{VERSION 2 3 "HP RISC UNIX" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 } {CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "2 D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 256 " " 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 271 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "W arning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE " " -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 256 "" 0 "" {TEXT 257 24 "The 4-D Measure Polytope" }} {PARA 256 "" 0 "" {TEXT 256 13 "The Hypercube" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }}{PARA 0 "" 0 "" {TEXT -1 402 " These are the procedures necessary to generate the various images of the hyperc ube. In order to keep the text uninterupted, placing the cursor on th e first line of commands (in red) will execute all procedures necessar y to define the polytope. Maple will then take you to the first line \+ of the next execution group. You may need to scroll back to read the \+ text which precedes each display." }}{PARA 0 "" 0 "" {TEXT -1 103 "For a detailed explanation of the procedures below, scroll through them a nd read the text and refer to " }{HYPERLNK 17 "Appendix A." 1 "proc.mw s" "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 " These are the libraries of commands needed to execute the procedures below." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "with(linalg): with(plots):" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " " }{TEXT 262 7 "Project" }{TEXT -1 54 " is a procedure whi ch takes a polytope P and a vector " }{TEXT 259 1 "v" }{TEXT -1 66 " a nd projects the vertices of P onto the hyperplane orthogonal to " } {TEXT 258 1 "v" }{TEXT -1 1 "." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 314 " project:=proc(P,v) local a,b,c,d,n,i;\na:=v[1];b:=v[2];c:=v[3];d:=v[4] ;\nn:=sqrt(a^2+b^2+c^2+d^2);\n[[seq([(P[1][i][1]*d+P[1][i][2]*c-P[1][i ][3]*b- P[1][i][4]*a)/n,\n(-P[1][i][1]*c+P[1][i][2]*d+P[1][i][3]*a- P[ 1][i][4]*b)/n,\n(P[1][i][1]*b-P[1][i][2]*a+P[1][i][3]*d- P[1][i][4]*c) /n],\ni=1..nops(P[1]))],P[2],P[3]];\nend:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " " }{TEXT 268 9 "Perspproj" } {TEXT -1 51 " is a procedure which takes a polytope P, a vector " } {TEXT 260 1 "v" }{TEXT -1 20 ", and a real number " }{TEXT 263 1 "f" } {TEXT -1 67 ", and projects the vertices of P onto the hyperplane orth ogonal to " }{TEXT 265 1 "v" }{TEXT -1 40 ", with a perspective distan ce factor of " }{TEXT 264 1 "f" }{TEXT -1 23 ". For large values of \+ " }{TEXT 266 1 "f" }{TEXT -1 158 ", the object will be perspected as i f it were viewed from far away (though no smaller). Thus, the effects of perspective will be minimal. Smaller values of " }{TEXT 267 1 "f " }{TEXT -1 180 " will generate a greater perspective effect in the pr ojection. Values must be greater than 1, and this often results in a \+ division by 0. Other values may create the same problem." }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 512 "perspproj:=proc(P,v,f) local a,b,c,d,n,i,p1 ,p2,p3,p4,g,q1,q2,q3,q4;\na:=v[1];b:=v[2];c:=v[3];d:=v[4];\nn:=sqrt(a^ 2+b^2+c^2+d^2);\nfor i from 1 to nops(P[1]) do;\np1:=P[1][i][1];p2:=P[ 1][i][2];\np3:=P[1][i][3];p4:=P[1][i][4];\np1:=p1-f*a/n;p2:=p2-f*b/n; \np3:=p3-f*c/n;p4:=p4-f*d/n;\ng:=-n*f/(a*p1+b*p2+c*p3+d*p4);\nq1[i]:=g *p1;q2[i]:=g*p2;q3[i]:=g*p3;q4[i]:=g*p4;\nod;\n[[seq([(q1[i]*d+q2[i]*c -q3[i]*b-q4[i]*a)/n,\n(-q1[i]*c+q2[i]*d+q3[i]*a-q4[i]*b)/n,\n(q1[i]*b- q2[i]*a+q3[i]*d-q4[i]*c)/n],\ni=1..nops(P[1]))],P[2],P[3]];\nend:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " " } {TEXT 269 10 "Polydecode" }{TEXT -1 185 " is a procedure which takes a polytope P and lists the coordinates of the vertices of each polygona l face. This defines the polytope by its faces, allowing polygonplot3 D to display it." