Addresses in the Koch curve

8.(c) Consider the point 1(3infinity) = 2(0infinity), the apex of the Koch curve.

To show this point has no tangent, we shall produce a sequence of points converging to the apex, with the chords from these points to the apex alternating over a 30 degree range.

Next conider the chord between the apex and 133(0infinity).

Now the pattern should be clear: both sequences of points

13...32(0infinity) and 13...3(0infinity)

approach the apex 1(3infinity)

Chords from 13k(0infinity) to the apex make an angle of 30 deg with the horizontal.

Chords from 13k2(0infinity) to the apex make an angle of 60 deg with the horizontal.

Combining these two sequences we obtain a sequence approaching the apex with chord angles alternating over a 30 degree range.

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