UKH

/ Maths help please - solid geometry

This topic has been archived, and won't accept reply postings.
Deadeye - on 09 Jan 2019

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.

 

 

tlouth7 on 10 Jan 2019
In reply to Deadeye:

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

Deadeye - on 10 Jan 2019
In reply to tlouth7:

Brilliant; thank you

elsewhere on 10 Jan 2019
In reply to Deadeye:

> 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.

Post edited at 11:28
tlouth7 on 10 Jan 2019
In reply to Deadeye:

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).

mike123 - on 10 Jan 2019
In reply to Deadeye:

out of intrest ........what are you making ?

Edit : (whatever it is , it sounds cool )

 

Post edited at 17:24
Boredoftheforum on 10 Jan 2019
In reply to mike123:

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.

Deadeye - on 10 Jan 2019
In reply to Boredoftheforum:

> 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.

Post edited at 20:01
deepsoup - on 11 Jan 2019
In reply to Deadeye:

Wow, some birthday present!  Sounds like a real labour of love.

Boredoftheforum on 11 Jan 2019
In reply to Deadeye:

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?

Deadeye - on 11 Jan 2019
In reply to Boredoftheforum:

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.


This topic has been archived, and won't accept reply postings.