Making and 3D Printing Vases: MathArt

img_20181202_200508027.jpgLast year, I got a 3D printer for Christmas.  My two kids found a vase on Thingiverse and printed it.  After looking at the vase, I told them that I could create a similar vase using math.  That started a journey that turned my house — and classroom — into a vase-making center for months.

This is my second year of making vases with my Honors Pre-Calculus class, and the students love it.  It takes about two class periods after we finish graphing transformations of trigonometric functions.   Most students create the vases by playing with transformations and domain restrictions of sine and cosine.  But some student take advantage of other techniques that we learned, such as variable amplitude equations like f(x) = x^2 sin(x)  or graphical addition.

The 3D prints are between 4 cm and 6 cm tall and print in about 30 minutes each.  They have a THIN shell.  I use filament that looks translucent when printed this way.  I also take advantage of special settings in the slicer to make the prints so thin.

This post describes:

  • How to design vases in Mathematica (requires a software license)
  • How to design vases in GeoGebra 3D (free software)
  • Slicer Settings for 3D printing the vases
  • Filament Suggestions
  • Extension Activities

Designing Vases in Mathematica (requires a software license)
I am fortunate that my school has a license for Mathematica, so this is how we make our vases.  I teach my students to use Mathematica throughout the year, and this project applies the syntax that they have learned.

  1. Create and plot an equation that has at least one sinusoidal part.  Play with the domain of the plot to get a shape you like.vase plot
  2. Rotate the equation around the x-axis.  Play with the domain to change the shape that is created.
    vase rotate
    If an object has very quick slope changes, like the design below, it will probably have stringy sections in the print.  If I see a vase design that has quick slope changes, I encourage the students to alter the equation so that the slope changes are more gradual.


  3. Mathematica defaults to creating a diamond-shape mesh on the exterior of the object.  This looks pretty cool when it 3D prints.  However, if you want the surface of the object to be smooth, you can add the Mesh command to your print, as below.  The value of 50 is arbitrary — I tried it, and it worked.vase mesh


diamonds vs smooth
The vase on the left has a diamond-like pattern on the surface. The vase on the right has a smooth surface.


4.  Export the vase to an STL file.vase export


Designing Vases in GeoGebra 3D (free software)

I realize that not everyone has access to Mathematica, so I’ve tried to come up with a free software option for this activity.  So far, I have 3D printed only two vases from a design I made in GeoGebra 3D.  Neither printed with surfaces as smooth as the vases from Mathematica.  If I had some extra time, I would tweak some settings in GeoGebra to see if I could make the vase export with a smoother surface.

I had never used GeoGebra to make a solid of revolution, so after my first few attempts came up short, I did some searching on the web and on Twitter.  I found this great Twitter post and video from Steve Phelps, @giohio, that describes how to create a solid of revolution in GeoGebra.   I followed his very clear directions to make this:

vase geogebra


Time to export the object.  First, go to the settings gear at the top right of the 3D graph.  UNCHECK the “Show Axes” and “Show Plane” options so that you see only the vase.  If you don’t do this, both the Axes and the Plane will export with the vase.geogebra export

Use the menu  “Download As”, and choose STL.

Slicer Settings for 3D Printing the Vases

Whether your vase was made in Mathematica or GeoGebra 3D, the vase exports as a solid shape.  The next step is to import the vase into a slicer and change settings to make it print as an empty vase.

  • Scale the vase: The Mathematica vases can export especially tiny, so I scale each vase to a size that will print in 25-30 minutes (this  varies between 300% to 1200%.)
  • Turn the solid object into an empty vase: Change the slicer settings so the object does not print solid.  I want a vase that has a bottom, a thin outer shell, an empty interior, and no top.  The key to printing a thin vase is to use a special mode called “Spiralize Outer Contour” mode or  “Vase Mode”.  A normal 3D printing mode lays down one layer of the vase and then moves the nozzle up to print the next layer.  This works, but it leaves a vertical line along the side of the vase where each layer ends, like this:
Notice the vertical line on the front of the vase, where each layer ended.

Using “vase mode” tells the printer to extrude filament in one continuous string, winding up and around as it creates the shell of the vase.  If you look closely at the top of a vase that has been printed in “vase mode,” you can see the end of the filament string.  There is no vertical line on the side of the vase, and the shell of the vase is very thin.

