Virtual Polyhedra

    The Encyclopedia of Polyhedra

    By George W. Hart


    Space is not a passive vacuum, but has properties that impose
    powerful constraints on any structure that inhabits it.
    ---Arthur Loeb

    Copyright

      Copyright 1996, 1997, 1998, 1999, 2000 George W. Hart

    Introduction

      Welcome to this collection of thousands of virtual reality polyhedra for you to explore. I hope you enjoy playing with them as much as I enjoyed making them. There are hundreds here which have never been illustrated in any previous publication.

      Polyhedra have an enormous aesthetic appeal and the subject is fun and easy to learn on one's own. One can appreciate the beauty of this image without knowing exactly what its name means --- the compound of the snub disicosidodecahedron and its dual hexagonal hexecontahedron --- but the more you know about polyhedra, the more beauty you will see.

      This site is a free self-contained easy-to-explore tutorial, reference work, and object library for people interested in polyhedra. You may choose to simply view the virtual objects for their timeless, serene aesthetics, or to read the related mathematical background material at various levels of depth. Of course, as an academic type, I feel obliged to include a few exercises. And as it says in the textbooks, you'll learn a lot more if you work on the exercises yourself before looking at the solutions.

      I believe the best way to learn about polyhedra is to make your own paper models or other models.  The second best is to play with a set someone else has made. You can do that here because you can look at, move, and spin these models which I have made for you. And in one respect, virtual models have an advantage: you can travel inside them to gain a perspective not possible in paper models. After exploring my virtual models, I hope you choose to make some of your own paper models.

      This work is complementary to my Pavilion of Polyhedreality, which you may wish to visit to find additional information on polyhedra and links to other related material.  You may also like my polyhedra-based geometric sculpture.

    About Virtual Reality Models

      Your web browser may already be set up for viewing virtual reality models. Find out: click on this picture and see if you can view a 3D version of this compound of five cubes.

      • If your web browser shows a message that it does not know what to do with files of mime type x-world/x-vrml then read about how to set your web browser to view virtual reality files.
      • If your VRML browser is operational and you see five crisp cubes, like the image here, try this experiment. First study the object, spin it around, and see it as five interpenetrating cubes, one in each color. Then, imagine what you would see if you were sitting at the center of this object. This is the same as asking what the intersection of the five cubes looks like. (Hint: how many sides must it have ?) After you imagine the view from the center, use the virtual reality viewer to travel to the inside of the compound of five cubes. What do you see ? Answer.


      Before going on too far, you may also want to read about:

      • viewing tips for a range of VRML browsers
      • background material on polygons
      • the coloring system I am using
      • those polysyllabic polyhedron names
      • glossary
      • classroom ideas
      • polyhedra and art
      • my polyhedral sculpture
      • complete list of articles
      • annotated bibliography of polyhedra references
      • SEARCH these pages


    Table of Contents

    Each of these top level sections contains some background information, a list of virtual reality models, and links to further topics. Try reading the background sections in the order listed here, being sure to work on the exercises.

    Platonic Solids (Regular Convex Polyhedra)

      Background --- List of models

    Kepler-Poinsot Polyhedra (Regular NonConvex Polyhedra)

      Background --- List of models

    Archimedean Polyhedra (Semi-Regular Convex Polyhedra)

      Background --- List of models

    Prisms and Anti-Prisms

      Background --- List of models

    Archimedean Duals

      Background --- List of models

    Quasi-Regular Polyhedra

      Background --- List of Models

    Johnson Solids (the remaining convex polyhedra with regular faces)

      Background --- List of models

    Pyramids, Dipyramids, and Trapezohedra

      Background --- List of models

    Compound Polyhedra --- Introduction

      Background --- List of models

    Stellated Polyhedra --- Introduction

      Background --- List of models

    Compounds of Cubes

      Background --- List of models

    Convex Deltahedra

      Background --- List of models

    Zonohedra

      Background --- List of models

    Uniform Polyhedra

      Background --- List of models

    Uniform Compounds of Uniform Polyhedra

      Background --- List of models

    Stellations of the Icosahedron

      Background --- List of models

    Stellations of the Rhombic Triacontahedron

      Background --- List of models

    Other Good Stuff

      Alphabetic list of articles

      Alphabetic listing of polyhedra models

      Glossary

      Acknowledgments

      References




    Notes:

      Please email comments, typos, errors, dead links, and any suggestions to george@georgehart.com. I continue to expand everything, so come back often and check!