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Morphism from the Associahedron to its dual, the Triaugmented Triangular Prism. by MEGL
by Thingiverse
Last crawled date: 4 years, 7 months ago
This was created by a student at George Mason University for Math 493, Mathematics Through 3D Printing. This is the third project for the class, prepared on Feb. 21st 2016, with the prompt of: choosing a bounded polyhedron (polytope) and exploring the morphism to its dual.
First, a few definitions. A polyhedron is an n-th dimensional object that can be described via number of vertices (0 dimensional), edges (1 dimensional), faces (2 dimensional) , ... , facets (n-1 dimensional) and finally the full object. Again, we restrict ourselves to polytopes since we actually have to print the object. That is, we could not print an infinitely-large object. Not easily, at least. Next, the choice of the object-
The associahedron (K5) is a polytope in which each vertex corresponds to a way of correctly inserting opening and closing parentheses in a word of 5 letters and the edges correspond to single application of the associativity rule.
Finally, taking the dual of a polytope. That is, for each facet (n-1 dimensional face) assign a vertex to it's center (1 dimensional face). Then, two vertices are joined by other faces, if and only the facets share an edge. This will create a new, n dimensional, polytope.
Doing so to the associahedron will generate the triaugmented triangular prism. Which is a triangular prism, where each face has a square pyramid poking out of it.
More information about the Associahedron can be found here-https://en.wikipedia.org/wiki/Associahedron
First, a few definitions. A polyhedron is an n-th dimensional object that can be described via number of vertices (0 dimensional), edges (1 dimensional), faces (2 dimensional) , ... , facets (n-1 dimensional) and finally the full object. Again, we restrict ourselves to polytopes since we actually have to print the object. That is, we could not print an infinitely-large object. Not easily, at least. Next, the choice of the object-
The associahedron (K5) is a polytope in which each vertex corresponds to a way of correctly inserting opening and closing parentheses in a word of 5 letters and the edges correspond to single application of the associativity rule.
Finally, taking the dual of a polytope. That is, for each facet (n-1 dimensional face) assign a vertex to it's center (1 dimensional face). Then, two vertices are joined by other faces, if and only the facets share an edge. This will create a new, n dimensional, polytope.
Doing so to the associahedron will generate the triaugmented triangular prism. Which is a triangular prism, where each face has a square pyramid poking out of it.
More information about the Associahedron can be found here-https://en.wikipedia.org/wiki/Associahedron