Drilling a Hexagonal Hole

By using a suitable roller restricted to rotate inside a hexagon, one can force a point on the roller to trace out an exact hexagonal path. This construction is due to the authors, inspired by a 1939 construction for a perfect square-hole drill. In this case, though, the roller does not touch all six sides of the hexagon, but it always touches sufficiently many sides to ensure rigidity within the ambient hexagon. As the roller rotates within the ambient hexagon, the center of the blue circle has a locus that is an exact hexagon. The red points are the points on the roller that touch the hexagon and enforce rigidity within it.
  • Contributed by: Barry Cox (University of Wollongong, Australia) and Stan Wagon (Macalester College, USA)


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


The construction of the roller proceeds as follows.
1. Draw a unit hexagon centered at .
2. Get the top arc using a circle of radius centered at (0,1).
3. Get the bottom arc using a circle of radius centered at (0,1). Let be the point of intersection of the bottom arc and the hexagon.
4. Get the northwest arc using center and radius . The northeast arc is similar.
5. Get the southwest arc using center and radius . The northeast arc is similar.
6. Use the intersection of the northwest and southwest arcs to define the upper termination of those arcs; the east side is similar.
As increases from 0 to the entire roller is rotated clockwise through and then translated leftward so as to place on the hexagon. For the first 0.049 radians of this rotation there are only four points of contact: the upper and lower points guarantee that there is no up-down motion, and the two points on the southwest and southeast sides guarantee rigidity in the left-right direction. After this critical value of there are five points of contact. As the rotation continues, symmetry governs the patterns and there are always four or five points of contact, sufficient to ensure rigidity.


Contributed by: Barry Cox (University of Wollongong, Australia) and Stan Wagon (Macalester College, USA)
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+