Visualizing Math

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Cubes fall through “Flatland”. On the left is a view of the cube in perspective; on the right is a view from directly above which represents what a two-dimensional person viewing the cube from within the plane would be able to perceive.

The top animation shows a square falling through flatland on its face. The slices are always squares. So our two-dimensional person would see “a square existing for a while”.

The second animation shows a square falling through flatland on one of its edges. The slice begins as an edge, then becomes a rectangle; the rectangle grows, becomes a square for a moment, and then gets wider than it is tall. At its widest, it is as wide as the diagonal of one of the square faces of the cube. The rectangle then shrinks back to an edge at the top of the cube.

The third animation is the coolest one! The cube passes through Flatland on one of its corners. In this case, the initial contact is a point, which then becomes a small equilateral triangle. This triangle grows until it touches three of the corners of the cube. At this point, the corners of the triangles begin to be cut off by the other three faces of the cube. For a short moment, the triangle turns into a certain regular polygon..As the cube progresses through the plane, the slice turns again into a cut-off triangle (but inverted with respect to the original one) and finally becomes an equilateral triangle once again as three more vertices pass through the plane. This triangle shrinks down to a point and disappears.

In the third animation, what regular polygon does the triangle turn into halfway through its fall? If you can’t figure out, maybe this artwork by Robert Fathauer will help. (Scroll to the bottom.)

If a 4D cube entered our dimension, what would we see? If you can’t figure this out, check out this awesome page. (Click the GIF links.)

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