Stellation of Icosahedron

SoapyJoe · 2006-07-19 22:01:12

The way to do this is to use a 'filter' in conjunction with Adobe Photoshop called Flexify2. This filter is quite amazing... One needs to purchase it but it is not expensive.

Some parts of the photograph are a bit distorted but if you select with care this can actually add to the effect.

I will post a copy of the filtered Babushka photograph so you can see what I mean.

SoapyJoe · 2006-07-19 22:05:46

Filtered Flat photograph

MathsIsFun · 2006-07-19 22:07:52

SoapyJoe wrote:

My first question is what would be the size of a 4V form ..... could I get it through my workshop door?

"So, why are you taking the roof of your workshop, Soapy?"
"Well, it all started when ... "

But seriously, I guess you can make it any size you want. Even the same size as the 3V, except the triangles will be smaller. Your diagram has the length multiples, and you can relate that to your finished model and the 3V diagram you posted here: http://www.mathsisfun.com/forum/viewtop … 217#p35217

I also think I could follow the same procedure as the 3V to calculate the dihedral angles, and your diagram has the length multiples, so you would be all set.

MathsIsFun · 2006-07-19 22:13:26

BTW That is a great-looking net. I have some nets I made of the platonic solids, for example Dodecahedron Net. How did you do the net? With Flexify2? (It gives me some ideas for some really cool class projects)

SoapyJoe · 2006-07-19 22:16:17

The white tabs are designed so that you could construct this model from paper and glue it all together..... I ignore them but put the photograph onto layers.... print onto 6x4 photographic paper..... laminate to protect the print .... then cut them out and stick them onto the model with doubles sided tape.... meaning I can quite easily change the photgraph when I should wish to.

If one selects the distorted bits and place them at say the North Pole.... they look quite good.... I rotate the whole thing with a micro-wave turntable motor.

Soapy

SoapyJoe · 2006-07-19 22:27:27

In Flexify 2 you can select any of 20 forms of input and filter them to 80 types of output.... lots of combinations! The output for this was 'football' or something like that..... all the usual sorts of mathematical shapes are listed

Just type 'Flexify 2' into 'Google' and follow the links.

Soapy

MathsIsFun · 2006-07-19 22:32:40

Just had a look at their page. Looks great.

I was thinking I could have some nets (the platonic solids plus more) with nice artwork like your babushka, and students could download, print and assemble as a class project.

John E. Franklin · 2006-07-19 23:53:05

Referring to post #27 photo of foldable football (soccerball) with 32 sides.

There are 10 columns of pentagons and hexagons.

The columns shown are 3,4,4,3,3,3,3,3,3,3 high.

Is the top of the 2nd column and the bottom of the 3rd column opposite sides when folded up??

And could they have been sticking out on other columns instead over by 2's and perhaps rotated??

SoapyJoe · 2006-07-20 01:15:09

You will need 'Photoshop' to sort of 'host' the filter .... It may work with the cut down version of Photoshop called 'elements' but I have not experimented with it.

The top of the 2nd column is normally the Nouth Pole and the bottom of the 3rd column is the South Pole.

However, you can arrange them differently like having the 'star' (top of the 9th column) as the North Pole and shuffling everything about.... but I can tell you from experience that this is quite a mind blowing excercise...... the best approach is to assemble, as presented in the net, starting at the North Pole then placing the matching pieces together like a jigsaw. Then, with some system of numbering, shuffle them around the 'sphere'.

Another way is to assemble the pictures before deciding which are the 'Poles'.

Another problem, which I experienced doing 'Babushka', was that I had a hexagonal face as N & S.
The 'net' always places a pentagon at N & S

I could not figure how to cope with this so started again.... I had to make another pentagon face to acommodate the rotational spindle.

Great fun!

Soapy

SoapyJoe · 2006-07-20 01:24:54

Re-reading your posting..... the north & south poles could not be placed at the top or bottom of other columns..... the picture would not match up. The individual pieces of the network need to stay firmly attached to their buddies. However, as I explained above, the whole thing can be pivoted as required.

Hope this helps!

Soapy

John E. Franklin · 2006-07-20 04:27:31

Neato man --- you got me interested in 3-D angles so I'm gonna pose a question in a new thread.
As for wood working, I'm afraid of power jig saws since I need all my fingers for clarinet playing.
But is there a way to cut accurate angles down to the degree with a hand jig saw or something?

SoapyJoe · 2006-07-20 18:59:05

Anything is possible!..... sure angles can be cut with hand tools to the degree of acuracy needed but if you consider that the 3V form of the Icosahedron is made up from some 180 triangles each consisting (in my method of construction)) of three pieces of wood . These individual components are mitred at each end and one face angled..... this adds up to a lot of individual joints.

I would equate this task as similar to 'fashioning a grand piano from jelly with chopsticks'.

Soapy

MathsIsFun · 2006-07-20 19:43:45

John, I need my fingers for typing! I am extra careful with power tools. I use a saw mounted on a workbench and have it just high enough to cut the wood.

I had to cut some small aluminum angles once, which required my fingers getting real close to the saw. I made one slip and cut a neat groove in my fingertip. But my policy of "just high enough" meant the cut was only millimeters deep and healed nicely.

I haven't used a bandsaw though.

BTW Soapy, did you want to go ahead with the 4V?

SoapyJoe · 2006-07-20 20:31:44

I would love to go ahead with the 4V form.

But honestly the maths is more than a little beyond me.