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "polydecode:=pr oc(P) local i,j;\n[seq([seq(P[1][P[2][i][j]],j=1..nops(P[2][i]))],\ni= 1..nops(P[2]))];\nend:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " " }{TEXT 270 8 "Multpoly" }{TEXT -1 178 " is a \+ procedure which take a polytope P and a rotation matrix M and multipli es each of the vertices (thought of as column matrices) by M, generati ng a rotation about some vector " }{TEXT 261 1 "v" }{TEXT -1 1 "." }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 363 "multpoly:=proc(P,M) local i;\n[[se q([M[1,1]*P[1][i][1]+M[1,2]*P[1][i][2]+M[1,3]*P[1][i][3]+M[1,4]*P[1][i ][4],\nM[2,1]*P[1][i][1]+M[2,2]*P[1][i][2]+M[2,3]*P[1][i][3]+M[2,4]*P[ 1][i][4],\nM[3,1]*P[1][i][1]+M[3,2]*P[1][i][2]+M[3,3]*P[1][i][3]+M[3,4 ]*P[1][i][4],\nM[4,1]*P[1][i][1]+M[4,2]*P[1][i][2]+M[4,3]*P[1][i][3]+M [4,4]*P[1][i][4]],\ni=1..nops(P[1]))],P[2],P[3]];\nend:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "The 16 vertices of t he hypercube:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 327 "p[1]:=[-1,-1,-1,- 1]: p[2]:=[1,-1,-1,-1]:\np[3]:=[-1,1,-1,-1]: p[4]:=[1,1,-1,-1]:\np[ 5]:=[-1,-1,1,-1]: p[6]:=[1,-1,1,-1]:\np[7]:=[-1,1,1,-1]: p[8]:=[1 ,1,1,-1]:\np[9]:=[-1,-1,-1,1]: p[10]:=[1,-1,-1,1]:\np[11]:=[-1,1,-1, 1]: p[12]:=[1,1,-1,1]:\np[13]:=[-1,-1,1,1]: p[14]:=[1,-1,1,1]:\np[ 15]:=[-1,1,1,1]: p[16]:=[1,1,1,1]:" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 85 "The 24 polygons of the hypercube are \+ each squares. They are given by their vertices." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 487 "f[1]:=[1,5,13,9]: f[2]:=[2,6,14,10]:\nf[3]:=[3,7, 15,11]: f[4]:=[4,8,16,12]:\nf[5]:=[1,3,11,9]: f[6]:=[2,4,12,10]: \nf[7]:=[5,7,15,13]: f[8]:=[6,8,16,14]:\nf[9]:=[1,2,10,9]: f[10]: =[3,4,12,11]:\nf[11]:=[5,6,14,13]: f[12]:=[7,8,16,15]:\nf[13]:=[1,3,7 ,5]: f[14]:=[2,4,8,6]:\nf[15]:=[9,11,15,13]: f[16]:=[10,12,16,14]: \nf[17]:=[1,2,6,5]: f[18]:=[3,4,8,7]:\nf[19]:=[9,10,14,13]: f[20]:= [11,12,16,15]:\nf[21]:=[1,2,4,3]: f[22]:=[5,6,8,7]:\nf[23]:=[9,10,1 2,11]: f[24]:=[13,14,16,15]:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "The eight cells of the hypercube are each cubes . They are given by their polygons." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 197 "c[1]:=[1,3,5,7,13,15]:\nc[2]:=[2,3,6,8,14,16]:\nc[3]:=[1,2,9,11 ,17,19]:\nc[4]:=[3,4,10,12,18,20]:\nc[5]:=[5,6,9,10,21,23]:\nc[6]:=[7, 8,11,12,22,24]:\nc[7]:=[13,14,17,18,21,22]:\nc[8]:=[15,16,19,20,23,24] :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 286 "The data structure used for the 4-polytope is a list of a list of vertice s, whose coordinates are defined above, a list of polygons given by th e number of their vertices, and a list of cells given by the number of their polygons. The list structure is: [[vertices],[polygons],[cells ]]." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "hypercube:=[[seq(p[i],i=1..1 6)],[seq(f[i],i=1..24)],[seq(c[i],i=1..8)]]:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for nor m" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace" } }}{PARA 257 "" 0 "" {TEXT 271 13 "The Hypercube" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 272 9 "Existence" }}{PARA 0 "" 0 "" {TEXT -1 605 " The so-called hypercube, or 4-cube, is un doubtedly the most familiar and frequently discussed polytope in a hig her dimension. It has even made its way into popular fiction in one f orm or another. In one piece, an architect builds a house which folds up into 4-space, so that its occupants can continuously walk in any d irection from one room into each of the other seven. Perhaps its popu larity is due to the easy regularity with which it is constructed. Th ough a simplex is, as the name suggests, a simpler polytope, it is the measure polytope with which we generally