiles are calling…

I want to start with an observation about the Common Core State Standards for mathematics.  In the past, I have been openly critical of many standards, but I find the central ideas behind this set quite appealing.  The key point is that these standards point the way toward student understanding of critical mathematical ideas, not just the memorization and regurgitation of facts quickly forgotten after the test.

Here are the key concepts:
  1. Make sense of problems and persevere in solving them.
  2. Reason abstractly and quantitatively.
  3. Construct viable arguments and critique the reasoning of others.
  4. Model with mathematics.
  5. Use appropriate tools strategically.
  6. Attend to precision.
  7. Look for and make use of structure.
  8. Look for and express regularity in repeated reasoning.

These are all wonderful ideas that directly relate to how mathematicians work.  The challenge is how these can be implemented in the teaching of mathematics.  One example I find interesting is to see how we can use 3D printers in the math classroom in support of these core ideas.  For example (and I have several), consider the construction of polyhedra using plastic struts and rubber bands.  These shapes are called tensigrity structures first popularized by Buckminster Fuller.  The idea is that a three-dimensional structure can be built with only two kinds of elements – those in compressions (the struts) and those in tension (the rubber bands.  For example, the following image shows an octahedron built with three struts and eight rubber bands.

This simple structure is easy to assemble, and numerous other polyhedra can be built the same way.  But you might be asking where the math comes in.  This polyhedron has 8 faces, 6 vertices (corners) and 12 edges.  What relation, if any, exists between the number of faces, edges, and vertices of polyhedra?  If you can find a relationship, how can you prove it works for any polyhedron?
These questions all derive from the design and construction of a kit for making polyhedra, and the questions we ask cover quite a few of the key concepts for the Common Core math standards.
Imagine how activities like this might make math exciting for kids who currently fail to see why math can be an interesting topic!

I start with a story:

Many years ago (in the Apple II days), before the Internet took off, and the online choices were bulletin boards and some ftp clients, I was headed to Toronto to speak at a conference. This was well before NAFTA was created. As I went through immigration, I was asked if I had any business materials. I showed the officer my floppy disks containing my presentations, at which time I was informed that I could not bring software into the country without an import license. Rather than argue, I said he could keep the discs. He then said, “Don’t you need them?”

“No,” I said, “because on the way to the hotel I will stop at a store, buy some blank discs and then download my software from the US where you can’t touch it.”

Now here’s the thing. First, never argue with an immigration officer who is holding your passport. Second, the words “where you can’t touch it,” were totally unnecessary. I had time to reflect on these things while sitting in a holding cell for an hour or so, after which the officer came in, gave me my passport and discs and said I could leave.

My point is that, even years ago, it was evident to some of us that a global network would render tariffs meaningless, especially in the digital domain. Many years later, when Nicholas Negroponte said “Bits are the new atoms,” he talked about a huge economy based on intangible goods that travel through the aether without regard to national boundaries. He was right, but the world of commerce is made of more than tangible goods. It will be a long time from now before someone figures out how to pass a bottle of a good Merlot through the Internet!

In the information domain, bits remain bits. A digitized movie can be sent through the internet and displayed without ever having to be put on physical media. An e-book can be downloaded and read without ever having to be printed. But, with 3D printers, instructions for building parts for a machine can be sent over the Internet, and turned into actual physical parts on a 3D printer. The same parts that would be subject to import duties can now bypass customs, and be completely undetected. Atoms are the new bits!

My comment, “where you can’t touch it,” is even more true today than it was when I first made the point. While we still aren’t talking about a good Merlot – yet – there are many other physical items of value that can cross borders without detection (and without import duties.) A quick perusal of Thingiverse ( shows a wide collection of things that can be made with inexpensive printers like those made by Afinia. They range from the practical to the artistic, and you are encouraged to post your own designs as well. Of course, if you have a proprietary part to build, you just design it using design software, and save the result as an STL (stereolithography) file. This file can then be encrypted and sent as an e-mail attachment to the person in another country who can then print it out of plastic or other materials locally.

