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 Post subject: How to calculate angles.
PostPosted: Thu Jan 02, 2014 6:55 am 
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This is for TP designs.

Let's say I want to put 8 pentagons around an octagonal face. If the octagon is laying flat, what angle should the pentagons make in relation to the octagonal face to fit exactly together? And most importantly, how do I go about calculating these kind of angles?

Thanks in advance!

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 Post subject: Re: How to calculate angles.
PostPosted: Thu Jan 02, 2014 7:53 am 
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You need to specify the overall shape/symmetry of the puzzle (not just one face) and explain how the pentagons lie with restect to the octagon: flat in the same plane, or at an angle between two faces, etc. Are the pentagons and octagon all regular? The angle within a regular octagon is 135, and that of a regular pentagon is 108. But how do the pentagons sit relative to each other, the octagon, and other adjacent sides, and how do those other sides sit relative to the octagon? Do they overlap, or share an edge or corner?

Can you post a sketch, or actually create the design in SolidWorks? The latter would also give you the answer.

EDIT: Am I right in thinking each corner is made by one regular octagon and 2 regular pentagons?

In any case, to calculate the answer you just need a bit of trig, plus pythagoras...

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 Post subject: Re: How to calculate angles.
PostPosted: Thu Jan 02, 2014 8:04 am 
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I'm sorry if my explanation wasn't clear in my opening post.

A regular octagon lies flat, 8 regular pentagons surround it (in 3D space), having one edge connected to the octagon. Every pentagon is touching the edges of it's 2 neigbouring pentagons too. I don't have solidworks, but Sketchup, and

Thus, at every vertex on the octagon, 2 pentagons meet it.

I want to create these kind of shapes, but need to know how to calculate the angles that the pentagons make in respect to the octagon.

I hope someone can make sense of this.

Thanks!

EDIT: I just saw you edit, and you are right, as explained above. I know I have to use pythagoras, I just don't know where to start. And to think that I was actally pretty good at math(s) in school... It's been too long!

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 Post subject: Re: How to calculate angles.
PostPosted: Thu Jan 02, 2014 8:11 am 
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OK, got it. The key is that you need to break everything into triangles:

1. Draw just one 3D corner with 2 pentagons and one octagon, assuming unit length of each edge and regular angles of 135 and 108 degrees within the octagon and pentagons, respectively.
2. Draw lines between between the midpoint of the octagon edge to the opposite corner of the adjacent pentagon
3. Add more lines to build various triangles
4. Calculate all the angles using basic trig and pythagoras
5. Basically you need the angle between the line that bisects the octagon (running between midpoints of opposite edges) and the line that bisects a pentagon (as in 2 above).

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 Post subject: Re: How to calculate angles.
PostPosted: Thu Jan 02, 2014 8:24 am 
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Thanks, I will tke a shot at that, however, it seems to me that there should be an easier way to calculate this.
I am now doing this stuff with trial and error. Which takes too long. Still, my gut tells me it could be done without drawing triangles... But hey, what do I know! :lol:

If we do the same as above, but replace the octagon with a square:

180-72° is too far to join the surrounding pentagons by the edges, the pentagons will intersect.
72° is the internal angle between vertices in a pentagon.
Because it would seem that these values should have something in common, that is what makes it so counter-intuitive to calculate this.

EDIT: the angle in the above example seems to be 180-71,05=108,95 degrees.

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Last edited by 1NSAN3 on Thu Jan 02, 2014 8:42 am, edited 1 time in total.

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 Post subject: Re: How to calculate angles.
PostPosted: Thu Jan 02, 2014 8:36 am 
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Is this something like what you were meaning?

Attachment:
File comment: OctagonPentagonsThingy
OctagonPentagonsThingy.png
OctagonPentagonsThingy.png [ 32.88 KiB | Viewed 1248 times ]


That is just a rough approximation though, the angle between the octagon and a pentagon is around 140 degrees.

Although I may be way off here :P

-Mark- :)

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 Post subject: Re: How to calculate angles.
PostPosted: Thu Jan 02, 2014 8:40 am 
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Door wrote:
Is this something like what you were meaning?

Attachment:
OctagonPentagonsThingy.png


That is just a rough approximation though, the angle between the octagon and a pentagon is around 140 degrees.

Although I may be way off here :P

-Mark- :)


That is exactly what I would like to draw, however, how did you calculate the angles? Or is this in SW and is there some kind of feature that allows you to do this in SW?

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 Post subject: Re: How to calculate angles.
PostPosted: Thu Jan 02, 2014 8:42 am 
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1NSAN3 wrote:
Door wrote:
Is this something like what you were meaning?

Attachment:
OctagonPentagonsThingy.png


That is just a rough approximation though, the angle between the octagon and a pentagon is around 140 degrees.

Although I may be way off here :P

-Mark- :)


That is exactly what I would like to draw, however, how did you calculate the angles? Or is this in SW and is there some kind of feature that allows you to do this in SW?


