Lesson
Challenge
Standards

Snubbing

What is it?

Archimedean solids obtained by snubbing Platonic Solids

  • Snub Cube
  • Snub Dodecahedron


snub_dod.png
Image of Snub Dodecahedron from 3dwarehouse


Image from Thingiverse member pmoews



Snubbing is a process of taking all the faces of the solid, pulling them outward so they no longer touch, then giving them each a small rotation on their centers (all clockwise or all counter-clockwise) until the spaces between can be filled with equilateral triangles.
  1. Separate all the faces
    snubbing1.png
    Image from http://www.math.nus.edu

  2. Consider two originally adjacent faces of the cube. Attach an equilateral triangle to each face such that the triangle is attached to the edge of the face that was originally the common edge between the two adjacent square faces. Repeat the process by attaching an equilateral triangle to every edge of the square faces. Now move the adjacent faces towards each other as indicated by the arrows to place the equilateral triangles side by side.
    snubbing2.png
    Image from http://www.math.nus.edu
    This movement causes equilateral triangle 1 to point to the left or to the right of triangle 2 and in the process the square faces are tilted. Throughout the whole snub cube the triangles in the pairs of equilateral triangles will either point to the left or right of each other. If the triangles point to the left it is called a left snub cube and if it points to the right it is called the right snub cube. Polyhedra related in this way are said to be enantiomorphic. Note that equilateral triangles are used because faces of Archimedean solids are regular.
    snubbing1.png
    Image from http://www.math.nus.edu
  3. Three square faces meet at each vertex of the cube. Pulling the faces of the cube apart forms empty spaces between the three faces sharing a common vertex. The common vertex splits into three vertices, one for each of the three faces that meet at the vertex. Place an equilateral triangle such that the vertices of the equilateral triangle meet the three vertices that were split from the common vertex. The green triangle represents the three-gon that replaced one of the vertices where the three purple square faces meet.
    snubbing4.png
    Image from http://www.math.nus.edu