How to calculate bevel angles / dihedral angles

[Calvin Glover]

I need to learn the mathematics of dihedral angles.  In particular I am looking at a 1V 5/8 Rhombicuboctahedron dome on the SimplyDifferently.org site's calculator.  

https://simplydifferently.org/Geodesic_Dome_Notes?page=16#The Rhombicuboctahedron

I'd like to build it with plywood clad triangles, but I have no idea how to calculate the bevel angles of each side.

Any help will be appreciated.

[TaffGoch]

Calvin,

When I try to go to simplydifferent's webpages, I get an SSL error (which wasn't, previously, a problem.)

Is this the tessellation about which your query applies?

Secondly, do you really want to learn the mathematics, or do you simply need the angles (and how to employ them?)

-Taff

[Calvin Glover]

That's the one.  Practically speaking, I need the angles, but I would like to know how to arrive at them as well.

 

Calvin

 

“A man sees in the world what he carries in his heart.” -Goethe

--
--
You received this message because you are subscribed to the "Geodesic Help" Google Group
--
To unsubscribe from this group, send email to GeodesicHelp...@googlegroups.com
--
To post to this group, send email to geodes...@googlegroups.com
--
For more options, visit http://groups.google.com/group/geodesichelp?hl=en

---
You received this message because you are subscribed to the Google Groups "Geodesic Help Group" group.
To unsubscribe from this group and stop receiving emails from it, send an email to geodesichelp...@googlegroups.com.
For more options, visit https://groups.google.com/d/optout.

[Calvin Glover]

Here is a better link

https://simplydifferently.org/Geodesic_Dome_Notes?page=16

 

Calvin

 

“A man sees in the world what he carries in his heart.” -Goethe

[Paul Kranz]

Taff:

Are the axial and dihedral angles of polyhedra related to the same in the discipline of organic chemistry? It looks like the dihedral angle in domes and molecules are a function of the axial angle (perhaps visa-versa).

Paul sends...

--

--
You received this message because you are subscribed to the "Geodesic Help" Google Group
--
To unsubscribe from this group, send email to GeodesicHelp...@googlegroups.com
--
To post to this group, send email to geodes...@googlegroups.com
--
For more options, visit http://groups.google.com/group/geodesichelp?hl=en

---
You received this message because you are subscribed to the Google Groups "Geodesic Help Group" group.
To unsubscribe from this group and stop receiving emails from it, send an email to geodesichelp...@googlegroups.com.
For more options, visit https://groups.google.com/d/optout.

--

Very high regards,

 

Paul sends...

ReviveTheFlame.com

[Gerry in Quebec]

Calvin,

In the Excel calculator I posted earlier today, the sequence of 5 chord factors -- the inputs -- doesn't match the highlighted pair of triangles in the diagram of your rhombicuboctahedron-based dome. 

Here's the correct sequence of inputs:

v: 0.52372

w: 0.52372

x: 0.71481

y: 0.71481

z: 0.71481

 

The total dihedral angle remains unchanged.

 

- Gerry in Québec, Canada

On Wednesday, May 20, 2015 at 7:40:13 AM UTC-4, Gerry in Quebec wrote:

Calvin,

Two files attached: Excel calculator and jpg screen shot of it. This is just one of several ways to calculate dihedral angles in domes and their component angles (partial dihedral angles). The example is from the dome you referenced, based on the rhombicuboctahedron. The edge-bevel angle (depending on your construction method) is 90 degrees minus the partial dihedral angle.

- Gerry in Quebec

[Calvin Glover]

Gerry,

Thank you so much!

 

Calvin

 

“A man sees in the world what he carries in his heart.” -Goethe

On Wed, May 20, 2015 at 5:40 AM, Gerry in Quebec <toomey...@gmail.com> wrote:

Calvin,

Two files attached: Excel calculator and jpg screen shot of it. This is just one of several ways to calculate dihedral angles in domes and their component angles (partial dihedral angles). The example is from the dome you referenced, based on the rhombicuboctahedron. The edge-bevel angle (depending on your construction method) is 90 degrees minus the partial dihedral angle.

- Gerry in Quebec

On Sunday, May 17, 2015 at 7:28:44 PM UTC-4, Calvin Glover wrote:

--

[Gerry in Quebec]

Hi again Calvin,

 

I see a glitch in the second half of the equation for dihedral angle GFE. The expression vwzx should be vwx. This doesn't affect the Excel calculations; it's just a typo in the text version.

 

I've attached the corrected Excel spreadsheet and jpg screenshot and will delete the two files posted earlier.

 

- Gerry in Québec

 

Original post to Calvin, May 20, 2015:

 

Calvin,

Two files attached: Excel calculator and jpg screen shot of it. This is just one of several ways to calculate dihedral angles in domes and their component angles (partial dihedral angles). The example is from the dome you referenced, based on the rhombicuboctahedron. The edge-bevel angle (depending on your construction method) is 90 degrees minus the partial dihedral angle.

- Gerry in Quebec

[Calvin Glover]

Thanks again.

If I may, I do have a couple of questions:

I had expected one of the bevel angles to be 22.50 (half of the angles that form a regular hexagon).  The 22.76+ from your model is close, but I would have expected an exact correlation.  Where have I gone wrong?

I am unable to visualize how second set of sides would lie on a circle.  In a search of the internet I came up with 2 sites that each put the bevel angle at 17.6322, but with no explanation of how they arrived at the result.  Your model's output of 90-88.49=1.51 seems counterintuitive.

I appreciate your willingness to help.  I'll keep working with the equations to see if I can get them to "soak in."

