# Determining dihedral angles of a solid from a net **URL:** \ **Category:** Factual Questions\ **Created:** [July 7, 2015, 1:29am UTC](https://boards.straightdope.com/t/determining-dihedral-angles-of-a-solid-from-a-net/724438 "2015-07-07T01:29:21Z")\ **Posts on this page:** 13\ **Page:** 2
**Author:** ![Dr.Strangelove](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dr.strangelove/32/6613_2.png) [@Dr.Strangelove](https://boards.straightdope.com/u/Dr.Strangelove)\ **Post date:** [July 8, 2015, 3:53am UTC](https://boards.straightdope.com/t/determining-dihedral-angles-of-a-solid-from-a-net/724438/21 "2015-07-08T03:53:58Z")
> [@Chronos](#): > > The torus would certainly not be convex, and I don’t think the infinite tube would be, either. Yeah, I realized the torus wasn’t convex after I posted (d’oh)… seems like a infinite tube would count as convex, though. It’s certainly convex at any point a finite distance from the start. Regardless, I came up with an example (I think). See [this picture](http://i22.photobucket.com/albums/b314/spcutler/20150707_202800.jpg). Each color represents an equal-length line (green and dotted green are different). Note that you can either fold the B triangles _or_ the C triangles to be next to A, and each results in a different shape. I believe that both shapes are convex, or can be made so with small tweaks (I’m not 100% certain about the angle between B and C). ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [July 8, 2015, 11:02am UTC](https://boards.straightdope.com/t/determining-dihedral-angles-of-a-solid-from-a-net/724438/22 "2015-07-08T11:02:52Z")
An infinite tube would count as convex, except that I think that the way you’re making it out of triangles would leave it “crinkled”. ---
**Author:** ![Lance\_Turbo](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lance_turbo/32/6156_2.png) [@Lance\_Turbo](https://boards.straightdope.com/u/Lance_Turbo)\ **Post date:** [July 8, 2015, 5:26pm UTC](https://boards.straightdope.com/t/determining-dihedral-angles-of-a-solid-from-a-net/724438/23 "2015-07-08T17:26:50Z")
Let x = sqrt((41 - sqrt(1105))/2). Then the following points: (x, 0, 0) (-x, 0, 0) (0, sqrt(16 - x^2), 0) (0, -sqrt(25 - x^2), 0) (0, -(x^2)/(sqrt(16 - x^2)), x) (0, -(x^2)/(sqrt(16 - x^2)), -x) Form an octahedron with each face a 3-4-5 triangle. Rational approximation: a -\> (1.969576061, 0, 0) b -\> (-1.969576061, 0, 0) c -\> (0, 3.481489644, 0) d -\> (0, -4.595733906, 0) e -\> (0, -1.114244263, 1.969576061) f -\> (0, -1.114244263, -1.969576061) ---
**Author:** ![Lance\_Turbo](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lance_turbo/32/6156_2.png) [@Lance\_Turbo](https://boards.straightdope.com/u/Lance_Turbo)\ **Post date:** [July 8, 2015, 8:57pm UTC](https://boards.straightdope.com/t/determining-dihedral-angles-of-a-solid-from-a-net/724438/24 "2015-07-08T20:57:08Z")
I think I have worked out a general solution to get six vertices of an octahedron with each face an a, b, c triangle, but I’m not sure if any right triangle assumptions snuck in there. I’ll post it in case anyone is interested in verifying it. Given a triangle with sides a, b, c name its angles alpha, beta, and gamma in the usual way and let A denote its area. Let x be the smallest real root of the polynomial z^4 - (b^2 + c^2) z^2 + 4A^2. Then the following points: (x, 0, 0) (-x, 0, 0) (0, sqrt(b^2 - x^2), 0) (0, -sqrt(c^2 - x^2), 0) (0, (ab cos(gamma)-(x^2))/(sqrt(b^2 - x^2)), x) (0, (ab cos(gamma)-(x^2))/(sqrt(b^2 - x^2)), -x) Form an octahedron with each face an a, b, c triangle. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [July 8, 2015, 9:15pm UTC](https://boards.straightdope.com/t/determining-dihedral-angles-of-a-solid-from-a-net/724438/25 "2015-07-08T21:15:41Z")
Ooh, thanks! That still doesn’t get me everything I want, but it gets me close enough that I should be able to get the rest of the way on my own (cross-products between edges to get normals, and dot-products between normals to get dihedral angles). Or maybe I should just figure out how to use those points directly (I’m sure that’s possible, in Autocad). How did you find those? EDIT: For the general case, you probably need to put in some stipulation about which sides are shorter or longer. Some arrangements won’t produce a valid solid (I think this would mean no real roots of your quartic). ---
