/ Maths help please - solid geometry
What is the difference in radius between the sphere that will just enclose a pentakis dodecahedron where the long edges are length a, and sphere just enclosed by the same pentakis dodecahedron?
I have found this: https://rechneronline.de/pi/pentakis-dodecahedron.php
Am I correct in interpreting the answer from that as being 2*(Rm-Ri)?
Not an idle question - I'm building one out of plywood with a view to turning to a sphere and need to know how big a ball I can reasonably make from 18 or 25mm ply before I'll turn a hole in the faces.
I picked that solid as all 60 pieces are the same and they all have the same dihedral angles, which will make manufacture a bit easier I hope.
The midsphere does not necessarily fall halfway between the inscribed and circumscribed spheres. I see nothing in your link giving the circumradius for a pentakis dodecahedron, but for the one you describe (with equal dihedral angles) I believe the circumradius should be equal to that of a conventional dodecahedron (the midradius certainly is).
So I get:
ri = 1.282a
rm = 1.309a
rc = 1.401a
Brilliant; thank you
> Not an idle question - I'm building one out of plywood with a view to turning to a sphere and need to know how big a ball I can reasonably make from 18 or 25mm ply before I'll turn a hole in the faces.
"hole in the faces" - did you type that wrong?
You will have hole at the vertices before you have a hole in the faces.
The interior surface of the plywood is a dodecahedron. As you reduce the exterior radius you'd first encounter that empty volume (creating a hole) at the vertices.
Have you considered a rhombic tricontrahedron? It has half as many faces, all identical diamonds. The disadvantage would be a bigger rc:ri ratio, so you would break through sooner (requiring greater ply thickness for a given edge length).
out of intrest ........what are you making ?
Edit : (whatever it is , it sounds cool )
My guess is it'll be constructed out of alternating pentagons or light and dark wood and will be used as an elaborate practical joke on some footballers.
Actually thinking about how the layers of ply will be exposed by turning it spherical it should look pretty stunning in ordinary ply.
> Actually thinking about how the layers of ply will be exposed by turning it spherical it should look pretty stunning in ordinary ply.
This really. Furniture grade baltic ply (which is rather expensive so I'm keen not to mess it up!).
I'm then adding some teeny magnets and a set of countries of the world in different woods with matching teeny magnets and hollowed in the backs to match the curve of the sphere..
It's intended as a birthday present.
Wow, some birthday present! Sounds like a real labour of love.
Sounds really interesting, out of interest how are you planning to clamp it together when gluing it up as some of the parts will be pretty awkwardly shaped?
I'll make pentagonal sub-assemblies in a frame that I can clamp round and press on the apex.
Then I'll glue the sub assemblies into two halves.
Or duct tape.
Probably duct tape. Lots of duct tape.
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