 

Calvin

 

“A man sees in the world what he carries in his heart.” -Goethe

--

[Gerry in Quebec]

Hi Calvin,

I'm having some trouble understanding your queries, perhaps because we're not using the terms such as "bevel angle" in the same way. Where does the 22.76 degrees come from? As for the 22.5 degrees you mention, that number can be found by carving up a regular octagon pizza-style, into 16 pieces, but I don't see its connection to a regular hexagon. And what does the 17.6322 degree angle refer to?

 

If you were to look at a physical or virtual model of the dome you want to build, I think you'd see that the angle between a pair of isosceles triangle faces, across their long edge, is very close to flat, namely 176.99 degrees, giving an edge-bevel angle of just 1.51 degrees for the long struts of the isosceles triangles.

 

- Gerry in Québec

[Calvin Glover]

I'm sorry to be so obtuse.  I was an accountant.  I'm not used to thinking in three dimensions. 

I misspoke when I said hexagon.  I should have said octagon.  When I say "bevel angle" I mean one half of (180 - dihedral angle).  The base of the dome appears to be a regular octagon, so the square-to-square faces would have a dihedral angle of 135 degrees (bevel angle of 22.5 degrees).

I'm lost when I try to visual the square-triangle abutment.  The 17.63 figure comes from wikepedia and an article by Richard Klitzing on bendway.org.  I'm sure I'm not understanding something very basic, but it seems to me that a bevel angle of 1.51 degrees would imply 178.21 edges circumscribed by the sphere.

 

Calvin

 

“A man sees in the world what he carries in his heart.” -Goethe

[Gerry in Quebec]

Hi Calvin and others with an acute interest in dihedral angles,

I should have mentioned a disadvantage of the dihedral angle Excel calculator I posted.... It doesn't distinguish between positive and negative dihedral angles. So, if two adjacent faces of a dome happen to form a valley instead of a ridge, the dihedral angle given by the calculator would still be less than 180 degrees. See the attached illustration.

 

For most domes, including the one based on the rhombicuboctahedron that Calvin is interested in, this isn't a problem as there are no valleys (the dome shell is fully convex). But for out-of-the-ordinary dome design work, where it might not be clear in advance whether a new geometric layout involves valleys or not, it's best to use a more sophisticated dihedral angle calculator/equation.

 

- Gerry in Québec

P.S. Calvin, do you use SketchUp? If so, I can post a bare-bones version of the rhombicuboctahedron-based dome (56 triangles).

[Calvin Glover]

Gerry,

A 3D Sketchup model should be very helpful.  Thanks again...  

 

Calvin

 

“A man sees in the world what he carries in his heart.” -Goethe

[Gerry in Quebec]

Here it is.

[Timothy Gregory]

hi all
just on this topic point
there was a book called "Hexadome"
back in the days of books like Shelter and Domebook II before PCs
hexadome was a house design that has 9 foot triangles 9'3" in the main hexagons
a cool feature was you simply mark the centre on the ends of your 4x4 lumber
then draw a line through the centre at seven degrees and rip saw the 4x4 in half on that angle
then flip one half and presto you get the right dihedral tilt for the dome too easy.
kind regards
tim

[Brian OConnell]

Hi Gerry,

I've just seen and used your calculator. Thanks for sharing! However, I've come across and inconsistency I would like to reconcile before proceeding.

It's my first dome. 3v icos. I have attached a spreadsheet where I did the calculations.

on this website, the given dihedral angle for the A-A is different. How am I to know which one is correct?

Thanks

[Gerry in Quebec]

Hello Brian,

Wasn't able to open your Dih q.ods spreadsheet file. Is that the Open Source software whose spreadsheet is very similar to Excel? If so, I haven't used that program in quite a few years.

I looked at the webpage you mentioned (http://web.comhem.se/p12dome/3vcalculator.html). The three dihedral angles it gives for the 3v icosa dome -- 165.6 degrees over A struts, 168.6 over B struts, and 166.4 over C struts -- are the same as the outputs from my dihedral calculator.

Cheers,

- Gerry in Québec

[Brian OConnell]

Thanks Gerry,

file attached as excel now so you should be able to open it

[Bryan L]

Hi Brian,

you have the C strut in the calcs for A'A not the B strut

[Gerry in Quebec]

Hi Brian with an i. Bryan with a y is right. Also, your dihedral angle Hex 1 – hex 2 (B'B') is correct.

Two suggestions: First, for greater accuracy use 0.4035 as your chord factor for B struts rather than 0.4036. Second, consider using the 4-chord-factor version of the 3v dome instead of the 3-chord-factor version. Here we've been calling it "Method Kruschke". It has three triangle types instead of two, so it's a bit more complex than the version you're planning (so-called method 1). But it has the distinct advantage of sitting flat under the 4th and 5th rows of triangles (4/9 and 5/9 truncations). The method 1 version does not sit flat at either of those truncations, unless you alter the bottom row of triangles. Use the Search function to find past threads here on the Kruschke method, including links to SketchUp models by TaffGoch.

- Gerry in Québec

[Gerry in Quebec]

Brian:

Here's some info, from 2015, on the 4-chord-factor version of the 3v dome, under the heading: "A bit of history: Fuller, Kruschke & the 3v icosa dome":

https://groups.google.com/d/topic/geodesichelp/CRMbxEszbKo/discussion

- Gerry

[Brian OConnell]

Hi Bryan and Gerry, thanks for the help there! what a basic blunder haha.

thanks for the nudge towards Kruschke. I think as this is my first dome, and the flat bottom isnt important to me, i'll keep it simple. project 2 will probably be a Kruschke, maybe 6/7m in diameter. 

ultimately I want to build a house, maybe 10-15m in diameter, so starting small now at 3m diameter.