feel most comfortable." }} {PARA 0 "" 0 "" {TEXT -1 496 " The reason this type of polyto pe is referred to as a \"measure\" is this: it is the unit in each sp ace in which we measure. The measure polytope in a line is a line-seg ment, which is the same as its simplex. (1-space is a rather boring t opic of discussion, but is included for consistency.) In 1-space, we m easure content in terms of length. The measure of any region of the l ine can be expressed as a real number times the unit length, that is, \+ the measure polytope of side-length 1." }}{PARA 0 "" 0 "" {TEXT -1 371 " In 2-space, the measure polytope is a square. 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). This sweeps out a square of side-length 1, w ith which we measure area." }}{PARA 0 "" 0 "" {TEXT -1 173 " \+ Sliding the square one unit in a direction perpendicular to its face s weeps out a cube, the measure polyhedron. We speak of volume as being in cubic units (e.g., " }{XPPEDIT 18 0 "cm^3" "*$%#cmG\"\"$" }{TEXT -1 241 "). Just as line-segments can be placed end-to-end to fill 1-s pace, and squares can be laid edge-to-edge to fill the plane, cubes ca n be placed together neatly to fill all of 3-space. (This is the reas on bricks are shaped the way they are.)" }}{PARA 0 "" 0 "" {TEXT -1 542 " If we look at the vertex of any measure polytope, we se e an edge coming from it parallel to each of the orthogonal bases of t he vector-space. There are two perpendicular edges meeting at each co rner of the square, and three perpendicular edges meeting at each corn er of the cube. If we take a unit cube, and slide it in a direction p erpendicular 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. This polytope is the famous hypercube." }}{PARA 0 " " 0 "" {TEXT -1 349 " Furthermore, since n-space is defined by n orthogonal bases, it should be clear that one can construct a polyt ope with n edges meeting at a vertex, each parallel to a base, and so \+ each perpendicular to one another. With this sort of argument, we can prove more rigorously that there is a measure polytope which exists i n every dimension. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 273 9 "Structure" }}{PARA 0 "" 0 "" {TEXT -1 366 " Th e construction of the hypercube is usually best thought of in the term s mentioned above. First a vertex sweeps out a line. That line then \+ sweeps out a plane. When it does so, each of the vertices sweeps out \+ another line. So the four sides of a square come from the starting an d stopping point of the line, and the two lines created by its end-poi nts." }}{PARA 0 "" 0 "" {TEXT -1 530 " When a square sweeps o ut a cube, each of the four corners of the square sweeps out new lines , which become four additional edges. Each edge of the cube sweeps ou t a new square. So the six faces of the cube come from the starting a nd stopping points of the original square, plus the four faces swept o ut by its edges. The 12 edges come from the 8 edges of the initial an d final square, plus the four edges generated by the four corners of t he square. The number of vertices is again doubled, so that a cube ha s eight." }}{PARA 0 "" 0 "" {TEXT -1 119 " Now, when we think of this cube moving perpendicular to itself, we are thinking of a dir ection orthogonal to " }{TEXT 276 3 "all" }{TEXT -1 652 " the faces of the cube. We certainly can't point in that direction. But we should be able to see that the cube will have a starting and stopping point, which are two of the cubic cells of the hypercube. Also, we should s ee that each of the six faces of the cube will sweep out cubes. So th e hypercube consists of eight cubes in all. The 12 edges sweep out 12 new faces. These, together with the 12 faces of the first and last c ube, total 24 in the hypercube. There is one new edge for each corner of the cube, plus 12 edges in the first cube and 12 edges in the last comes to 24. And again, the vertices are doubled, so that a hypercub e has 16." