{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "Python for Everyone!
[Oregon Curriculum Network](http://4dsolutions.net/ocn/)\n", "\n", "## Polyhedrons\n", "\n", "Polyhedrons make a great entry point into mathematics, because they're not only pretty to look at, they're graphs (as in Graph Theory), a kind of data structure, made of nodes and edges. \n", "\n", "The edges border openings we call \"faces\" though in wireframe polys, these may be more like \"windows\", not filled in.\n", "\n", "Consider: in modern data analysis and machine learning, we stack up the vectors, or features, in our X rectangle, corresponding to y labels. A vector is both a geometric object and a pointer to anywhere in an n-dimensional phase space.\n", "\n", "Because n-dimensional polytopes and data analysis go together, it makes sense to introduce them in tandem, with Polyhedrons in relational data tables.\n", "\n", "Here in Oregon, we have a predeliction to introduce a scheme of nested polyhedrons of more whole number volumes than you may be expecting. We make use of the so-called Concentric Hierarchy based around a central Tetrahedron of unit volume. Other volumes get measured in tetravolumes.\n", "\n", "Consider this table:\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "
 \n", "Shape\n", " \n", "Volume\n", " \n", "Scale\n", " Tetrahedron \n", "1.0000\n", " \n", "1.0000\n", " Cuboctahedron \n", "2.5000\n", " \n", "0.5000\n", " Icosahedron \n", "1 \n", "~2.9180\n", " \n", "~0.5402\n", " Duo-Tet Cube \n", "3.0000\n", " \n", "1.0000\n", " Octahedron \n", "4.0000\n", " \n", "1.0000\n", " Rhombic\n", "Triacontahedron \n", "5.0000\n", " \n", "~0.9995\n", " Rhombic Triacontahedron 2 \n", "~5.0078\n", " \n", "1.0000\n", " Pentagonal Dodecahedron 3 \n", "5.4271\n", " \n", "~0.7071\n", " Rhombic Dodecahedron 4 \n", "6.0000\n", " \n", "1.0000\n", " Pentagonal Dodecahedron 5 \n", "~15.3500\n", " \n", "1.0000\n", " Icosahedron \n", "~18.5123\n", " \n", "1.0000\n", " Cuboctahedron \n", "20.0000\n", " \n", "1.0000\n", " Duo-Tet Cube \n", "24.0000\n", " \n", "2.0000\n", " \n", "1 faces flush with\n", "volume 4 octahedron's\n", " 2,4 \"shrink-wrapped\" around\n", "unit-radius sphere\n", " 3 face diagonals = edges of\n", "volume 3 cube\n", " 5 structural dual of Icosa (Icosa =\n", "jitterbugged Cubocta)\n", "