Topological Geometry can be defined as the study of geometric objects that have topological composition. Thus, the ring and star polyhedra found here are examples of annulated polyhedra or polyhedra constructed with polygonal annuli; an annulus being a two space analog of a torus. The Crinkled and Crown polyhedra also are included in this category as Crinkled and Crown annuli represent the operations of addition and subtraction on annuli which are then used to build polyhedra. The Foil annuli are just generalizations of the annulus as are the Foil polyhedra constructed from them.
Including the Exploded polyhedra, the total classes of polyhedra depicted at this site comes to five, however, by creating a table with each class of polyhedra in rows and columns and then integrating them, its possible to create 25 subcategories of polyhedra. For example, the Exploded Crown polyhedra represent the integration of (obviously) Exploded polyhedra and Crown polyhedra. The rest follow…
The models depicted on this web site serve a dual purpose as both molecular and mathematical models. Many of them are novel to Geometry/Materials Science. They are in part inspired by the work of Allan Holden and especially his book Shapes, Space and Symmetry as well as B.M. Stewart’s Adventures Among the Tori not to mention Robert Williams’ The Geometrical Foundations of Natural Structure.
In Holden’s text, he presents a sampling of nolids. The difference between his and mine is that his are continuous (i.e. no gaps or holes between the polygons that make up the polyhedra) while mine are discontinuous (They are of positive genus).
All the polyhedra found here are based upon or derived from the five Platonic and thirteen Archimedean polyhedra. They are designed and constructed by Albert P. Carpenter and are freely available to anyone and everyone.
My aim in presenting these structures is to inspire materials scientists i.e chemists, nanochemists and nanoscientists as well as geometers to study them with the hope that they will one day be synthesized/printed in laboratories and find a wide range of applications out side of them.
HOW TO USE THIS SITE:
Each page has a two part title representing its geometric name and chemical/nanoscientific name. The number to the right of each title represents the number of photos and the total number of objects shown. Titles for each photo include the geometric name of the object with its chemical/nanoscientific name. Links are provided as deemed helpful for further research.
Please See: What’s New (s)/ Geometry Journal for updates.