Normally I would use 3D sketching to make this strange shape. Then you can use a dimension to get the angle.

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 Post subject: Re: How to calculate angles.
PostPosted: Thu Jan 02, 2014 8:48 am 
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This is in Alibre design, which I use for my puzzle designing. I made the model extremely rough, just changing the angle on the pentagons until they met up along the edges. Then I used a build-in feature to measure the angle of my model:

Attachment:
File comment: Angle
OctagonPentagonsThingy2.png
OctagonPentagonsThingy2.png [ 40.43 KiB | Viewed 1237 times ]


-Mark- :)

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 Post subject: Re: How to calculate angles.
PostPosted: Thu Jan 02, 2014 8:50 am 
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Mark, this means your pentagons do not meet exactly at their edges, there overlapping actually.
By trial and error, I found out the exact angle: it's 141,7 degrees.

However, I'd still like to know a simpler way to calculate this, without drawing anything, just using numbers.

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 Post subject: Re: How to calculate angles.
PostPosted: Thu Jan 02, 2014 8:59 am 
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1NSAN3 wrote:
Mark, this means your pentagons do not meet exactly at their edges, there overlapping actually.
By trial and error, I found out the exact angle: it's 141,7 degrees.

However, I'd still like to know a simpler way to calculate this, without drawing anything, just using numbers.


As I said, this was just a rough hack job, to visualise the object at hand. :P Unless I got lucky, my angle is nothing more than an approximate.

There should be a fairly straight forward way to calculate the angle - perhaps someone could come up with a formula where as long as the polygons are regular (and they can actually be fitted around each other in such a way) you can input the number of sides of each, and get the angle as a result.

-Mark-

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Tony Fisher wrote:
A rare puzzle is one that is only lightly cooked.

Kelvin Stott wrote:
Squiggle is such a funny word to say out loud. Squiggle! :lol:

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 Post subject: Re: How to calculate angles.
PostPosted: Thu Jan 02, 2014 9:07 am 
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Door wrote:
1NSAN3 wrote:
Mark, this means your pentagons do not meet exactly at their edges, there overlapping actually.
By trial and error, I found out the exact angle: it's 141,7 degrees.

However, I'd still like to know a simpler way to calculate this, without drawing anything, just using numbers.


As I said, this was just a rough hack job, to visualise the object at hand. :P Unless I got lucky, my angle is nothing more than an approximate.

There should be a fairly straight forward way to calculate the angle - perhaps someone could come up with a formula where as long as the polygons are regular (and they can actually be fitted around each other in such a way) you can input the number of sides of each, and get the angle as a result.

-Mark-


Thank you, for finding the words that I could not find, to describe exactly what I would love to have. A formula to calculate these angles. 8-)

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 Post subject: Re: How to calculate angles.
PostPosted: Thu Jan 02, 2014 9:26 am 
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If you follow the process as I described above (with trig and pythagoras on basic triangles) then you will also see how successive angles (each labeled by a different letter) depend on each other, as a series of simple formulae. Then if you substitute these formulae into each other by simple algebra you will end up with an overall precise formula, as a complex function of the two basic angles (108 and 135 degrees). There is no other way to get such a formula, no short cuts, and you just need to make the effort to work through all the math.

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 Post subject: Re: How to calculate angles.
PostPosted: Thu Jan 02, 2014 9:34 am 
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KelvinS wrote:
If you follow the process as I described above (with trig and pythagoras on basic triangles) then you will also see how successive angles (each labeled by a different letter) depend on each other, as a series of simple formulae as a complex function of the two basic angles (108 and 135 degrees). Then if you substitute these formulae into each other by simple algebra you will end up with an overall precise formula. There is no other way to get such a formula, no short cuts, and you just need to make the effort to work through all the math.


Could you please give an example? That way I could more easilyunderstand it then when I have to visualize it. Thanks.

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 Post subject: Re: How to calculate angles.
PostPosted: Thu Jan 02, 2014 9:36 am 
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I'm writing this from my iPhone so can't attach pictures, sorry. Perhaps someone else can explain better what I have in mind?

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 Post subject: Re: How to calculate angles.
PostPosted: Thu Jan 02, 2014 9:49 am 
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KelvinS wrote:
I'm writing this from my iPhone so can't attach pictures, sorry. Perhaps someone else can explain better what I have in mind?


No need to apologize Kevin :wink: thanks for showing interest in this topic!

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 Post subject: Re: How to calculate angles.
PostPosted: Thu Jan 02, 2014 9:51 am 
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Using some vector math, if there are three faces at a vertex with angles a, b, and c, the cosine of the dihedral angle between the faces with angles a and c is (cos(b)-cos(a)cos(c))/(sin(a)sin(c)). In your case, this comes out to Arccos((((2 + sqrt(2)]) (1 - sqrt(5)))/(2 sqrt(5 + sqrt(5)))) which is about 141.668 degrees, very close to the approximation. I can give more details on deriving the formula upon request.