With my personal 3D printer, I use a free slicer called Cura.  I set the bottom thickness at 0.8 mm and select the Special Mode called “Spiralize Outer Contour”.  As Cura slices the object, it estimates the print time.  I aim for a print time between 25-30 minutes.cura settings

The 3D printers at my school run through the Polar Cloud, which I really like because it is easy to use and manages student objects well.  However, the Polar Cloud slicer does not currently have a “vase mode” or “spiralize outer contour” mode.  I adjust the print settings so the vase has 0% infill, a shell thickness of 1, a bottom layer count of 3, and a top layer count of 0.  The vases are not as thin as with “vase mode,” and they have a vertical line up the side where each layer ends, but they still look very nice.  See the picture of the green vase above.

Filament Suggestions

I’ve had good luck using CCTREE PLA filament that I purchased on Amazon.  The green, orange, and fluorescent yellow filament looks fantastic when printed really thin.  This year, I added Purple Haze filament to the mix, made by a company called COLORME3D (also available on Amazon).  I liked it so much that I just purchased the Hawaiian Green Haze color as a Christmas present for my kids…and, well, for me!

While I’m on the subject of filament, I should add that it’s a good idea to store your filament in an airtight container to extend its life.  I store mine in a bin that I made somewhat airtight by sticking weatherstripping to the lid.  I put some silica gel packs in the bin with the filament to prevent moisture buildup.

Extension Activities

  • Calculus!  Obviously, this would be a great activity to do with a Calculus class when learning about solids of revolution.  But I also plan to spend some time brainstorming with my Honors Pre-Calculus classes, which have not yet been exposed to calculus, to discover how to find the volume of their vase using math they already know.
  • Rotations!  Being able to envision the shape created by rotating a curve around an axis is an important skill for a student of calculus.  My students use Microsoft OneNote, so I ask them to snip (using the Windows Snipping tool) and share at least three other plots and rotations that they made.  This forces them to play with other equations and domain restrictions and predict and view the shape of the resulting solid of revolution.
  • Bigger Vases!  I’ve printed a bunch of taller vases (up to 15 cm tall) — they are all over my house!  I fill them with fake flowers … or nothing at all!  The larger vases take 2-3 hours to print, so plan accordingly.vases at home
  • Display!  Show off these student creations!  Does your school have an art gallery?  Or somewhere to display student work?  I thought about hanging each vase from the classroom ceiling using fishing line.  One of my students suggested stringing the vases together and hanging them like Christmas lights.  That’s what we plan to try this year.Happy designing and 3D printing!

Making Connections with Diamond Paper

Perhaps you have seen the diamond paper graphical organizer.  Teachers can pose a problem or question inside the diamond and ask students to justify or explain their work in different ways in the four rectangles.  Here’s a version that I created for my Honors Pre-Calculus class:

blank diamond paper inverse trig

I learned about “diamond paper” from Jo Boaler — I either read about it in her book Mathematical Mindsets or saw it in her online course.  When I attended one of Jo’s workshops, I worked through a diamond paper problem from the perspective of a student and was immediately hooked.  I often ask students to look at a problem graphically, analytically, numerically, and in written form, but this graphical organizer furthers the discussion by visually organizing their work.

For these activities, my students work in groups to solve the problem in the center.  They record their work in Microsoft OneNote.  When discussing their results, I either project a copy of work from one group or ask each group to use markers to fill in one of the sections on a whiteboard.  We discuss and annotate the work as we go.  I take a picture at the end to recap our discussion.

I created the “diamond paper” activity shown in the picture at the beginning of this post to introduce inverse trigonometric functions.  My hope was that students would:

  • Recall AND connect the different methods to find an angle whose sine is 1/2.
  • Discuss how many solutions to provide.

Here is a picture of student work that we reviewed in class:

diamond paper inverse trig

Some groups found one solution for the equation, some found two solutions, and some found many solutions.  This led to a discussion about how to restrict the domain of sine so that it has an inverse.  Students unanimously agreed that the toughest part of the diamond paper to complete was the section that asked them to explain with words.  We had a productive discussion about the difference between explaining their method in words versus explaining their solution in words.