**Author:** ![Lance\_Turbo](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lance_turbo/32/6156_2.png) [@Lance\_Turbo](https://boards.straightdope.com/u/Lance_Turbo)\ **Post date:** [July 8, 2015, 9:23pm UTC](https://boards.straightdope.com/t/determining-dihedral-angles-of-a-solid-from-a-net/724438/26 "2015-07-08T21:23:32Z")
Yeah, I know you were specifically looking for the dihedral angles, but I didn’t feel like doing all that crossing and dotting. > [@Chronos](#): > > How did you find those? There was a 90 minute all hands meeting at work today. ---
**Author:** ![Lance\_Turbo](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lance_turbo/32/6156_2.png) [@Lance\_Turbo](https://boards.straightdope.com/u/Lance_Turbo)\ **Post date:** [July 8, 2015, 9:54pm UTC](https://boards.straightdope.com/t/determining-dihedral-angles-of-a-solid-from-a-net/724438/27 "2015-07-08T21:54:04Z")
> [@Chronos](#): > > For the general case, you probably need to put in some stipulation about which sides are shorter or longer. Some arrangements won’t produce a valid solid (I think this would mean no real roots of your quartic). The quartic is guaranteed to have four real roots for all triangles a, b, c. This I can prove. I don’t know if there are constraints on a, b, c. I sketched a few cases and they all seemed to work, but this is pretty far from a proof. One thing I should note is that all four edges of length a are adjacent while there are two pairs each of adjacent b’s and c’s. I’m not crazy about the way I worded the previous sentence, but I hope you get the idea. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [July 8, 2015, 10:03pm UTC](https://boards.straightdope.com/t/determining-dihedral-angles-of-a-solid-from-a-net/724438/28 "2015-07-08T22:03:09Z")
Well, if nothing else, angle alpha must be acute, since you have four copies of it meeting at each of the “pole” vertices. Trying it with alpha as a right angle would make some faces coplanar and turn it into a tetrahedron, and an obtuse alpha just wouldn’t fit for a convex figure. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [July 9, 2015, 1:21am UTC](https://boards.straightdope.com/t/determining-dihedral-angles-of-a-solid-from-a-net/724438/29 "2015-07-09T01:21:19Z")
OK, I just took a stab at doing those crosses and dots, and there’s clearly something wrong, here. For your x = sqrt((41 - sqrt(1105))/2) , I get 4.93753277 …, compared to your figure of 1.969576061 . On the other hand, if we can fix that, the relevant products promise to be quite simple, owing to the edge lengths being small integers. ---
**Author:** ![jimbuff314](https://avatars.discourse-cdn.com/v4/letter/j/c57346/32.png) [@jimbuff314](https://boards.straightdope.com/u/jimbuff314)\ **Post date:** [July 9, 2015, 2:24am UTC](https://boards.straightdope.com/t/determining-dihedral-angles-of-a-solid-from-a-net/724438/30 "2015-07-09T02:24:35Z")
You didn’t follow the parentheses order. You took sqrt(1105)/2 and subtracted it from 41 instead of (41-sqrt(1105))/2 ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [July 9, 2015, 11:08am UTC](https://boards.straightdope.com/t/determining-dihedral-angles-of-a-solid-from-a-net/724438/31 "2015-07-09T11:08:08Z")
Ah, yes. Nested parentheses can be nasty. I’ll try again this afternoon. ---
**Author:** ![Lance\_Turbo](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lance_turbo/32/6156_2.png) [@Lance\_Turbo](https://boards.straightdope.com/u/Lance_Turbo)\ **Post date:** [July 9, 2015, 9:38pm UTC](https://boards.straightdope.com/t/determining-dihedral-angles-of-a-solid-from-a-net/724438/32 "2015-07-09T21:38:06Z")
Here’s the angle associated with each edge for the 3-4-5 case. Points are labeled as in post 23. ac -\> 82.0708236 ad -\> 110.3011706 ae -\> 139.248521 af -\> 139.248521 bc -\> 82.0708236 bd -\> 110.3011706 be -\> 139.248521 bf -\> 139.248521 ce -\> 110.3011706 cf -\> 110.3011706 de -\> 82.0708236 df -\> 82.0708236 Some of those might actually be the supplements of the angle you’re looking for. I’m having a little trouble visualizing. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [July 10, 2015, 12:47am UTC](https://boards.straightdope.com/t/determining-dihedral-angles-of-a-solid-from-a-net/724438/33 "2015-07-10T00:47:30Z")
> [@](#): > > I’m having a little trouble visualizing. [This might help](https://www.tinkercad.com/things/jJffYEbl2w6-d8). In other words, mission accomplished. Thanks so much! [Previous page](https://boards.straightdope.com/t/determining-dihedral-angles-of-a-solid-from-a-net/724438.md?page=1)