clifford_a-_pickover_surfing_through_hyperspacebookfi-org
clifford_a-_pickover_surfing_through_hyperspacebookfi-org
clifford_a-_pickover_surfing_through_hyperspacebookfi-org
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100 <strong>surfing</strong> <strong>through</strong> hyperspace<br />
In this chapter we've discussed how a cube would appear to Flatlanders as it<br />
penetrates their world. As a cube moves corner-first <strong>through</strong> Flatland, the Flatlanders<br />
first see a point, then a series of growing triangles, then hexagons, then<br />
triangles, and finally a point (Fig. 4.9). Similarly, a variety of shapes would be<br />
produced as a hypercube penetrated our world (Fig. 4.10). In four dimensions,<br />
slicing with a 3-D knife produces an array of strange shapes ranging from distorted<br />
cubes, various prisms, and polyhedra often in unexpected arrangements.<br />
What would the hypercube look like as we rotated it? One way to visualize<br />
this is to begin with the "central projection" in Figure 4.5 and watch the wireframe<br />
change as the hypercube rotates (Fig. 4.11). Starting from the left and<br />
going clockwise, the large exterior cube opens toward the top, flattens out, and<br />
opens inward to form an incomplete pyramid. Meanwhile, the smaller interior<br />
shaded cube opens toward the bottom to form another incomplete pyramid. If<br />
we continue to rotate the hypercube, the unshaded exterior cube will become<br />
the small cube, and the small cube will flatten out and come back to become the<br />
large cube. As Thomas Banchoff notes, each of the eight cubical faces takes its<br />
turn holding all the various positions in this projection. As each of the cubes<br />
flattens out and opens up again during the rotation, it changes orientations. If a<br />
cube contained a right-handed object before the flattening, the object would<br />
become left-handed afterwards.<br />
On the Trail of the Tesseract<br />
In this chapter, we've also used lower-dimensional analogies to help contemplate<br />
the mathematics of higher spaces. Throughout history, mathematicians have<br />
used interdimensional analogies. For example, if mathematicians understood a<br />
theorem in plane geometry, they were often able to find analogous theorems in<br />
solid geometry. (Theorems about circles provide insight into theorems about<br />
spheres and cylinders.) Similarly, solid geometry theorems have suggested new<br />
relationships among plane figures. If history shows that knowledge can be<br />
gained by going to higher dimensions, imagine what we might learn by contemplating<br />
4-D geometries.<br />
The most often-used analogy for contemplating shapes in higher dimensions<br />
involves moving objects perpendicular to themselves. If we move a 0-D point<br />
with no degrees of freedom, we generate a line, a 1-D object with two end<br />
points (Fig. 4.3). A line moved perpendicular to itself along a plane generates a<br />
square with four corners. A square moved perpendicular to itself forms a cube