One of the unexpected outcomes was a discussion of the various graphs that students made for the “justify with a graph” section.  Some students graphed y = sin x and y = 1/2 in various software (see an example from Mathematica below):

mathematica sine one half graph

Some students graphed sin x = 1/2 in software (see an example from Desmos below):

sinxequalsonehalf graph

The students who created this graph were perplexed as to why the graph did not look like a typical sine wave.  Students shared their thoughts about how to interpret this graph and explained why its shape was not sinusoidal.

I created several versions of the activity below for the topic of solving trigonometric equations.

blank diamond paper solving trig equation

diamond paper

Students were especially excited to see the connection between the solutions of this equation and the x-intercepts of the graph.  We spent a substantial amount of time on the “justify in words” section, again discussing the difference between describing a method and justifying a solution.

Diamond paper is perfect for calculus as well.  I use it to help students see connections in calculus topics like continuity.  Here’s an example of work from a group of my students a few years ago.  As an aside, the graph in this picture was made using FluidMath graphing software.  I prefer to use FluidMath when working with rational functions and continuity because it shows holes in the function, as you can see if you look closely at the graph below.

continuity twitter

As a bonus, diamond paper is easy to make.  I created a template in PowerPoint so that I can simply change the words for new activities.  Here’s a blank graphical organizer (diamond paper) to get you started.


How I Introduce 3D Printing in My Classes

For the last five years, I have integrated 3D printing into my classes as part of student learning.  When I started, I only 3D printed with my Honors Geometry classes.  But in recent years, I’ve expanded 3D printing to my Honors Pre-Calculus classes.  (I teach honors classes, but I would do the same projects in non-honors classes.)  One of my most important learnings:  students create better designs if they understand how 3D printing works.

Here’s how I introduce 3D printing to my Honors Geometry classes.  I use a shortened version of this process with my Honors Pre-Calculus classes.  Note that this process has been five years in the making.  If you have never 3D printed with your classes and you have only a few days of class time to give it a try, just do whatever makes most sense for you!

Explore 2D Cross-Sections of 3D Objects

Note:  This is a topic that I explore more deeply in my Honors Geometry classes than with my Honors Pre-Calculus classes.  The explanation below details what I do in Honors Geometry.

This lesson uses Play-Doh and something that we call a “hi-tech cutting device”: a piece of sewing string for cutting through the Play-Doh. Of course, I start the lesson by asking students to create the ground rules for working with Play-Doh in the classroom.  Every year, the first rule students come up with is, “Don’t eat the Play-Doh!”

Students first make a sphere out of Play-Doh, then use their hi-tech cutting device to cut through the sphere, parallel to the base.  I explain the concept of 2D cross section, and we look at its shape. Does the shape of the cross-section change if we cut the sphere perpendicular to its base?  We repeat this with a cone and with a torus.   This takes about one full class period.  Students LOVE working with Play-Doh, but fun takes time.  It also takes time for them to create, cut, and analyze their shapes.

What is 3D printing and where is it used?

I show these short videos to all of my classes before we 3D print. I look for updated videos each year, but these videos work well:

This usually leads to a productive discussion of 3D printing applications in medicine,  food manufacturing, and construction of homes.

What is the Vocabulary of 3D Printing?

I want my students to speak with confidence about 3D printing.  Using the 3D printer at my school, I point out the extruder, nozzle, build plate / print bed, x-axis, y-axis, and z-axis.

I have a few 3D prints that failed during printing, and I display pictures of those prints to discuss these terms: filament, shell, infill, raft, and supports.  I also talk about the difference between PLA and ABS filament and the cost of a typical roll of filament.   I also pass the failed prints around the room so students can explore them more closely.

I show about the first 2 minutes of this video to introduce supports:  3D Printing Using Support Materials – 3D Printing Tech Tips

In my Honors Geometry classes, we do one final activity:  I ask them to draw 2D cross-sections of the top layer of various 3D objects as they would 3D print.  Years ago, I found this fabulous lesson on the Mathematics Assessment Project website.  The lesson asks students to imagine 3D objects filling with water and draw 2D cross-sections of the surface of the water.  I altered this lesson (and some of the images in the exercise) for my 3D printing needs:  I have students draw 2D cross-sections of the objects as if the objects were 3D printing.  I also ask students if the objects would be able to 3D print in the given orientation without supports.  Of course, we bring out the Play-Doh again so students can check their thinking with a “real” object.