I am not quite sure what polyhedron you are trying to construct, but there is no convex polyhedron with exactly two pentagons and one octagon at each vertex as can be seen by trying to place all of the faces adjacent to the pentagon. There is a polyhedron that is close (some vertexes have three pentagons), but the pentagons and octagons cannot be regular at the same time.

Attachment:
File comment: close
polyhedron.png
polyhedron.png [ 67.3 KiB | Viewed 1209 times ]

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 Post subject: Re: How to calculate angles.
PostPosted: Thu Jan 02, 2014 10:36 am 
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Yikes, the formula for calculating a dihedral angle from the angles surrounding a 3-edged vetex is a monster, or it might just seem more monstrous than it really is due to my screen reader being crap at reading mathematical formulas. I would be interested in the derivation, though I doubt it could easily be written in a way that both my screenreader and myself could comprehend. Is their a similar formula for when four or more faces meet at a vertex, or is it easier to just figure out the angle between the edges of the faces you want the dihedral angle of?

Is the shape posted in that image the shape generated by replacing the polar pentagons of a dodecahedron with Octagons, adding extra pentagons to complete the two bands and dissorting the angles so everything fits together? Or alternatively, equavlent to a octagonal trapozohedron with its polar vertexes truncated? If you restrict the Octagons to being relar and the pentagons to being equilateral, how far from having all equal angles would the pentagons have to be?

t

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 Post subject: Re: How to calculate angles.
PostPosted: Thu Jan 02, 2014 10:53 am 
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Wow contrabass, now that is helpful! I am not trying to construct anything, I'm just experimenting with mixing shapes together that look nice, to get somewhat symmetrical polyhedra, mostly using regular polygons and replacing them with irregular ones where neccesary. Very interesting way to spend your time!

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 Post subject: Re: How to calculate angles.
PostPosted: Thu Jan 02, 2014 11:05 am 
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Hi Eric,

If you didn't already, you might want to check the Wolfram pages on Platonic, Arcimedean en Johnson solids
http://mathworld.wolfram.com/JohnsonSolid.html

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 Post subject: Re: How to calculate angles.
PostPosted: Thu Jan 02, 2014 1:16 pm 
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maarten wrote:
Hi Eric,

If you didn't already, you might want to check the Wolfram pages on Platonic, Arcimedean en Johnson solids
http://mathworld.wolfram.com/JohnsonSolid.html


Yeah, I already did, but thanks anyway :wink:

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 Post subject: Re: How to calculate angles.
PostPosted: Thu Jan 02, 2014 2:01 pm 
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Thanks!
The polyhedron shown was constructed by replacing each edge of a cube with two pentagons or truncating the 8-fold vertexes of a triakis octahedron, although other constructions are possible. With the coordinates shown, the canonical ones, the octagons are regular and the pentagons have four angles measuring 105.7 degrees and on of 117.2 degrees, which can't be made regular but possibly could be made closer. There are also two different edge lengths, with one being about 1.344 times the length of the other.
As for where the formula comes from, if you put the vertex at the origin and have three vectors A, B, and C which are unit vectors along the edges pointing away from the vertex, then to calculate the dihedral angle on the edge with vector B, first you need to find a vectors orthogonal to B and on each adjacent face. These vectors are (using a.b to indicate the dot product) A-(A.B)B and C-(C.B)C (look up projections to see why). Then the cosine of the dihedral angle is (A-(A.B)B).(C-(C.B)C)/(|A-(A.B)B|*|C-(C.B)C|) which simplifies to (A.C-(A.B)(C.B))/sqrt((1-(A.B)^2)(1-(C.B)^2)). Using A.B=cos(c) because A and B are of unit length, this gives the formula I listed above, (cos(b)-cos(a)cos(c))/(sin(a)sin(c)). Hopefully this is decipherable.
Another way of writing the coding of the angle that I listed is cot(108)tan(67.5) (arguments in degrees).

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 Post subject: Re: How to calculate angles.
PostPosted: Thu Jan 02, 2014 3:02 pm 
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Perhaps pillowing could make all the pentagonal angles the same?

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 Post subject: Re: How to calculate angles.
PostPosted: Thu Jan 02, 2014 3:06 pm 
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KelvinS wrote:
Perhaps pillowing could make all the pentagonal angles the same?

Maybe externally but pillowing doesn't change the internal geometry so making them the same wouldn't help the mechanical workings of the puzzle or allow for jumbling between the different dihedral angles.

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 Post subject: Re: How to calculate angles.
PostPosted: Thu Jan 02, 2014 3:21 pm 
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Very true, changing the internal geometry would require fudging, but since these are only surface features and not cuts, there isn't any corresponding internal geometry to fudge.

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