Up: Math 53 Selected Course Notes

# Orthographic Views of the Sliced Cube:

 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 half-way 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.

Up: Math 53 Selected Course Notes