What is the Process to 3D Print an Object?

Student don’t always understand that the process to 3D print an object is very different than the process to print a document with a laser printer.  I created this diagram so that we can discuss what software is used for each part of the process and which person (student or teacher) is responsible for each part of the process.

An image that I created for my classes to discuss the 3D printing process.

Student Reflection

I am trying to integrate more student reflection into my classes, and I love using the free website for student reflection.  First, I create a prompt for my students.  As students respond to my prompt, they can add mindset-inspired sentiments such as:  grew my thinking, learned something new, challenged my ideas, or made me think.  After they respond to my prompt, they are able to read and reply to the responses of their classmates using words and reactions such as: I agree, I have a question, moved my thinking, tell me more, etc.

Here is the reflection prompt that I gave my students this year:

grokspot 3dprinting

Introduce The First 3D Printing Project

My student goals for this project in Honors Geometry are:

  • Learn to use Tinkercad to create objects with holes and objects with multiple aligned parts
  • Learn to export objects as STL files and view the file to identify any gaps between parts
  • Determine the best orientation for 3D printing an object
  • Determine whether or not supports are needed
  • Draw 2D cross-sections of their 3D design

This year, to bring a little cross-curricular work from our Outdoor Education Program’s pollinator garden, I asked my students to design a honeycomb from a bee hive with at least 7 chambers for holding honey.  Maybe we can use these designs again later in the year when we reach the topic of hexagons.

More Info

Looking for some tips to make your 3D printed projects run smoothly? Or more info on the 3D printing process?  Check out this article about 3D printing that I wrote for

Happy 3D Printing!


What Worked & What I Would Change With The History of Math Activity

As promised, here’s a recap of what worked and what I would change with the history of math card activity   The whole activity (including discussion) took about 30 minutes.  We could have spent more time discussing the events, but we had a shortened bell schedule that day.

What Worked

  • I was impressed by the conversations that students had with each other while trying to figure out the order of events.  They tried to analyze specific words as they fit the events into a historical context.
  • One of my history colleagues, who is an expert on gaming in the history classroom (Twitter:  @gamingthepast) suggested that I project the dates of some major world events to give students perspective while they worked.  That was helpful.  Here are the events I projected:

    Some important world events that you might want to consider as you sort:

    50,000 BCE — Neanderthal Man

    2,500 BCE — Construction on the Great Pyramid in Egypt

    750 BCE — Founding of Rome

    1215 CE — Magna Carta

    1492 CE — Columbus discovers America

    1776 CE — Declaration of Independence

    1914 CE — WWI begins

    1969 CE — Moon landing

    1977 CE — First Star Wars movie released

  • My students worked in groups of two, with a few groups of three as needed.  This was the perfect group size.
  • When groups finished sorting, I asked them to visit other groups to see what other groups did.  This worked REALLY well.  I enjoyed hearing friendly debates about card placements (“Can we use the word BOOK before the invention of the printing press?”) and discussions of potentially related world events.  Groups were allowed to change their order of events after this visit, and some students were convinced to make changes by their peers.
  • Students enjoyed checking the dates on the back of the card and joked about how the answer was in front of them the whole time.  A few groups put all the cards in the right order on the first try.
  • I asked students which events were most surprising to them.  The date of the first use of the symbol Pi in a book was a popular answer.
  • I asked students to take pictures of the cards, in order, and put the pictures into their notes.  (We use Microsoft OneNote at my school.)
  • I have some pictures of the Mandelbrot set in my classroom, and the card on fractals led us into a quick discussion of how operations on complex numbers are used to create these images.

What I Would Change

  • How, oh how, did I forget to include the first algebra book and the origin of the word algebra?  My deepest apologies to Islamic mathematician Al-Khwarizmi, who some call the Father of Algebra.  I’ll add this for next year.  I’ll also include a fact from a Chinese and/or Indian mathematician.  Adding three more cards might turn into too many to sort, so I may decide to cut out a few of the original cards.  I’ll repost the cards when I make the changes.
  • I asked students to take pictures of the cards AFTER we discussed the events.  However, some students had already put their cards away and had to re-sort them to take pictures.  I’ll mention the picture-taking step earlier next time.
  • Since we reflected during class, I did not assign a reflection activity as homework.  I still think this would be a useful assignment, so I’ll keep it on the “to do next year” list.