I understood everything (from a practical point of view) that you so kindly worked out for me for the 3V and I believe that the photograph that I posted is more than proof that your maths was spot on..... I was very impressed with your working methods.

I am presently constructing the small stelation of the dodecahedron which I believe will look quite stunning .... I am actually 'skinning' it with mirror acrylic and incorporating a LED light show in the base...... When this is finished I would love to construct something a little more difficult.

The 4V form... If I made every individual triangle based on B = 2 inches (from the Icosahedral symetry network that I posted about two days ago) then I estimate that I would end up with a sphere some 20 inches in diameter..... I think I can get this through the workshop door.....

When I was working on the 3V form I made myself a working drawing based on your figures (which I will post shortly)

If you would so kindly provide me with the set of figures, from which I could construct a similar working drawing, I would be very appreciative.

Soapy

SoapyJoe · 2006-07-20 20:34:34

This really is the only drawing that I needed for the 3V

MathsIsFun · 2006-07-20 20:57:19

OK, give me a day or two

SoapyJoe · 2006-07-20 21:30:23

Thanks,

It is going to be a few weeks before I can start this new project so no rush.

But I do like to plan everything in advance.

Working out how much timber I need.

What I am going to skin it with

Arranging the motor drive

And, above all, the 'Dihedral Angles' or the way that I cut them using these figures.

Soapy

SoapyJoe · 2006-07-21 05:37:09

Thinking about the 4V project ....

This afternoon I decided to map out the 4V form onto the 'rough' model of a Icosahedron that I made when I was working on the 3V form.

When I made the 3V .... after producing all the individual triangles I sub-assembled the pentagons and hexagons prior to full assembly.

Looking at this model the above procedure will not work.... in my opinion.

What I will probably do is sub-assemble the 12 pentagons and sub-assemble the remaining 13 triangles that constitute a 'face'

Just thinking about it at the moment.

Soapy

SoapyJoe · 2006-07-21 05:39:28

Sorry needs rotating

MathsIsFun · 2006-07-21 11:47:00

There is also a "central" triangle that in theory should be "parallel" to the base nodel's face.

SoapyJoe · 2006-07-22 18:24:28

Yes, This triangle is an equilateral designated the letters FFF.

I will most probably be building my sub-assemblies around this particular triangle.

Soapy

SoapyJoe · 2006-07-24 01:51:14

An interesting shot show the construction of the small stellation of a dodecahedron.

The Pentagonal pyramids are all attached to a central Dodecehedron core.

Note the faces of the pyramid have been rebated to take the 'skin'.

I will make a base to house the drive motor and will construct in a similar manner but think I will make the base in the form of a pyramid but with an angle that corresponds to the angle that the tripod legs make with the base.

Need to work this all out.

Soapy

MathsIsFun · 2006-07-24 09:36:03

It works!

Was our thinking on it right?

Yes, I think it would look good for the stand to "fit" the shape of the solid.

SoapyJoe · 2006-07-24 20:39:16

Yes, The thinking was quite correct.

The only angle that I needed (except, of course, constructing the triangles) was setting the bandsaw to 21.7degrees. This angle was used to construct the pyramids and the dodecahedron that the pyramids sit onto.

Just messing about with the design for the base which will contain the microwave turntable drive motor and the wiring for the blue and white LEDs.

When I skin the shape with small triangles of mirror acrylic it should look quite different.

The base ......My thinking at the moment is a triangular pyramid with base of say 8 inches sides and the height of the apex about 2.25 inches...... a very quick drawing of this shape seems to fit quite nicely with the shape of the 'stellation'.

I should be able to work out the chord lengths and the dihedral.

Soapy

MathsIsFun · 2006-07-29 20:07:27

OK, Done!

         1
       2   3
     4   5   6
   7   8   9  10
11  12  13  14  15

These are the angles I calculated:

2-3 7.80°
2-5 10.38°
4-5 10.02°
5-8 10.36°

I think all the other angles can be worked out by symmetry.

I also made a Dihedral Angle Calculator in case you happen to have some coordinates of two planes and want to get the angle.

Are you going to test it in MDF first?

Math Is Fun Forum

#26 2006-07-19 22:01:12

Re: Stellation of Icosahedron

#27 2006-07-19 22:05:46

Re: Stellation of Icosahedron

#28 2006-07-19 22:07:52

Re: Stellation of Icosahedron

#29 2006-07-19 22:13:26

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#30 2006-07-19 22:16:17

Re: Stellation of Icosahedron

#31 2006-07-19 22:27:27

Re: Stellation of Icosahedron

#32 2006-07-19 22:32:40

Re: Stellation of Icosahedron

#33 2006-07-19 23:53:05

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#34 2006-07-20 01:15:09

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#35 2006-07-20 01:24:54

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#36 2006-07-20 04:27:31

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#37 2006-07-20 18:59:05

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#38 2006-07-20 19:43:45

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#39 2006-07-20 20:31:44

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#40 2006-07-20 20:34:34

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#41 2006-07-20 20:57:19

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#42 2006-07-20 21:30:23

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#43 2006-07-21 05:37:09

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#44 2006-07-21 05:39:28

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#45 2006-07-21 11:47:00

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#46 2006-07-22 18:24:28

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#47 2006-07-24 01:51:14

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#48 2006-07-24 09:36:03

Re: Stellation of Icosahedron

#49 2006-07-24 20:39:16

Re: Stellation of Icosahedron

#50 2006-07-29 20:07:27

Re: Stellation of Icosahedron

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