 


116K 

This movie shows the orthographic view of a cube looking directly at a
face of the cube while the cube is being sliced parallel to a face. A
perspective view is also included here (and in all the moveis below) to
help you determine the position of the slices within the square.
The first sequence shows slices parallel to the face we are looking at:
they start at the back face and move toward us. We can see this
progression in the perspective view, but in the orthographic view, we see
"a square for a while". This was our initial method of explaining a cube
to the Flatlanders. The second and third sequences show slices from the
bottom and the side of the cube. Since our line of sight is parallel to
these faces, the projected slices appear as line segments. We see these
segments sweep out the square, and this represents our original means of
understanding the square that A Square used to explain himself to the King
of Lineland.




116K 

This movie shows the orthographic view of a cube looking directly at a
face of the cube while the cube is being sliced starting at an edge. We
start at the back left edge and end at the front right edge. The halfway
point is a rectangle with width equal to the diagonal of one of the square
faces, but it appears as a square in the projection since it is slanted in
comparison to our viewpoint. Starting at any of the edges that form the
square in the projection of the cube will give this same sequence.
But there are four other edges in the cube: they are parallel to our
direction of sight, and so are projected to the corners of the square.
Slice from one of these gives the second sequence in the movie. Since the
slice is parallel to our line of sight, we see only a segment. We have the
illusion of seeing a square swept out by a line segment, corner first.




119K 

This movie shows the orthographic view of a cube looking directly at a
face of the cube while the cube is being sliced starting at a corner. The
sequence from any corner looks essentially the same. Here we see the
slice start as a triangle, grow to a larger triangle, and when this hits
the three corners of the cube, it becomes a truncated triangle. Half way
through, the slice is a perfect regular hexagon (though it appears
irregular in this projection). After this the sequence reverses itself,
and the hexagon becomes a truncated triangle, then an equilateral
triangle, and finally shrinnks to a point.




113K 

This movie shows the orthographic view of a cube looking directly at an
edge of the cube while the cube is being sliced starting at a face. The
first sequence shows the slices if the face is one of the sides of the
cube: it simply appears as a rectangle sliding through the projection. The
second sequence shows the slices parallel to the top and bottom of the
cube. Since these faces are parallel to our line of sight, the projections
of the slices appear a line segments.




164K 

This movie shows the orthographic view of a cube looking directly at an
edge of the cube while the cube is being sliced starting at an edge. We
begin at the back edge and end at the front one. Half way through, the
silce is as wide as possible, forming a rectangle that has width equal to
the diagonal of one of the square faces of the cube. In the projection,
the rectangle seems to grow out of the central edge and than then shrink
back to it again; but remember that both the front and the back edges are
projected to this same location. The second sequence shows what the slices
look like if we start at one of the edges at the top or bottom of the
projection. Here we see parallelograms, but they actually are rectangles
in the cube. Finally, the last sequence shows the slices starting at one
of th eside edges. Here, the slice is parallel to our line of sight, so
the slice appears as a single line segment in the projection.




179K 

This movie shows the orthographic view of a cube looking directly at an
edge of the cube while the cube is being sliced starting at a corner. We
begin at the lower back corner and end at the front top corner. The
equilateral triangles and regular hexagon look distorted due to the
projection. Starting at the top back corner would look essentially the
same. The second sequence shows what happens if the slice begins at any
of the other corners. Here, the slicing plane is parallel to our
direction of sight, so we see only a line passing through the projection.




111K 

This movie shows the orthographic view of a cube looking directly at a
corner (along its long diagonal) while the cube is sliced starting at a
face. In this view of the cube, every face looks the same (it appears as
a rhombus. As the slice progresses across the cube, the rhombus seems to
move across the hexagon to the other side.




180K 

This movie shows the orthographic view of a cube looking directly at a
corner (along its long diagonal) while the cube is sliced starting at an
edge. The first set of slices starts at the back edge and moves toward
the front. Half way through, the slice is a rectangle that is as wide as
the diagonal of one of th efaces of the cube. The slices starting at any
of the six edges that meet at the center of the hexagon will look
essentially the same as this. The second sequence shows the slices when
we start at one of th edges taht forms the circumference of the hexagon.
In the projection, we are looking parallel to the slicing plane, so we
just see a line.




302K 

This movie shows the orthographic view of a cube looking directly at a
corner (along its long diagonal) while the cube is sliced starting at a
corner. The first set of slices starts at the back corner and ends at the
front corner. The central slice is a regular hexagon, and is clearly
visible in this view. The second sequence shows the slice from one of the
other corners; this time the hexagon does not appear as a regular hexagon
in the projection since it is titled compared to the direction of sight.
The slices from any other corner will be similar to this sequence.