And, as for the wine, food replicators are in the process of being designed, and it is just a matter of time before this challenge is addressed.

Tariffs are dead, and the first countries to realize this have the chance to look at the positive impact on local economies when every business – no matter how small – is a global business. The 3D printer is an economic leveler and the trigger for a new industrial revolution.

Just don’t hand a copy of this blog to the immigration officer the next time you enter a foreign country.

Well, this is my current attempt to distill some of the core ideas in the Next Generation Science Standards (NGSS) in the form of an infographic.  Let me know if you find it interesting or useful.

I used Inkscape to create the graphic over a period of a few weeks as we were preparing for one of our NGSS workshops.

ngss infographic

Over the years, I’ve gotten to know some Flash programmers who have mastered an arcane art:  How to use a tool so cumbersome that it provides full employment on projects that could be done in a fraction of the time in Hyperstudio and then exported to HTML5.

This all came home to me when I was asked to help edit some e-books using Adobe Flash Pro.  Those of you who remember Macromind Director, and the adage that a camel is a horse designed by a committee, can imagine what the interface looks like.  You have libraries of resources that you import to either frames or libraries (and be sure you never get them confused).  Animations made from still frames can be made in only five times the time it takes to do this in Quicktime.  But the real time sink comes when attaching everything together.  Heaven forbid you need to duplicate a frame without duplicating the frame’s inheritance since once you edit the “new” frame it will replace the contents of the one you copied it from as well.  Of course this is all fixable by a series of crafty maneuvers that only take a half-hour to learn.  Next comes the linking of sounds to images, but I’ve made my point.  Once you have your e-book project finished in a month or so you then need to compile it for every platform on which you want it to run.  “Voila!” (which is French for “my wrist hurts,”) you are done at last.  See, full employment for even the simplest of projects!

Hyperstudio, on the other hand, handles the same project this way.  First, create a background screen with the major navigation buttons and a text frame.  Make as many cards with this background as you need (this takes 20 minutes, if you include a 15 minute coffee break.)  Next, and this is the tricky part – drag and drop the images and text for each page onto each card.  If your images are movies, then drag and drop works just as well. Of course, if you are in a hurry, all you need to do is create the original template card with the navigation buttons and the text object marked as group objects, you can then do one “New Group Card” action (Edit menu) to start a group, and then just drag and drop the entire folder of images on to the HyperStudio icon in the dock.  This will instantly create as many additional group cards as you have images in this folder.  This skips the many steps of dragging and dropping each image individually on to each new group card as you make it.  What makes this step tricky is how you create the illusion that it takes days, not minutes, to assemble these pages.  Next, attach sounds as needed so that when (for example) you click on some text, the narration plays.  Finally, there is the matter of exporting to HTML5 which is done by choosing Export and HTML5 from the menu.  The result plays on virtually anything.  Put a fork in it, you are done.

Now how you explain the next 29 days of R&R is up to you.

Staff development

Now that the next generation science standards (NGSS) have been released, about half the states have adopted them.  The missing element seems to be staff development.  Because these new standards go way beyond content to address the very methodologies of education, it seems essential that all teachers have the support they need to implement the new standards.  According to a recent article in the journal Science  (vol. 340, p. 1391, 21 June, 2013), while 83% of science teachers think NGSS will improve learning, only 38% think they will get the training they need.

Now if the only thing that changed was the order of existing content, this wouldn’t be a problem.  But these standards go way beyond that.  For example, Engineering is now specified as a K-12 subject – a field that teachers in general need to know more about.

In general, the standards move us from nouns to verbs – to activities students explore to enhance their understanding of core concepts.  For example, the standards state, with reference to the application of math in science:  “Emphasis is on assessing students’ use of mathematical thinking and not on memorization and rote application of problem-solving techniques.”

Now at first glance, coming off a generation of NCLB-driven education, this shift is overwhelming.  In fact it is delightful – provided teachers get the support they need.  This support can take many forms, but our own approach is to engage educators in hands-on experiences that enhance learning and rekindle the joy that drove teachers into education in the first place.

Contact us for more information:


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