Note that another way of getting the same result, using a technique we've already used before, is unfolding the cube and looking at the "raw material" of our new irregular polyhedra. The faces became their dual, and the corners became whatever one got by slicing half way through an edge. Thus, our edge-dual polytope has 6 regular diamonds and 8 re For a cube, this process transforms each square into its "dual" diamond and each corner into a triangle (we sliced off a regular tetrahedron). "shaving" or slicing off the corners (vertices). I then realized that in the process I was doing two things: i) creating a new shape out of each original face, and ii) creating a new shape by effectively Originally I drew a cube, marked the centers of edges and connected them. Take note of the interesting way that this type of duality includes the original type of duality in a dimension one less than the one we're working in. This type of duality involves connecting the center of edges rather than faces. In three space another type of polyhedral duality exists, creating irregular polyhedra (polyhedra composed of more than one type of regular two dimensional polygon) rather than regular ones. More tetrahedra will be created by connecting to the eight triangles on the outside of the "inner octahedron". When we add the last cube with its centered vertex, 8 Trahedra as i) half of the 16-cell, the tetrahedra of which will become regular when the cubes are folded into a hypercube, or ii) a skewed perspective view looking into the bottom half of a 16-cell. But we know now, from both analogy and process, that the 16-cell is also ii) two octahĮdrons sharing the same "base surface" (base surface being the outer crust / shell / armor of the octahedron, analogous to the "base diamond" before, which was the outer edges of the "diamond based pyramid" base diamond). We know from experience (and from "Beyond the Third Dimension") that the dual to the hypercube is the 16-cell, which is i) a composition of 16 regular tetrahedrons. When we connect the centĮrs of the cubes to create the dual (just the seven cells first), we get eight right tetrahedra sharing the same vertex, which is the center point of the "inside" cube. The dual to the hypercube will have a vertex at the center of each cell in the hypercube, so we mark those points. The folded out hypercube will haveĨ cubes aligned in a very similar cross pattern with one internal cube that is "opposite" the last external cube. We can now do this for other three dimensional polyhedra, but I'd rather move directly to the hypercube instead (haven't checked out the other 3D polyhedra yet, but would assume similar results would be achieved). When we add the last square with its centered vertex, four more triangles will be created (which are difficult to represent in the plane) by connecting to the four outer squares (just like the inside square did). So we can look at our four right triangles as i) half of the octahedron, the triangles of which will become equilateral when the squares are folded into a cube, or ii) a perspective view looking into the bottom half ofĪn octahedron. We know that the dual to the cube is the octahedron, which is i) eight equilateral triangles, and ii) two "diamond" (note: diamond and square are same thing in 2 space, and in this case diamond is more appropriate) based pyra Rs of the squares to create the dual (just the five squares first), we get four right triangles connected in the plane sharing a common vertex. The dual to the cube will have a vertex at the center of each face, so we mark those with points. A cube fold out will have six squares aligned in the cross pattern. Start with a cube because of its level of intuitive quality. An interesting strategy I stumbled upon is one that deals with examining duality by looking at the figures fold outs. It can be difficult to get a good look at what is really going on when dealing with polytopes in three and four dimensions. I want to continue discussing duality in both dimensions here. Ting centers of 2D faces (squares in cube), while in four dimensions a dual is usually created by connecting centers of 3D cells (cubes in hypercube). One of the main points was how duality changes context when it is considered in different dimensions: in three, duals are created by connec Much of my discussion for chapter five (last week) focused on polyhedral duality in three and four dimensions. Michael Matthews: Reflections Michael Matthews: Reflections
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