With this sort of argument, we can prove more rigorously that there is a measure polytope which exists in every dimension. This polytope is the famous hypercube.įurthermore, since n-space is defined by n orthogonal bases, it should be clear that one can construct a polytope with n edges meeting at a vertex, each parallel to a base, and so each perpendicular to one another. If we take a unit cube, and slide it in a direction perpendicular to all six faces, we would generate the measure polytope for 4-space, having four edges meet at a vertex, each perpendicular to the other three. There are two perpendicular edges meeting at each corner of the square, and three perpendicular edges meeting at each corner of the cube. If we look at the vertex of any measure polytope, we see an edge coming from it parallel to each of the orthogonal bases of the vector-space. (This is the reason bricks are shaped the way they are.) Just as line-segments can be placed end-to-end to fill 1-space, and squares can be laid edge-to-edge to fill the plane, cubes can be placed together neatly to fill all of 3-space. We speak of volume as being in cubic units (e.g., ). Sliding the square one unit in a direction perpendicular to its face sweeps out a cube, the measure polyhedron. This sweeps out a square of side-length 1, with which we measure area. You can think of generating it by first moving a point one unit in one of the base directions (say, along the x-axis), then turning perpindicular to it and sliding the whole line segment one unit in the orthogonal base direction (the y-axis). In 2-space, the measure polytope is a square. The measure of any region of the line can be expressed as a real number times the unit length, that is, the measure polytope of side-length 1. (1-space is a rather boring topic of discussion, but is included for consistency.) In 1-space, we measure content in terms of length. The measure polytope in a line is a line-segment, which is the same as its simplex. The reason this type of polytope is referred to as a "measure" is this: it is the unit in each space in which we measure. Though a simplex is, as the name suggests, a simpler polytope, it is the measure polytope with which we generally feel most comfortable. Perhaps its popularity is due to the easy regularity with which it is constructed. In one piece, an architect builds a house which folds up into 4-space, so that its occupants can continuously walk in any direction from one room into each of the other seven. It has even made its way into popular fiction in one form or another. The so-called hypercube, or 4-cube, is undoubtedly the most familiar and frequently discussed polytope in a higher dimension. The data structure used for the 4-polytope is a list of a list of vertices, whose coordinates are defined above, a list of polygons given by the number of their vertices, and a list of cells given by the number of their polygons. The eight cells of the hypercube are each cubes. They are given by their vertices.į:=: f:=: The 24 polygons of the hypercube are each squares. Multpoly is a procedure which take a polytope P and a rotation matrix M and multiplies each of the vertices (thought of as column matrices) by M, generating a rotation about some vector v. This defines the polytope by its faces, allowing polygonplot3D to display it. Polydecode is a procedure which takes a polytope P and lists the coordinates of the vertices of each polygonal face. Other values may create the same problem. Values must be greater than 1, and this often results in a division by 0. Smaller values of f will generate a greater perspective effect in the projection. Thus, the effects of perspective will be minimal. For large values of f, the object will be perspected as if it were viewed from far away (though no smaller). Perspproj is a procedure which takes a polytope P, a vector v, and a real number f, and projects the vertices of P onto the hyperplane orthogonal to v, with a perspective distance factor of f. Project is a procedure which takes a polytope P and a vector v and projects the vertices of P onto the hyperplane orthogonal to v. These are the libraries of commands needed to execute the procedures below. You may need to scroll back to read the text which precedes each display. Maple will then take you to the first line of the next execution group. In order to keep the text uninterupted, placing the cursor on the first line of commands (in red) will execute all procedures necessary to define the polytope. These are the procedures necessary to generate the various images of the hypercube.
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