Hirano Pentagon

[Chainring]

Greetings geniuses. I have an interesting little puzzle. I found a bit of geometry and I don't know, I was compelled to draw it. I had to fully express it I want to know if this drawing can be turned into an equation. Thank you very much in advance.

 

[Dr.Peterson]

Are you asking for proof that this constructs a regular pentagon, or what?

If so, it isn't too hard. What have you tried?

 

[Chainring]

Thank you for the reply, Dr. Please forgive my ignorance. I have not tried anything except the drawing. What I would like to do is express the pentacle and the 13 circles in an equation. Can this be done?

 

[Dr.Peterson]

How would one equation express all that? Do you mean you want the entire figure to be the graph of one equation? Why?

I'm not sure whether you know what "equation" means! (Many people don't, just as they think every problem can be solved by a "formula".)

 

[Chainring]

You are probably right. Math isn't my thing. I'm an artist. I want to know if it can be written out. Why? Curiosity, maybe. Thanks for the reply!

 

[Dr.Peterson]

You are probably right. Math isn't my thing. I'm an artist. I want to know if it can be written out. Why? Curiosity, maybe. Thanks for the reply!

I'd say many paragraphs could be written to describe it, but nothing very compact.

 

[Chainring]

Ok. Thank you. I did not know if super complicated shapes like that could be written out in some sort of notation. Plot points or something. I dont know. I mean, if you can do that with a triangle, why not this? Thanks again, Dr.

 

[Steven G]

Actually you can not write out an equation for a triangle. You certainly can write the equation of the three sides of a triangle.

In your diagram, although I did not look at it too carefully, I suspect that you can write the equation for each circle, but not one equation for all the circles.

 

[Chainring]

Thank you very much, both of you.

 

[Mrjackp]

Hey there Chainring. I have a particular love for Hirano's Pentagon, I have it inked on me forever as my first tattoo. It was the first time I ever saw maths as truly beautiful, and I ended up getting it permanently on me after I finished my maths masters, so yeah, I would say it's pretty special to me.
I love art, and especially maths as art or combined with art. If you have anything further you want to ask, please do, I promise to do my level best to help. I suspect my having read the whole thread and my sentimentality about this specific construction means I'm ideally suited