Stellation of Icosahedron

SoapyJoe · 2006-07-01 18:53:06

Following the very sucessful completion of my 3V form of a Icosahedron I now wish to construct on of the 'stellations'

Any help or advice would be very welcomed.

Soapy

SoapyJoe · 2006-07-01 18:55:36

Drawing of Stellation

MathsIsFun · 2006-07-02 00:03:20

I am sure it can be done ...

We must first work out the coordinates of the vertices, then from the coordinates we can work out the dihedral angles using the same technique as for the previous model.

So ... how do we work out the coordinates?

Can you tell us the details of the polyhedron you have chosen?

SoapyJoe · 2006-07-02 18:41:15

As you have probably guessed my constructional skills are considerably better than my mathematics.

Looking at various shapes posted on the internet I wish now to construct a stellated form of an Icosahedron.

I have not made up my mind which... but did wonder if the Small Triambic Icosahedron as shown at http://mathworld.wolfram.com/SmallTriambicIcosahedron.html would be a suitable project.

MathsIsFun · 2006-07-02 22:44:21

It looks like this, right?

I think it is made by just taking an icosahedron and putting a little pyramid on each face. I think.

Also see this page: http://www.prospero78.freeserve.co.uk/icosa/stell02.htm

MathsIsFun · 2006-07-02 22:50:40

If that is indeed the case, then I already have the coordinates of the icosahedron, and all we need do is add the apex of the pyramid, which is √(15) / 15 high according to mathworld

In fact, I may even be able to animate it like this: Spinning Icosahedron

Yes?

SoapyJoe · 2006-07-07 01:36:32

One very quick question...

If I were to construct a 'Small Sellated Dodecahedron'

How can I work out the size of the individual sides of the pentagons to arrive at a model that will be around 14 inches diameter?

MathsIsFun · 2006-07-07 23:06:09

It has 60 faces, each "pointy bit" (call it a pentagonal pyramid) has 5 faces, and there are 12 of them. So, "underlying" it is a Dodecahedron (12 pentagonal faces).

The height of the little pyramids is √( (1/5)(5 + 2√5) ) = 1.37638 times the edge length of the "underlying" dodecahedron.

The radius of a sphere that would touch the tips of the pyramids is (1/4)5[sup](1/4)[/sup] √(2(√5 - 1)) = 0.587785 times the edge length of each pentagram (according to mathworld). The pentagram is the 5 pointed star, that is visible on your image above.

In fact, that pentagram is very important. If you take a dodecahedron, and extend the edges you end up making this shape. In other words, take each pentagon and make into a pentagram.

MathsIsFun · 2006-07-07 23:58:49

So, the whole thing will be 2*0.587785 = 1.17557 times as large as the edge length of the pentagram.

And the pentagram is amazing because it contains lots of ratios all based on the "golden mean", which is approx 1.618

The ratio of the pentagram edge length to the pentagon edge length is 1.618*(1+1.618) = 4.235924

So, the overall size of the polyhedron will be 2*0.587785*4.235924 = 4.98, or approx 5 times.

So, for a 14" model the pentagon edge will be 2.8"

Please give me time to think about this, as I may have made a mistake somewhere (maybe someone could check my work?)

SoapyJoe · 2006-07-08 02:23:58

Thanks ever so much.

I had guessed about a 3inch edge.

I am, at present, just about to make a trial Pentagonal pyramid .... will make from MDF to start with ..... Am I correct in assumimg the the pyramids should all go together with a dihedral angle that is directly in relation to 116.56 degrees (the dihedral angle for a dodecahedron)

Soapy

SoapyJoe · 2006-07-09 01:14:50

No real problems in constructing the trial Pentagonal Pyramid.... fairly straightforward!

But how do I work out the dihedral angle that 12 of these join up together?

soapy

SoapyJoe · 2006-07-09 01:17:27

The Pentagonal Pyramid

MathsIsFun · 2006-07-09 11:22:47

Nice! I think your guess would be right, because each face of that pentagonal pyramid is like an "extension" of a dodecahedron face, in a similar way that each line of the pentagram is an extension of one of the pentagon's sides.

So, I am not using any great mathematical insights here, just my 3D sense (which may not be as good as yours!) to visualize the inner angle of the dodecahedron being the same as the angle between the face on one pyamid and another.

I am not *sure* this is right. To be sure I would need to work out the coords of each face and do the sums.

George,Y · 2006-07-09 17:32:43

Great! A foot ball!

SoapyJoe · 2006-07-09 20:24:26

Good thinking!

