Jan.27, 2013

(portrait of Archimedes)

In the near future, 3D printer will be used in school to help students learn core Science, Technology, Engineering and Mathematics (STEM) principles.

Mathematician Oliver Knill and Elizabeth Slavkovsky of Harvard University write a paper about 3D printing and its practical use in education. They combine Archimedes and 3D printing: to illustrate Archimedes' method using models produced with 3D printers.

This approach allowed us to create physical proofs of results known to Archimedes and illustrate ideas of a mathematician who is known both for his for his mechanical inventions as well as his breakthroughs in geometry and calculus. We use technology from the 21st century to trace intellectual achievements from the 3rd century BC. While we celebrate the 2300th birthday of Archimedes (287-212 BC) in 2013, we also live in an exciting time, where 3D printing is becoming popular and affordable.

In history, many great Mathematicians have worked to use mathematics to communicate the 3D world, one of them is Archimedes. The paper introduces briefly Archimedes and his works. Archimedes was a great Greek mathematician, physicist, engineer, inventor in 2300 years ago. He was "a great thinker in terms of 3D math" writes Slavkovsky. Archimedes designed many practical apparatus using mathematics to solve problems, one example is Archimedes screw, a machine that could raise water efficiently and is still in use today.

When students move from two-dimensional (2D) geometry to three-dimensional geometry, options for working in 3D are often limited by materials or standard designs. When Slavkovsky was a geometry student in 1993, she spent a long time trying to mentally imagine geometry shape, for example the Platonic solids, a regular, convex polyhedron. Then she started modeling them out of clay and eventually get a real solid object in hand.

When price of 3D printer gets affordable, Slavkovsky purchased an UP! Plus 3D printer for getting hands-on experiences. She created 3D models in Google Sketchup and then printed it out on UP! Plus. For complex geometry shape she uses Shapeways' 3D printing service.

Here are some nice examples of Archimedes' mechanical inventions they have made on a 3D printer:

1. A model of the "cochlias" or Archimedes screw. The model is not a connected body, but it is printed with connectors, which are then broken.

2. Two attempts to visualize cutting a parallelepiped into 6 pieces of equal volume.


3. The drinkable Archimedes Proof. The Mathematica model and the printout. As a demonstration, one can fill the spherical reservoir on top with water. After it has dripped down, it fills the complement of a cone in a cylinder. The volumes match.

(Images credit: Oliver Knill and Elizabeth Slavkovsky)

These physical models are important for hands-on active learning. At Harvard teachers have already used 3D printer to illustrate concept. "3D printing stimulates a student's mechanical-spatial awareness in ways that textbooks cannot." says Timothy Jump, a teacher at Benilde St. Margaret's Highschool.

The two authors conclude the article with some tips for printing in 3D for anyone who wants to explore the 3D printing technology.

First of all, the technology needs patience. Don't expect that things work immediately. Especially, when printing on your own printer needs a lot of time and ability to overcome obstacles.

 

Printer parts can fail, calibration can be subtle, software glitches occur frequently. Also commercially available printing surfaces can refuse to print certain parts. It can happen for example that objects pass initial tests but cannot be printed in the end.

Download the PDF file of ther paper "Thinking like Archimedes with a 3D printer" on arxiv.org.

 

 

 

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