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I'm curious if there's any historical link between the kanji for "gate", 門{かど}, and the kanji for "circle" or "yen", 円{えん}. If 門 is gate, 円 looks like a closed gate. Am I being whimsical and seeing patterns in the wind, or is there a substantive link here?
Please use furigana when writing kanji; I'm a beginning student and don't know many kanji.
As noted in the question comments, the kanji 円 was originally 圓. The nutshell version of the article Yang Muye linked is that monks developed a shorthand version of 圓 that looked like a box with a vertical line through it: 
. Over time, the shape of the surrounding box changed, likely due to the same anatomical and mechanical processes that inform any change in handwriting. 
became 
, and then that became the modern simplified shinjitai character 円.
門, meanwhile, is a pictogram of a gate. You might run across alternative form 鬥, but otherwise, this character is much more straightforward, both in semantic development and graphical evolution.
Ultimately, the resemblance between 円 and 門 is purely accidental, and has much more to do with the constraints placed on kanji shapes and the kinds of strokes used in handwriting.
