Archive of posts filed under the 3D Printing category.

## Henry Segerman at MoMath

Mathematician, artist, and 3D-printing virtuoso Henry Segerman will be speaking at the Museum of Mathematics on October 5th, 2016.

Henry is currently a professor at Oklahoma State University, where he researches geometry, topology, and mathematical visualization.  His mathematical 3D printing is truly amazing:  to start, you can check out his triple gears, stereographic projections, and Hilbert curves.  And he has recently published a book, Visualizing Mathematics with 3D Printing, that includes companion 3D prints that readers can download for free and print themselves!  He is also involved in fascinating work in spherical video (see this spherical droste video for an example) and virtual reality, and has been featured in several Numberphile videos.

Henry will be giving a talk as part of MoMath’s Math Encounters series.  His talk is titled “3D Shadows: Casting Light on the Fourth Dimension”, and is sure to make for a fascinating evening.  I am proud and excited to be introducing Henry, whom I’ve known for many years, and whose work inspires me both as a mathematician and as a teacher.

You can find out more about the event and register here.

UPDATE:  The museum has made Henry’s full talk is available here.

## 3D Printing in Math Class

We were fortunate to receive a 3D printer for use in our math class midway through the last school year.  Figuring out how it best worked was fun, and often frustrating.

We enjoyed a variety of successes throughout the spring, printing simple surfaces and some complicated ones, too.  It was fascinating to uncover how the printer, and its software and hardware, tackled certain engineering obstacles, like how to print in mid-air!

Ultimately I got comfortable enough to start producing some lesson-specific mathematical objects.  This trio of solids I designed worked perfectly as an introduction to Cavalieri’s principle:  seeing and holding the objects immediately initiated the conversations I wanted students to have.

By the end of the school year, I felt comfortable enough with the process to run our first official student project.  It was fairly open-ended, with options for students, but essentially the idea was to design an object for printing using equations and inequalities.

The project was a success, and here are some of the student designs.

I’m looking forward to exploring some new ideas and projects this year.  It’s clear to me that this technology, which is fundamentally mathematical in concept and design, can play a valuable and meaningful role in math class.

## Cavalieri’s Principle and 3D Printing

These 3D printed objects served as an excellent starting point for a classroom conversation on Cavalieri’s Principle.

Each shape above is an extrusion of the same square.  In the middle, the square moves straight up; at left, the square travels in a complete circle from bottom to top; and at right, the square moves along a line segment and back.

The objects all have the same height.  Since at every level they have the same cross-sectional area, by Cavalieri’s Principle they all have the same volume!

Cavalieri’s Principle is a simple but powerful idea, and one that can be easily demonstrated around the house.  Here are some other examples using CDs and CD cases.

## 3D-Printed Chmutov Surface

My most successful 3D print so far:  a Chmutov surface!

Chmutov surfaces are algebraic surfaces that have interesting properties involving their singular points.  This surface has equation

$2[x^2(3-4x^2)^2 + y^2(3-4y^2)^2 + z^2(3-4z^2)^2]=3$

And in addition to being quite beautiful, this Chmutov surface is already coming in handy!

## 3D Printing a Cube Frame

I’ve been having a lot of fun exploring mathematics through 3D printing.

Recently, I had the idea to 3D-print a “cube frame”, that is, the edges of a cube.  My first mathematical task was to figure out an equation whose graph was such a cube frame.

It took a little work, but I ended up with this.I exported the graph into the appropriate file format and successfully printed my cube frame!

Just as fun as the mathematical challenges of producing the graph are the engineering challenges, and mysteries, of the 3D printing process itself.  For example, notice the scaffolding that the software automatically adds in order to print the top square of the cube.

There are some interesting mathematical and structural consequences of the scaffold-building algorithm, but even more amazing was that one “face” of the cube contained no scaffolding at all.  This meant the 3D printer printed one edge of the top square into thin air!  And it succeeded!

I’m excited and inspired by 3D printing, and I’m looking forward to finding more ways to integrate into our math classrooms.