Dihedral Angles

SoapyJoe · 2006-05-28 05:06:20

In this particular model all I did was to cut out equilateral triangles from thin hardboard .... (I did not bother with any angles)... then I glued the triangles to a piece of thin fabric and allowed it to dry.... I followed a 'net' from the web

Once dry I then folded it all together and tacked the joints with 'Duck Tape'

Then I glued along each joint.

Before assembling the 'finished ' product I will bandsaw along all edges that need to be cut at half the dihedral angles for say 1-2 I like to do all angles at one go.... so I need to colour code everything.

I love making these things.

I am enclosing a previous working model I used for the Truncated Icosahedron....

I actually measured all the angles on this model and then worked out averages.... I now know how to do this mathematically .... but I had a lot of fun doing it!

Soapy

SoapyJoe · 2006-05-28 05:08:01

The model

SoapyJoe · 2006-05-28 20:43:51

When measuring the angles on the model of the truncated icosahedron it very quickly became apperent that I was dealing with two quite distinct sets of angles....
1 The angle measured between ajoining hexagons.
2 The angles measured between hexagons to pentagons.

I arrived at a practical figure of 21 degrees Hex to Hex and 18 degrees and 40 mins Hex to Pent. I use a vernier protractor to actually set and measure angles.

Mathematicaly the figures are ...... (180-138.1833)/2 = 20.908 degrees Hex to Hex.
and (180-142.6166)/2 = 18.692 degrees Hex to Pent.

Soapy

MathsIsFun · 2006-05-28 23:37:04

Good - well that part makes sense.

Now, fingers and toes crossed for the 3-frequency dihedrals.

Last edited by MathsIsFun (2006-05-28 23:40:16)

SoapyJoe · 2006-05-31 01:34:22

Just a quick progress report

I have constructed two trials, so far, using the angles that you so kindly assisted with.

I am posting two photographs

1 A sample hexagon from MDF
2 A sample Pentagon.

Personally I felt that they both went together quite well.

I feel that the pentagon is slightly better than the hexagon but in the finished model the glue line against the dark mahogany is not as prominent.

One or two more trials then will have a go with the Mahogany.

Soapy

SoapyJoe · 2006-05-31 01:41:48

Where did the other image go to?

SoapyJoe · 2006-05-31 02:19:27

View of the 'bulge'

MathsIsFun · 2006-05-31 09:43:32

Looking good so far!

This is good fun (just so long as the angles are right!)

(BTW You could use the MDF pieces as little bowls - just add a few feet.)

SoapyJoe · 2006-06-04 05:56:43

Well, thanks to you, the numbers worked out fine.

SoapyJoe · 2006-06-04 05:58:55

As I wrote previously this is only a trial in medium density fibreboard.

But I think stained and polished it sould look quite nice.

soapy

MathsIsFun · 2006-06-04 10:43:20

It looks Grand!

You can see hexagonal patterns then the pentagonal one.

A clear finish would look nice, but may show up the glue lines.

BTW you could colour each polygon using only 4 colours. A proven theorem in maths is that every plane map needs no more than 4 colours so that no two adjacent regions have the same colour. And I think that Hemisphere would qualify, as it could be flattened out without affecting which regions are adjacent.

[Just had a thought - you could fill the inside with expandable foam to give it greater strength]

SoapyJoe · 2006-06-04 18:47:06

Good idea about the four colours .... but will use only different shades of Mahogany.

'A proven theorem in maths is that every plane map needs no more than 4 colours so that no two adjacent regions have the same colour.'

Where can I read about this?

The MDF is 12 mm thick so absolutely no problems about strength .... It is so very strong.

Soapy

MathsIsFun · 2006-06-04 23:21:46

I am fairly sure the "Hemisphere" is four-colorable. And a Sphere also (but not a Torus).

It may need some trial-and-error to discover a workable colour layout for this polyhedron. And you may even be able to use less than 4 colours.

Wikipedia has a good article on it, with references at the end: http://en.wikipedia.org/wiki/Four_color_theorem

SoapyJoe · 2006-06-17 06:43:06

Just a quick update on progress.

I have just made the 3V form of an Icioahedron using the angles that without the assistant of this forum I doubt if I wouls ever have worked out by trial and error.

SoapyJoe · 2006-06-17 06:49:46

My next job is to sand and polish then place some nice pieces of card into each facet.... mount a suitable panoramic photograph.... divided into 120 pieces..... then cover each photograph with a piece of 2mm clear plastic sheet .... then motorise so that it will revolve about 1rev in 2 minutes .... incidently I have seen some slow revolving motors that work from solar power.... but not using these for this model.

soapy

MathsIsFun · 2006-06-17 12:43:48

Amazing! Stunning!

