9758

Enantiomorphs of the Truncated Icosahedron

The oldest studies of polytope composition predate Kepler's Harmonices Mundi, which included the famous stella octangula. Recent studies [1] have shown the existence of a surprising number of regular compositions of regular polytopes. This Demonstration shows the composition of a regular polytope, the truncated icosahedron, from a nonregular "quasiprism". Like the dodecahedron, another polytope with icosahedral symmetry, the truncated icosahedron displays enantiomorphism. With this polytope, however, the enantiomorphism requires not only two non-equivalent halves, but additionally requires two non-equivalent compositional elements.

THINGS TO TRY

SNAPSHOTS

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

DETAILS

The composition shown above diverges from the historical trend of composing regular polytopes from other regular polytopes. These unsightly pentagonal quasiprisms remained undiscovered for thousands of years, but the enantiomorphism they display is different from that of any regular composition.
A numerical calculation of the angle of offset between pentagonal faces gives approximately 8.772 degrees. Can the offset angle be given as a ratio of ? If not, then the projected vertices of stacked pentagonal quasiprisms will approximate a continuous circle. Alternatively, a stacking of pentagonal quasiprisms will be defined by an exact period if the angle can be rationalized.
Other polytopes must have irregular compositions as well. Do other nonregular compositions of regular polyhedra exist so that the composition uses a maximum of two nonregular polytopes? Specifically, do compositions of the icosahedron with an enantiomorphism exist? This last question may already be answered in the extensive literature about the icosahedron.
More information about uniform composition of uniform polytopes can be found on this website on polyhedra compounds, which contains useful information from [1].
Reference
[1] J. Skilling, "Uniform Compounds of Uniform Polyhedra," Mathematical Proceedings of the Cambridge Philosophical Society, 79, 1976 pp. 447–457.
    • 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.









 
RELATED RESOURCES
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 © 2014 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+