Hope you enjoy the cards, if you can use them!

Making 3D Printed Flowers from Polar Graphs

This year, I decided to decorate my classroom/office with hanging 3D printed “flowers” that are made from graphs of polar equations.  I am also excited to do this activity with my Pre-Calculus students when we get to polar equations in February.  For now, I made my own flowers.  They look like this:

Polar Flowers

I created my graphs in Mathematica and prepared them for printing with Tinkercad.  Since not all teachers have access to Mathematica,  I also figured out how to create these same flowers in GeoGebra.   I tried with Desmos and had some issues.   The end of this article outlines the GeoGebra, Mathematica, and Tinkercad procedures.  I also quickly discuss what happened when I used Desmos.

How long does this take?  Some of my bigger flowers took about 2-3 hours to print, while the smaller ones took 40-60 minutes.  Once I figured out and mastered the process to create a flower, I could create one and have it ready to print in 15-20 minutes.

How will I turn this into a project for my students?  Once we graph some polar equations, I will give students a day or two to create their own equations and explore any patterns that they observe.  When students find interesting patterns, they become motivated to help each other figure out what is causing the pattern.  Their enthusiasm is contagious!  I’ll ask them to turn one of their favorites into 3D printed artwork, such as a flower.  I’ll also ask them to write a quick reflection about what they observed and learned in this activity.  Giving students free rein AND a 3D printed object that they can keep makes the learning more fun, memorable, and student-centered.

What kind of filament did I use?  I printed with PLA filament in green, orange, yellow, and purple.  But if you only have white, just go with it!

Directions for creating the flowers are below.  My next post will describe how I introduce 3D printing to my freshmen Honors Geometry class.  Have a great week, and let me know how this activity works if you try it!

Directions with GeoGebra & Tinkercad (BOTH ARE FREE)

I used the GeoGebra Graphing Calculator.  I tried LOTS of different settings/file type combinations in GeoGebra before I found a method that worked:

  1.  Enter the equation for your graph, such as r = sin (2θ), to get a polar graph like this:graph1
  2. Click on the gear in the top right corner of the graph.  UNCHECK Show Axes. Click on Show Grid and change it to No Grid.   settings on graph 2
  3. Right click on the graph and choose Settings.  On the Style tab, change the Line Thickness to the maximum value (13).  Change the Filling to Hatching, and change the Spacing to the minimum value (5).
    graph settingsClose this window so that you see a solid-filled graph on a white screen:final geogebra image
  4. Use the menu on the top left of the screen to select Download as -> png.
    download settings
  5. Tinkercad needs the file to be in SVG format.  I used a free online converter to convert the PNG file to SVG format.  In the converter:  choose your file, scroll down and click on Convert File, then download the SVG file to your computer.

    Note for those scratching their heads as to why I didn’t export an SVG file right from GeoGebra: I tried this many times.  No matter which settings I chose for the graph (line thickness, hatching style, no fill, etc.), the file imported into Tinkercad as either a solid block with some sort of hole in the middle, or as a discontinuous jagged-edged outline that would not print well.   

  6. Your work in GeoGebra is finished;  it’s time to open Tinkercad. Create a new object and select Import.tinkercad export
  7. Choose the SVG file you just created.  If you get a red error message like I did below, alter the scale (I used 20%).
    clover import
  8. Your imported file should look something like the image below.  If your flower is too big, use the Tinkercad keyboard shortcuts for 3D Scaling to scale the object to a better size.
    tinkercad image
  9. You may want to change the height of the flower (my height defaulted to 10 mm).  To do this, click on the image to see the measurements, find the height measurement, and type in the new height.  Note of caution:  I had a hard time removing thin flowers (3 mm) from my printer’s build plate without breaking them.
  10. Some flowers meet in the middle at a point.  If this happens, you should reinforce the center of the flower so that you can remove it from the build plate without breaking any petals.  I added a cylinder of the same height as the flower in the middle of the shape to support the petals:
    circle in middle
  11. If you want to hang your flower: use the cylinder hole shape to create a hole in one of the petals.  On the image above, there is a hole through one of the lower petals.
  12. When you are finished editing your flower, export your file as an STL, and send it to your printer or slicing software!  Done!


