Enneadecagon (regular polygon with 19 sides) is also not constructable in theory. But there is some classic construction method with error 0.02 ° for the angel. Consider the accumulation error,the total erroe will be 19x0.02=0.38° which can be found by regular compass.I had tried times and found a way to construct Enneadecagon with very little error. See right picture for details.
The idea is to construct a angel of 6 time Enneadecagon angel(360/9). Since 19 is a primer, so it is easy to use this angel to construct a perfect Enneadecagon if we can have a perfect 6 times angel.
The angel I found has error of 1E-5, the constructed Enneadecagon angel has error of 2E-5,so even for a big compass, you will be not able to find out the error as the max error is 0.00017° after construction the whole Enneadecagon。
If you not familiar for the construction with this 6time angel, see next section for details.
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