Tiling a Sphere

by pmoews, published

Tiling a Sphere by pmoews Jan 27, 2013
2 Share
Download This Thing! Customize Order This Printed Tools & Utilities

Thing Info

3574Views 1523Downloads Found in Math Art
Report Thing


Here is a simple illustration of the Pythogorean tiling of a sphere. The idea comes from a web page by Willian E. Wenger which explains how to carve a periodic tiling on the surface of a sphere.


A cube was used for this example. A simple two dimensional pattern with 4 fold rotation axes was generated. Nine copies of the pattern, which extends to infinity, are shown in the image above. Each cell has a 4 fold rotation axis at the center and a 4 fold rotation axis at each corner. One interesting thing is that the 4 fold axes at the corners are converted to 3 fold axes as the pattern goes from 2 to 3 dimensions.

To tile the sphere six copies are transferred to a cube - see images. The conversion to three dimensions changes the 4 fold axes at the corners to 3 fold axes on the cube. The 3 fold axes are preserved when the cube is converted to a sphere.

Three dimensional objects are the cube with the pattern transferred to its faces:


and two versions of the cube converted to a sphere:


Look at the corners of the cube. Now consider what would happen if the cube were flexible and inflated to a sphere. On the sphere the centers of the cube faces are represented by 6 hypocycloids and the corners by 8 distorted triangles.

OpenSCAD creates the correct symmetry for the "tiling" of all the Pythogorean solids. See things 40276, 39977, 39818, and especially 39424 which has a similar example.


Cube_with_pattern.stl and sphere_with_pattern_70.stl can be printed directly on the build surface. Sphere_with_pattern_50.stl needs external support.

The openSCAD code that generates the two dimensional dxf files is included as is the code that makes the three dimensional stl files. Dxf files and pdf files are in p4_patterns.zip.

A pdf file that shows the cube in 2 dimensions is included - p4_pattern_6.pdf. It can be printed, cut out, and folded to form a paper version of the cube.

More from Math Art

view more

Thing Info

3574Views 1523Downloads Found in Math Art
Report Thing


Liked By

View All



Tiling a Sphere by pmoews is licensed under the Creative Commons - Attribution license.

Give a Shout Out

If you print this Thing and display it in public proudly give attribution by printing and displaying this tag. Print Thing Tag

All Apps

This App connects Thingiverse with Makeprintable, a cloud-based mesh repair service that analyzes, validates and repairs most common mesh errors that can occur when preparing a 3D design file for p...

App Info Launch App

Kiri:Moto is an integrated cloud-based slicer and tool-path generator for 3D Printing, CAM / CNC and Laser cutting. *** 3D printing mode provides model slicing and GCode output using built-in...

App Info Launch App
KiriMoto Thing App

With 3D Slash, you can edit 3d models like a stonecutter. A unique interface: as fun as a building game! The perfect tool for non-designers and children to create in 3D.

App Info Launch App