- Link:
- http://hdl.handle.net/1721.1/32531
- Collection:
-
- Creator:
- Radul, Alexey
- Contributors:
- Mathematics and Computation Gerald Sussman
- Format
- 111 p.
- Format
- 1054739 bytes
- Format
- 8185809 bytes
- Format
- application/pdf
- Format
- application/postscript
- Language
- en_US
- Relation
- Massachusetts Institute of Technology Computer Science and Artificial Intelligence Laboratory
- Description
- I devised and implemented a method for constructing
regular andsemiregular geometric objects in n-dimensional Euclidean
space.Given a finite reflection group (a Coxeter group) G, there is
a standard way to give G a group action on n-space.Reflecting a
point through this group action yieldsan object that exhibits the
symmetries specified by G. If the pointis chosen well, the object
is guaranteed to be regular or semiregular,and many interesting
regular and semiregular objectsarise this way. By starting with the
symmetry group, I can use thegroup structure both to simplify the
actual graphics involved withdisplaying the object, and to
illustrate various aspects of itsstructure. For example, subgroups
of the symmetry group (and theircosets) correspond to substructures
of the object. Conversely, bydisplaying such symmetric objects and
their various substructures, Ifind that I can elucidate the
structure of the symmetry group thatgives rise to them.I have
written The Symmetriad, the computer system whose name thisdocument
has inherited, and used it to explore 3- and 4-dimensionalsymmetric
objects and their symmetry groups. The 3-dimensionalobjects are
already well understood, but they serve to illustrate thetechniques
used on the 4-dimensional objects and make them morecomprehensible.
Four dimensions offers a treasure trove of intriguingstructures,
many of which have no ready 3D analogue. These are what Iwill show
you here.
- Description
- MEng thesis
- Relation
- Massachusetts Institute of Technology Computer Science
and Artificial Intelligence Laboratory
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