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 274 15 "Schl\344fli Symbol" }}{PARA 0 "" 0 "" {TEXT -1 504 " \+ The Schl\344fli symbol of the measure polytope in 2-space, the square, is \{4\}. The next order measure polytope is generated by folding th ree of these squares about a point. So the symbol for the cube is \{4 ,3\}. Just as four squares about a point lie in a plane, four cubes a bout an edge lie in a hyperplane. We form the hypercube in roughly th e same way we form the cube: by removing one of the cubes about the ed ge and folding the other three. Hence, the symbol for the hypercube i s \{4,3,3\}. " }}{PARA 0 "" 0 "" {TEXT -1 385 " As we can now tell from the symbol, the vertex figure of the hypercube is a tetrahe dron. Just as the vertex figure of the 3-measure is a 2-simplex, the \+ vertex figure of the 4-measure is a 3-simplex. Likewise, the vertex f igure of the n-measure will be a (n-1)-simplex. So the symbols for al l the measure polytopes have a first number 4 and the rest of the entr ies are 3's." }}{PARA 0 "" 0 "" {TEXT -1 153 " Also, the dual of \{4,3,3\} is the polytope represented by \{3,3,4\}. This is the s o-called hyperoctahedron, which we discuss in the next section." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 275 11 "Coordina tes" }}{PARA 0 "" 0 "" {TEXT -1 528 " In order to generate or dered quadruples of coordinates for the hypercube, it helps to underst and how we might find the coordinates of other measures. In a line, t he endpoints can simply be expressed by any two numbers, so let's say \+ (1) and (-1). In the plane, we can think of connecting a point from e ach quadrant. Thus, the ordered pairs for the vertices of a square lo ok like combinations of 1 and -1. These possible combinations are cal led permutations. The corners look like (1,1), (1,-1), (-1,1) and (-1 ,-1)." }}{PARA 0 "" 0 "" {TEXT -1 469 " Moving to the cube, w e can think of sliding the square in the xy-plane down to z = -1 and u p to z = 1. Thus, the coordinates of the cube are the permutations of (\2611,\2611,\2611). From this, we can see that the coordinates for \+ the corners of a hypercube should come from permuting (\2611,\2611, \2611,\2611) and there are, in fact, 16 possibilities. Now we can eas ily induce that the coordinates for any n-measure polytope (with sidel enght 2) can be found by permuting n \2611's." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 " " {TEXT -1 256 " Here is a projected image of the hypercube. \+ Its cells can be easily distinguished as cubes if you imagine that the cubes are slightly slanted. That is, the angles are distorted. You \+ can change the projection vector to generate different images." }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "polygonplot3d(polydecode(project(h ypercube,[1,3,4,2])),scaling=constrained,style=wireframe,thickness=2); \n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 13 "" 1 "" {INLPLOT "6&-%) POLYGONSG6:7&7%$!+
FNFOFA7&F'F;FDF97&FFFMFPFK7&F.F>FAF47&FIFNFOFJ7&F'FFFIF.7&F;FMFNF>7&F9
FKFJF47&FDFPFOFA7&F'F;F>F.7&FFFMFNFI7&F9FDFAF47&FKFPFOFJ7&F'F;FMFF7&F.
F>FNFI7&F9FDFPFK7&F4FAFOFJ-%(SCALINGG6#%,CONSTRAINEDG-%*THICKNESSG6#\"
\"#-%&STYLEG6#%%LINEG" 3 376 376 376 6 0 1 2 2 1 0 1 1 1.000000
45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 101 207 190 101 207 190 1 2
0 0 0 -26888 16415 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 549
" Maple can display a series of images from different points \+
of view by varying the projection vector. The sequence below in gener
ated by looking at the hypercube from the direction of (1,0,0,0), vary