Please explain how you cut the pieces.

Another idea is etched glass - possibly with dark glass, and a light inside.

SoapyJoe · 2006-06-17 18:57:31

The wooden triangles, in this case Mahogony, are individually made from strip that has been first routed to take the photograph and perspex.

If you look at the low resolution photograph you will see one of the individual triangles at the base of the sphere..... this is where the mathematics come in.... each triangle has to be very carefully made to conform to the figures that you kindly worked out..... if the figures are correct and the manufacture is accurate then the shape goes together ..... any slight error is cumulative and I know from bitter experience that the shape will not 'close'.

Even if you get the maths and manufacture correct assembling the last few components is always 'exciting' ..... It is very satisfactory when the last triangles all lock into place and would hold together without glue.

I have been toying with the idea of making another similar model.... not quite sure what mathematic shape

(ANY IDEAS ON SOMETHING A BIT MORE CHALLENGING)

And have been considering in making the 'inserts' from natural Onyx.

I can buy onyx bathroom tiles 150mmX 150 mm X 3mm, imported from Pakistan, these are a bit expensive, and I have not tried cutting them yet.... I believe a wet diamond saw is what bathroom tilers use!

But the Onyx is fantastic and can be illuminated from inside

Another idea is to print the photographs onto clear film, illuminating these from inside.

BUT I AM REALLY LOOKING FOM ANOTHER INTERESTING AND CHALLENGING SHAPE .... ANY IDEAS ?

The finished model should be around say 14 to 15 inches diameter .....

Any larger and my wife will kill me.

Soapy Joseph

MathsIsFun · 2006-06-17 20:18:24

Ahh, thank you. Obvious now you tell me!

Shame on us, we should have a collection of polyhedra for you to browse through

But in the meantime try this wonderful website: http://www.georgehart.com/virtual-polyhedra/vp.html

Another, half-formed, idea also springs to mind: some kind of optical illusion! I think you could create one with an open structure like your image above, but I am not sure about what shape. Possibly something that only reveals itself at certain angles. Something to cogitate on, anyway.

luca-deltodesco · 2006-06-17 21:20:54

you could create a torus

Last edited by luca-deltodesco (2006-06-17 21:23:16)

SoapyJoe · 2006-06-17 23:50:41

Hi,

I had been thinking about a Torus .... thanks!

I have previously looked at George Hart's website ... some quite wonderful stuff.

Now the optical illusion ..... that sounds just me!

Wonder where I can find something suitable.... kept to a suitable size I can make more or less anything ....

I tend to treat wood as just another raw material .... when I was working I used to work with tolerances in microns .... mostly making aluminium prototypes for the electronics industry .... but now I do not have CNC (Computer Numeric Control) equipment but still like to keep my tolerances tight.

Soapy Joseph

SoapyJoe · 2006-06-20 05:49:57

I have now placed some card into each triangle.

In time I will be covering these with a panoramic photograph.

Soapy

SoapyJoe · 2006-06-20 05:55:20

My wife uses this stainless steel vase for flowers.

Can this sort of thing be constructed 'mathematically' ?

I would be looking for something with a bit more of a 'waist'.

I wonder if I could construct it fom say Pentagons, Hexagons, or Triangles.

Soapy

MathsIsFun · 2006-06-20 10:24:41

You coloured them nicely!

Most 3D modelling programs will do "wireframe" models, but you don't have control of the shapes used. For example a cylinder will have a circle at top and bottom, and lots of tall rectangles to make up the sides.

So it may be totally up to us to construct something interesting. You could wrap a piece of paper around it and start sketching triangles, that morph into diamonds, then pentagons or whatever.

As for the shape, it can be modelled mathematically as a "surface of revolution". Basically one simple curved line that is revolved around the central axis of the vase. Like my animation of this cone (down where I say "A Cone is a Rotated Triangle"): Spinning Cone

So, somwehere between defining that "revolved line", and your sketch of triangles we can hopefully come up with a set of 3D points, and hence angles.

I hope

luca: I was wondering if your ray-tracer has anything in it that would help out here?

SoapyJoe · 2006-06-22 06:13:22

The base, for my 3V Icosahedron, will need to be about 4 inches high and about 11 inches in diameter. Inside the 'sphere' is a very slow running electric motor.

This will be in the shape of a plain cylinder.

To be in keeping with the 'sphere' I want to construct it from a layer of 30 assembled triangles and a layer of 15 retangles..... see very rough sketch

My question is this.... I know that the 15 rectangles will all have to be mitred to produce 360/15 = 24 degrees but I feel I am getting myself confused about the angles for the 30 triangles.

Can you please help me?

SoapyJoe · 2006-06-22 06:14:34

The rough sketch

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Re: Dihedral Angles

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