Kock's kinky curve is created by starting with a straight line segment and replacing it with a four segments, each 1/3 as long as the original segment. So, at the second stage the curve has three bends, At the next stage, each segment is replaced by four segments, and so on. How many bends does this curve have at the third stage? the fourth stage? The nth stage?

Question

Bryan Maynard

Mathematics

1 January, 04:10

Kock's kinky curve is created by starting with a straight line segment and replacing it with a four segments, each 1/3 as long as the original segment. So, at the second stage the curve has three bends, At the next stage, each segment is replaced by four segments, and so on. How many bends does this curve have at the third stage? the fourth stage? The nth stage?

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Rudy Walls · Answer 1

Here is how this is solved.

First stage one segment

second stage 3 segment each 1/3 of the original (like dividing the first one into 3) this creates three bending

3rd stage each one of the 1/3 long segments is now divided into 4 making a total of 12

low and behold there is 12 bendings in total

the idea here is each division is considered a bend