Medieval Pierced Panel Geometry Problem. (create a model from just outer radius)

R

Rob Edwards

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Aug 22, 2020
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Hello

I am a traditional joiner who has just had a fascinating day exploring the geometry of this medieval church panel.

English_Church_Woodwork_p264_llananno.jpg
p264 LLananno, English Church Woodwork, Howard and Crossley

I would like to create a model that allows me to define all the geometry from the larger outer radius alone.

The first step was analysing the layout and I was delighted to see that the design consists of 19 identical circles inside a larger circle

I have made a schematic where each small circle is equal in size and every contact is tangent.

1.png

I would like to obtain the smaller radius for any given perimeter radius.

I have simplified the problem down to this little 30 degree slice of pie.

2.png
I can model this inside my CAD package but I was very interested about a mathematical solution.

I imagine I could use trigonometry but in homage to the craftsmen of old I would love to find a solution using only a straight edge and compass.

What do you think?
Is it possible?
Must be I presume, but not by me lol.

Thanks for looking

All the best

Rob
 
btw I also discovered that the pattern can be repeated indefinitely. I thought it was very beautiful1.png
 
An equivalent way that the problem could be specified is to find the radius of a circle which fits exactly 12 times inside the perimeter of the larger circle as shown here. Maybe that is an easier problem?1.png
 
Maybe construction via a hexagon split into triangles is a lot easier.1682362355141.png
 
Last edited:
I've been messing about and got a solution. Pretty sure its not very clean and I don't show all the steps but I think its working

maybe.gif
 
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