ing it to (1,1,1,1) and back again. Notice that because the initial p
rojection is along an axis, we are looking directly at one of the cubi
c cells, so it completly covers the others. (Six of them are hidden b
y the sides of the cube, and the seventh is directly behind the one we
see.) As the view changes, the other cells become visible." }}{PARA
0 "> " 0 "" {MPLTEXT 1 0 205 "display([seq(polygonplot3d(polydecode(pr
oject(hypercube,[1,evalf(sin(Pi*k/40),2),evalf(sin(Pi*k/40),2),evalf(s
in(Pi*k/40),2)]))),k=0..40)],insequence=true,thickness=2,scaling=const
rained,style=wireframe);\n" }}{PARA 13 "" 1 "" {INLPLOT "6&-%(ANIMATEG
6K7#-%)POLYGONSG6:7&7%$\"\"\"\"\"!$!\"\"F.F,7%F,F,F,7%F/F,F,7%F/F/F,F*
7&7%F,F/F/7%F,F,F/7%F/F,F/7%F/F/F/F47&F+F5F8F3F97&F1F6F7F2F:7&F+F+F3F3
7&F5F5F8F87&F1F1F2F27&F6F6F7F77&F+F5F6F1F?7&F3F8F7F2F@7&F+F+F1F17&F5F5
F6F67&F3F3F2F27&F8F8F7F77&F+F+F5F57&F1F1F6F67&F3F3F8F87&F2F2F7F77#-F(6
:7&7%$\"+]G*p8*!#5$!+]G*p8*FPFN7%$\"+Wu.\"f(FP$\"+E[Ho5!\"*FV7%$!+m.*G
A\"FXFNFN7%$!+E[Ho5FXFgnFT7&7%FVFgnFV7%FNFN$\"+m.*GA\"FX7%FgnFTFV7%FQF
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fnF_oF^oFY7&FfoF[pFjoFeo7&FMFjnFhoFao7&FSF[oFioFdo7&FfnF_oF[pFfo7&FYF^
oFjoFeo7#-F(6:7&7%$\"+:^!\\4)FP$!+:^!\\4)FPFeq7%$\"+Q<86]FP$\"+\\y'y6
\"FXF\\r7%$!+(=XiU\"FXFeqFeq7%$!+\\y'y6\"FXFbrFjq7&7%F\\rFbrF\\r7%FeqF
eq$\"+(=XiU\"FX7%FbrFjqF\\r7%FgqF_rFeq7&7%F\\r$!+Q<86]FPFbr7%FeqFgrFgq
7%FbrF\\rFbr7%FgqFgqF_r7&7%FgrFgqFgq7%F\\rF\\rF]s7%FgqFeqFgq7%F]sFbrFb
r7&FdqF\\sFasFar7&FerFcsFfsFjr7&FiqF_sF`sF^r7&FfrFdsFesFir7&FdqFerFjrF
ar7&F\\sFcsFfsFas7&FiqFfrFirF^r7&F_sFdsFesF`s7&FdqF\\sF_sFiq7&FerFcsFd
sFfr7&FarFasF`sF^r7&FjrFfsFesFir7&FdqFerFfrFiq7&F\\sFcsFdsF_s7&FarFjrF
irF^r7&FasFfsFesF`s7&FdqFerFcsF\\s7&FiqFfrFdsF_s7&FarFjrFfsFas7&F^rFir
FesF`s7#-F(6:7&7%$\"+]*zK:(FP$!+]*zK:(FPF`u7%$\"+G>*)zGFP$\"+(zmE9\"FX
Fgu7%$!+*f0+d\"FXF`uF`u7%$!+(zmE9\"FXF]vFeu7&7%FguF]vFgu7%F`uF`u$\"+*f
0+d\"FX7%F]vFeuFgu7%FbuFjuF`u7&7%Fgu$!+G>*)zGFPF]v7%F`uFbvFbu7%F]vFguF
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7960 16419 1 1 0 0 100 0 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 535 " \+
Here is a traditional perspective projection of the hypercube. T
his is the image most people associate with it, though they may not un
derstand it. If you imagine looking at a large cube very close to you
r face, what would the image be? The front face would be a large squa
re, and you could see all of the other edges and vertices inside this \+
square. The back square would be smaller, and its corners would be jo
ined to the corners of the front face. Thus, the top, bottom, and sid
e squares would appear to be trapezoids." }}{PARA 0 "" 0 "" {TEXT -1
643 " The image below is an analogy to this sort of view. Im
agine viewing the hypercube very close to one of its cubic cells. The
n you would see the \"front\" cube as large and containing the rest of
the object. The \"back\" cube would be smaller and in the middle, an
d its corners would be connected to the corners of the front cube. Th
e six cells in between appear as truncated pyraminds, just as the side
s of the perspective cube appear as \"truncated\" triangles. There is
a cubic cell on top, bottom, front, back, left, and right, which are \+
each distorted in this way. So in all, the display below shows us eig
ht cubes (not just two)." }}{PARA 0 "" 0 "" {TEXT -1 247 " In
terms of the construction of the hypercube as sweeping a cube through
4-space, this view shows a cube which has been swept outward in every
direction. This of course is indistinguishable from the sides of the
cube being swept inward." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "polyg
onplot3d(polydecode(perspproj(hypercube,[1,0,0,0],3)),\nscaling=constr
ained,style=wireframe,thickness=3);" }}{PARA 13 "" 1 "" {INLPLOT "6&-%
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