From a constructional point of view a very easy solution would be to make a dodecahedron from pentogons exactly the same size as the bases of the pyramids.... then mount the 12 pyramids onto this 'framework'.

I could use a less expensive wood for the framework because once the pyramids are in position nothing will be seen of the framework.

This model will be made from Honduras Walnut .... still thinking about what I choose as the 'skin'.

It's great to know that other like minded people are out there.

Will keep you posted.

Soapy

SoapyJoe · 2006-07-18 00:21:15

Just a quick follow on..... Photograph of the truncated Icosahedron with photograph mounted.

I call it 'Babushka'

soapy

John E. Franklin · 2006-07-18 19:24:39

Nice idea to add the photograph that coincide. How do you get all the photographs to go together all the way around??

MathsIsFun · 2006-07-18 21:15:24

Beautiful, Soapy! Great contrast of cuteness and fine woodwork.

George,Y · 2006-07-18 23:33:52

How did you deal with errors accured while construction?

MathsIsFun · 2006-07-19 00:08:05

If it was me I would just tighten the clamps a little more

SoapyJoe · 2006-07-19 18:48:11

The selection of a suitable photograph is really the most important consideration. I could almost write a book on it.... one thing is that the left hand side of the print needs to look very similar to the right otherwise where they join will be very obvious. The photograph dose not have to exactly match... but if you had say part of a house at the left hand edge then it would look a mess.

If anybody is familar with Photoshop then it is possible to get a range of 'filters' that assist in sorting out photographs for different mathematical shapes .... If anybody is interested I will go into it further.

SoapyJoe · 2006-07-19 18:49:29

This is the photograph that I used....

SoapyJoe · 2006-07-19 19:01:21

Errors in construction .... even a few minutes out in the 'Dihedrals' is a disaster... the sphere will not 'join' but if one sets the angles on the bandsaw with a good quality vernier protractor and makes all the component bits as accurate as possible then 'tightening the clamps' is really all that is neededto make minor adjustments.

Life gets a bit exciting when one gets to the last few pieces ..... then number of times that I have thought .... this is never going to join!

If anybody would like any assistance in constructing one of these models please contact me .... I find it is one of the most satisfactory projects that i have embarked on in years.

soapy.

SoapyJoe · 2006-07-19 19:15:36

I would love to construct the 4V form of a Icosahedron but I know that the maths is very difficult.

I feel that I can cope (the woodwork) with triangles with say approx 1.5 inch sides.

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

Soapy

John E. Franklin · 2006-07-19 21:05:32

So photoshop can take a flat photograph and clip portions that go around a soccer ball???
How can this be done without repeating portions of the photo or stretching, which would look distorted??

Math Is Fun Forum

#1 2006-07-01 18:53:06

Stellation of Icosahedron

#2 2006-07-01 18:55:36

Re: Stellation of Icosahedron

#3 2006-07-02 00:03:20

Re: Stellation of Icosahedron

#4 2006-07-02 18:41:15

Re: Stellation of Icosahedron

#5 2006-07-02 22:44:21

Re: Stellation of Icosahedron

#6 2006-07-02 22:50:40

Re: Stellation of Icosahedron

#7 2006-07-07 01:36:32

Re: Stellation of Icosahedron

#8 2006-07-07 23:06:09

Re: Stellation of Icosahedron

#9 2006-07-07 23:58:49

Re: Stellation of Icosahedron

#10 2006-07-08 02:23:58

Re: Stellation of Icosahedron

#11 2006-07-09 01:14:50

Re: Stellation of Icosahedron

#12 2006-07-09 01:17:27

Re: Stellation of Icosahedron

#13 2006-07-09 11:22:47

Re: Stellation of Icosahedron

#14 2006-07-09 17:32:43

Re: Stellation of Icosahedron

#15 2006-07-09 20:24:26

Re: Stellation of Icosahedron

#16 2006-07-18 00:21:15

Re: Stellation of Icosahedron

#17 2006-07-18 19:24:39

Re: Stellation of Icosahedron

#18 2006-07-18 21:15:24

Re: Stellation of Icosahedron

#19 2006-07-18 23:33:52

Re: Stellation of Icosahedron

#20 2006-07-19 00:08:05

Re: Stellation of Icosahedron

#21 2006-07-19 18:48:11

Re: Stellation of Icosahedron

#22 2006-07-19 18:49:29

Re: Stellation of Icosahedron

#23 2006-07-19 19:01:21

Re: Stellation of Icosahedron

#24 2006-07-19 19:15:36

Re: Stellation of Icosahedron

#25 2006-07-19 21:05:32

Re: Stellation of Icosahedron

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