From Math to Magic: The Greeks behind 3D Printing

The name Guido Fubini (1879-1943) may not ring a bell, but practically all of us have heard of 3D printing, a seemingly magical process where one literally makes a three-dimensional solid object using layers of 2- dimensional materials fed into the printer guided by software [1]. Well, Mr. Fubini invented mathematics which (unbeknownst to him) proved to be useful in writing 3D printing software, and that happened way back in 1907!

The Magic of 3D Printing

3D printing starts with a digital file derived from computer aided design (CAD) software. Once a design is completed, it must then be exported as a standard tessellation language (STL) file, meaning the file is translated into triangulated surfaces and vertices. The STL file is then sliced into hundreds – sometimes thousands – of 2-D layers. A 3D printer then reads the 2D layers as building blocks which it layers one atop the other, thus forming a three-dimensional object.” (Source: https://www.stratasysdirect.com)

It may come as a surprise to some that 3D printing isn’t that new: the technology has been around since the late 1980s where it was mainly used in Japan, the US and Europe to make prototypes for industrial parts. Two decades later, it took the world by storm, becoming more mainstream and powered by increasing sophisticated computing hardware and software. Today, hobbyists with a modest budget of less than \$5,000 can have access to the technology to make pretty cool toys and decorative accessories in the comfort of their homes. Indeed, the range of 3D-printed objects is staggering. They include fashion items such as dresses, handbags, shoes and jewellery, architecture scale models, medical prosthetics, aircraft components, automobile parts and even whole automobiles [2]. The only limit is the capability of the printer and the computer software used to convert digitally designed shapes into 3D pixels for printing.

The Math Behind 3D Printing

Most people – including those who use 3D printing – have no idea why 3D printing is possible. It defies belief that a printer can be designed to melt plastic or biodegradable materials, then fuse layers of the melted material to create a solid object. This magical process would not possible if not for a theorem proved by Fubini, an Italian mathematician who worked in many different areas of mathematics, including analysis, the calculus of variations and group theory. But he is best remembered for the theorem that bears his name. Fubini’s theorem establishes a link between multiple integrals and a repeated one. The theorem may be stated as follows:

What this statement implies is that any object of n dimensions can be represented as an agglomeration of n-1 dimensional shapes. In other words, one can manufacture a 3D object by sticking together a bunch of 2D sheets, which is what 3D printing essentially does. Given the right software, the type of shapes that can be 3D-printed is limited only by human imagination. In some cases, what is printed is truly beautiful and awe-inspiring like the shapes created by mathematician Henry Segerman (pictured below)

Listen to Henry Segerman on how he and collaborator Craig Kaplan created the bunny sculpture in this YouTube video.

Notes:

[1] Many different materials can be used for 3D printing, such as ABS plastic, PLA, polyamide (nylon), glass filled polyamide, stereolithography materials (epoxy resins), silver, titanium, steel, wax, photopolymers and polycarbonate.

[2] In 2014, Swedish supercar manufacturer, Koenigsegg announced the One:1, a supercar that utilizes many components that were 3D printed. In the limited run of vehicles Koenigsegg produces, the One:1 has side-mirror internals, air ducts, titanium exhaust components, and complete turbocharger assemblies that were 3D printed as part of the manufacturing process (source: wikipedia)