# 
# What part of a cube is nearer its center than any of its corners?
# (Half).
# Visualization and representation. 
# There's a beautiful conceptual proof, but you don't need Maple sheets
# for the concepts. In a nutshell, the proof is this: the points nearest
# one corner, and those nearest the center, split their octant in half
# by symmetry. QED.
> with(plots):
> with(plottools):
> xs:=proc(h) if h<1/2 then
> RETURN(polygon([[1/2-h,1],[1,1/2-h],[1,h-1/2],[1/2-h,-1],[h-1/2,-1],[-
> 1,h-1/2],[-1,1/2-h],[h-1/2,1],[1/2-h,1]],color=blue)) else
> RETURN(polygon([[3/2-h,0],[0,h-3/2],[h-3/2,0],[0,3/2-h],[3/2-h,0]],col
> or=blue))fi end;

xs := proc(h)
    if h < 1/2 then RETURN(polygon([[1/2 - h, 1], [1, 1/2 - h],
        [1, h - 1/2], [1/2 - h, -1], [h - 1/2, -1], [-1, h - 1/2],
        [-1, 1/2 - h], [h - 1/2, 1], [1/2 - h, 1]], color = blue))
    else RETURN(polygon([[3/2 - h, 0], [0, h - 3/2], [h - 3/2, 0],
        [0, 3/2 - h], [3/2 - h, 0]], color = blue))
    fi
end

# This shows the cross section in the plane z=1/3 of the part of the
# cube nearer the center than any of the 8 corners. For any particular
# point, there is only one corner which is even a contender for
# "nearest". The boundary between center and corner nearness is a plane:
# one of these planes, between the center and (1,1,1), has equation
# x+y+z=3/2. The shape we want is what's left of the cube after slicing
# off everything on the far side from the origin of this plane, and of
# the other seven like it.
> display(xs(1/3));

# The animation below shows the cross sections as h runs from 1 to -1.
# This shows what portion of the cube is nearest the center, cross
# section by cross section, as the cross sectioning plane passes through
# the cube. 
# 
# 
> rr:=[seq(xs(abs(k)/30),k=-30..30)]:
> display(rr,insequence=true);

> xs3d:=proc(h) if abs(h)<1/2 then
> RETURN(polygon([[1/2-abs(h),1,h],[1,1/2-abs(h),h],[1,abs(h)-1/2,h],[1/
> 2-abs(h),-1,h],[abs(h)-1/2,-1,h],[-1,abs(h)-1/2,h],[-1,1/2-abs(h),h],[
> abs(h)-1/2,1,h],[1/2-abs(h),1,h]],color=blue)) else
> RETURN(polygon([[3/2-abs(h),0,h],[0,abs(h)-3/2,h],[abs(h)-3/2,0,h],[0,
> 3/2-abs(h),h],[3/2-abs(h),0,h]],color=blue))fi end;

xs3d := proc(h)
    if abs(h) < 1/2 then RETURN(polygon([[1/2 - abs(h), 1, h],
        [1, 1/2 - abs(h), h], [1, abs(h) - 1/2, h],
        [1/2 - abs(h), -1, h], [abs(h) - 1/2, -1, h],
        [-1, abs(h) - 1/2, h], [-1, 1/2 - abs(h), h],
        [abs(h) - 1/2, 1, h], [1/2 - abs(h), 1, h]], color = blue))
    else RETURN(polygon([[3/2 - abs(h), 0, h],
        [0, abs(h) - 3/2, h], [abs(h) - 3/2, 0, h],
        [0, 3/2 - abs(h), h], [3/2 - abs(h), 0, h]], color = blue))
    fi
end

> display({xs3d(0),xs3d(1/4),xs3d(1/2),xs3d(3/4),xs3d(1)});

> xssq3d:=proc(h)
> polygon([[1,1,h],[1,-1,h],[-1,-1,h],[-1,1,h],[1,1,h]],color=yellow)end
> ;

xssq3d := proc(h)
    polygon(
    [[1, 1, h], [1, -1, h], [-1, -1, h], [-1, 1, h], [1, 1, h]],
    color = yellow)
end

# This next display is best viewed using the options wireframe, and the
# projection option, constrained. You can see how the central region
# fits into the cube. 
> display({seq(xs3d(k/5),k=-5..5),seq(xssq3d(k/5),k=-5..5)});

# This next object should also be viewed using the projection option,
# constrained. 
> display({seq(xs3d(k/20),k=0..20)});

# To see the object itself, and not through the lens of a slice-by-slice
# dissection, execute the lines below. 
> with(plottools);

[arc, arrow, circle, cone, cuboid, curve, cutin, cutout, cylinder,

    disk, dodecahedron, ellipse, ellipticArc, hemisphere, hexahedron,

    hyperbola, icosahedron, line, octahedron, pieslice, point,

    polygon, rectangle, rotate, scale, semitorus, sphere, stellate,

    tetrahedron, torus, transform, translate]

> with(plots):
> top:=polygon([[0,1/2,1],[1/2,0,1],[0,-1/2,1],[-1/2,0,1],[0,1/2,1]],col
> or=blue):
> bottom:=polygon([[0,1/2,-1],[1/2,0,-1],[0,-1/2,-1],[-1/2,0,-1],[0,1/2,
> -1]],color=blue):
> spinit:=proc(pt)[pt[2],pt[3],pt[1]]end;

             spinit := proc(pt) [pt[2], pt[3], pt[1]] end

> side1:=map(spinit,[[0,1/2,1],[1/2,0,1],[0,-1/2,1],[-1/2,0,1],[0,1/2,1]
> ]):
> side2:=map(spinit,side1):
> side3:=map(spinit,[[0,1/2,-1],[1/2,0,-1],[0,-1/2,-1],[-1/2,0,-1],[0,1/
> 2,-1]]):
> side4:=map(spinit,side3):
> s1:=polygon(side1,color=blue):s2:=polygon(side2,color=blue):s3:=polygo
> n(side3,color=blue):s4:=polygon(side4,color=blue):
> f1:=[[1,0,1/2],[1,1/2,0],[1/2,1,0],[0,1,1/2],[0,1/2,1],[1/2,0,1],[1,0,
> 1/2]]:
> f2:=-f1:
> hex1:=polygon(f1,color=green):hex2:=polygon(f2,color=green):
> f3:=map(u->[-u[1],u[2],u[3]],f1):f4:=-f3:hex3:=polygon(f3,color=green)
> :hex4:=polygon(f4,color=green):
> f5:=map(u->[u[1],-u[2],u[3]],f1):f6:=-f3:hex5:=polygon(f5,color=green)
> :hex6:=polygon(f6,color=green):
> f7:=map(u->[u[1],u[2],-u[3]],f1):f8:=-f7:hex7:=polygon(f7,color=green)
> :hex8:=polygon(f8,color=green):
> display({top,bottom,s1,s2,s3,s4,hex1,hex2,hex3,hex4,hex5,hex6,hex7,hex
> 8},style=PATCH,axes=BOXED);

>