Directions with Mathematica (NEED A LICENSE) and Tinkercad (FREE)

  1.  Create your graph with no background.
    mathematica clover
    Note:  Sometimes the import is a little jagged.  If that happens, I find it helps to use a sightly thicker line, such as Plotstyle->Thickness[.02].
  2. Use the more dialogue to export the image as SVG:mathematica export
  3. Import the image into Tinkercad (go back to step 7 in the process above for details).  I find it helpful to change the scale to at least 50%.   Edit the image in Tinkercad as needed (step 8 in the process above).

Desmos (FREE) and Tinkercad (FREE)

In theory, you should be able to use Desmos (instead of GeoGebra) to create and export your graph.  Of course, creating the graph in Desmos works beautifully.  I only had problems with exporting the graph.  Desmos exports PNG files.  I tried changing various graph settings,  converting the PNG file into SVG files, and importing into Tinkercad.  None of my attempts imported a usable image into Tinkercad.  However, the technology-lover in me believes that it can be done.  It’s just a matter of finding the right graph settings.



A Short History of Math Activity

It’s close to start of another school year, and I find myself thinking about what I should focus upon during the first few weeks of school this year.  I always spend time discussing mindset and the “Power of YET” with my students.  I love to show the Week of Inspirational Math videos and run a few of the Week of Inspirational Math activities from my favorite resource,  I survey my students about their feelings about working in groups so that we can share, and then we embark on some activities to help everyone learn and practice how to work in groups.  I also start some number talks. (If these topics sound like they are inspired by Jo Boaler, they most certainly are!)

In comes the topic of History of Math.  Geometry textbooks rarely mention Euclid’s Elements, let alone any other interesting tidbits about the history of math.  So every year, I start with a small activity to help students put the subject of geometry in perspective.   Where did the topics in our textbook come from?  Did people always use the π symbol to represent pi?  Did humans always use the decimal point?  What accomplishments have women made in the field of mathematics?

It’s not easy to introduce the history of math in one class period.  Sure, I could take longer, but I do lots of activities and 3D printing that also take time later in the year.  So for me, it’s a matter of hitting the highlights and inspiring students to learn more on their own.  I’ve tried lecture (boring!), web quests (just didn’t work for me and took longer than I hoped), and interactive worksheets (they were OK, but not very inspiring).  This year, I have a new idea:  I created some classroom sets of history of math timeline cards.

The front of each card has a history fact that is interesting or important about the development of geometry or other mathematics.  On the back of each card, hidden from the students by the card protector, is a (sometimes approximate) date of the fact.  My plan is to ask my students to work in groups to try to put the facts in chronological order.   When they finish, they can look at the back of the card to see the actual date.  We can discuss what they got right, what they got wrong, what surprised them, and what other facts they would like to know.  Homework for the night will be a quick reflection on the activity (which I have not made yet).

When I posted pictures of the cards I made on Twitter, I received some requests to share them.  I dutifully found the source of each fact so that I could update the cards with sources.  I also changed the pictures to make sure that the pictures I used were either in the public domain or had a creative commons license.  Hopefully I didn’t forget anything!  Here is a PDF of my history of math timeline cards.

Here’s how to make the cards:

  1. Print the cards (color is nice, but not required).  Each row has three columns.  The first two columns are the front and back of the card.  The last row is the source of the image, which is just FYI.
  2. Cut out the first two columns of each row, then fold the card on the vertical column line so that it has a front and back.  Just discard the last column of the card, unless you want to fold it on the inside of the card.
  3. Put each card in a card protector (I bought these on Amazon). If you can see the date on the back of the card through the card protector, just slip a piece of cardboard (or these from Amazon) in the card protector.   After sorting the cards, students can temporarily lift this cardboard to check the date.
  4. You’re ready to go!

If you don’t have the budget for card protectors, you can just fold and laminate the cards (or fold and tape them closed) and ask students not to look at the date on the back until they finish sorting.

I am sure I will decide to alter this activity after my students work through it in August, but learning what worked and what didn’t work is often my favorite part!


Update:  See my post about what I would change with the activity next time, which includes some new cards I’ll be adding to reflect the important contributions from mathematicians in other parts of the world.



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