Visualising mathematics with 3D printing

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There are too many maths books out there that repeat the same old boring stories about Pythagoras drowning people for inventing irrational numbers, Gauss adding up numbers quickly, or Dirichlet eating too many grapes[citation needed]. Luckily, Visualising Mathematics with 3D Printing (Amazon UK, USA) is not one of these: its author, Henry Segerman, has achieved the seemingly impossible and written something refreshingly original and different.

segerman1In this book, Segerman takes you on a tour of modern geometry, exploring symmetry, hyperbolic surfaces, four-dimensional shapes, knots and much more. These concepts are hard to visualise. While pictures in a book can help, there is only so much that can be shown in a 2D picture. This is where this book comes into its own: it focuses on using 3D objects to visualise these concepts, with instructions for 3D-printing the objects, as well as interactive simulations, available on the book’s website. For many objects, such as the tiled genus 3 surface shown below, the ability to explore these objects from multiple angles is key to understanding them.

One of my favourite concepts in the book (and one of Segerman’s favourites; perhaps this is why it comes across as the most interesting) is the stereographic projection. By placing a light source at the top point of a sphere, this projection maps the 2D surface of a 3D ball onto the 2D plane. Incredibly, this projection preserves angles: a right angle on the sphere is mapped to a right angle on the plane, as seen in the object below.

Stereographic projection of a dodecahedron

Stereographic projection of a dodecahedron

A 3D shape can be represented by lines drawn on a sphere; these lines can then be stereographically projected onto the 2D plane. The result of this projection for a dodecahedron is shown on the left.


The same process can be followed to project a 4D solid onto the 3D “plane”. Below, you can see a model of half of the 120-cell, a regular 4D shape made by connecting dodecahedra. Without a 3D model of this or the simulation below, it is impossible to get an idea of what this projection looks like.

There is so much more in this book that I don’t have space here to show you. My best advice is to go out and buy yourself a copy of the book. Be warned though, that you may find yourself also buying a very expensive 3D printer.

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Matthew Scroggs is a PhD student at UCL working on finite and boundary element methods. His website,, is full of maths and now features a video of him completing a level of Pac-Man optimally.
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