Shape

297 Calculating and Continuity
This looks natural enough—and, no doubt, it is—unless you start thinking about
what’s going on. The final shape in the series is a rotation of the initial one about its
center—that’s the vertex shared by the three triangles in both shapes. But the transformation
is surprising because the centers of the triangles in the initial shape change
position when it’s rotated
The rule doesn’t move the centers of triangles, but they move just the same. What kind
of paradox is this?
The answer is easier to see than to say. But let’s give it a try. The rule can be
applied to the fourth shape in the series in two ways. In one way, the rule picks out
the three triangles that correspond to the three triangles in the initial shape. Only
none of these triangles is resolved in the other way. Instead, the rule divides the fourth
shape into two triangles—the large outside triangle and the small inside one—that
have sides formed from sides of the three triangles that come from the ones in the
initial shape. The rule rotates the two triangles in turn to get the fifth and sixth
shapes. Now the rule can be applied in alternative ways to the sixth shape in the series.
Either the rule resolves both of the triangles that correspond to the ones in the fourth
shape, or it resolves the three triangles—one at each corner of the sixth shape—that
have sides formed from segments of sides of the triangles in the fourth shape. The
rule rotates the three corner triangles one at a time to complete the series. This isn’t
about thinking in a combinatorial scheme, but about seeing when lines fuse and
divide.
The nine shapes in the series