intersecting cylinders

[Ulrich]

Hi Ulrich,

> This is the core of six intersecting cylinders (displayed like
> 60-fold prisms).

Maybe you'd consider writing a post about it and how to create it on the
Stella forum? I'm sure others would be interested.

Thanks,
Rob.

It's quickly made: Take a 60-fold prism and facet 5-fold prisms from it:



Then you augment them to each face of a dodecahedron (excavation
does it as well). Now you facet new 60-fold prisms, using the vertices
from two parallel planes:



This model is not too complex, so you can dare to activate the stellation
view and watch the core:



If you apply this procedure on the icosahedron, the result is much more
complex and you can't stellate it. If you take a 30-fold prism it
works, but it isn't that pretty. If you use the option "create
convex core" in the "Poly"-menu, it works fine even with the 60-fold
prisms.

Ulrich

[robertw]

Thanks Ulrich!

Yes, rather than stellating the model to find its core, which requires the resource-heavy stellation process, better to just use "Poly->Create Convex Core", which will be more or less instant.

I thought of another way to do it. I'll demonstrate it for the icosahedron below.

Here are the steps:

• Start with an icosahedron (Ctrl+N, i, Enter)
• Set the edge length to 1 (Scale->Base Polyhedron Edge Length)
• Remember the distance from the centre to a face plane using "Scale->Base Polyhedron Inradius" (0.755761314076171).
• Create a 60-fold prism (Ctrl+N, 60.4.4, Enter)
• (We will create a 3-fold antiprism by faceting).
• From the rotational symmetry drop-down select "3-fold Dihedral (A)"
• Show reflection planes (hit "r")
• From the reflection symmetry drop-down, select "Vertical Reflection"
• Enter Faceting Mode (click ) and make sure you've got a Faceting Preview open (click in the view you want to change and hit ).
• Facet a large equilateral triangle from the top 60-sided face. Choose points where the reflection planes cross. A symmetrical repeat will automatically be added below like this:
  
• Facet one more triangle (will be auto-repeated for a total of 6) to complete a shallow antiprism.
  
• Put the completed faceting in a memory slot (click in the Faceting Preview view and type "m1")
• Select the base view again and use "Edit->Add/Blend from Memory->Memory 1". You now have a compound of the 60-fold prism and the new shallow antiprism.
• Use measurement mode to find the length of the large equilateral triangle's edge, and set it to 1 with "Scale->Measured Distance"
• Leave measurement mode (Esc) and select the large triangle.
• Copy the result shown with "Scale->Base Polyhedron Inradius" (0.0302161784251633).
• Now stretch the model with "Scale->Non-Uniform Scale". Use a calculator to divide the values we remembered earlier (divide by the current height and multiply by the height we want) and enter this value as the scale (25.01181001257231527 = 0.755761314076171 / 0.0302161784251633)
  
• Put the model in another memory slot (m2) and reload an icosahedron.
• Extract the model in memory (hit "a", choose Memory 2 from the drop-down and select Extract). Click OK. Hit Enter to accept the augmentation preview, and select Yes when asked whether to blend coplanar faces.
• Colours will come from the model in memory, so tell Stella to do its normal auto-colouring again, which will colour each part of the compound separately (hit Tab twice, or "Color->Basic Color Scheme->Auto Color"). Makes a nice-looking model:
  
• There's one more trick though before we're done. If you create the convex core now, it just comes out as a dodecahedron! Why? Because there's a great dodecahedron hiding inside as part of this compound of approximate cylinders. How did that get in there? Its pentagons were created when the lateral triangles of the antiprisms were blended together. Anyway, just explode the faces apart a bit till you spot a pentagon (Ctrl+Shift+Left-drag), then select a pentagon and hit Delete (to delete one part of a compound, same as "Poly->Delete One Part of Compound").
• Now at last we can use "Poly->Create Convex Code" to see the intersection.
  

Probably no easier in the end, although it looks more complicated because I've detailed every step for the beginners our there

Even with 60 sides the division is not really enough though, so more would be required for a better approximation to the true cylinder version.

Rob.

[robertw]

I did the same thing with a 240-fold prism!

Here are the cylinders:


And here is the intersection:


Even at 240, some of the edges are a bit wiggly!

However, if you try to create a 240.4.4 in Stella you will get an error due to the way they're created (I'll fix it sometime). Instead you should create something like 240/89.4.4 and then take the convex hull.

Rob.

[robertw]

I have moved this topic to the Stella forum, as it's about how to use Stella to create these intersections, and not about physical models.

Please post any further replies there.

Rob.

[Ulrich]

There's a way to produce such models still more easily by using the
"put models on faces"-function (a feature which could be enhanced
a lot). The picture shows the core of 30 intersecting 360-fold prisms.



To produce it, you just have to put the prisms on all faces of a strombic
hexecontahedron and create the convex core. The only problem is the
colouring of the prisms.

This one (31 prisms) is looking even better. It' s available from the
rhombicosidodecahedron using the same steps like above.



Ulrich

[robertw]

Ah, of course! A much easier way.

What are your suggestions for enhancing that feature? Maybe start a new post on the Feature Requests forum.

Rob.

[Ulrich]

I made some pictures of the cores of intersecting cylinders, see:

http://www.polyedergarten.de/intersecti ... index.html

This was easily done with stella 4D by putting 500-fold prisms on the
faces of uniform polyhedra or their duals. This can take a while if there
are many faces. In the second step, you colour as a compound, then you
create the convex core. In some cases you have to rearrange the colours
a bit because you have stripes of different colours where there should be
only one.
The naming in the gallery is according to the number of "cylinders", the
origin polyhedron ("d" means the dual) and the kind of the starting prism.

Best wishes

Ulrich

[robertw]

Very nice! The snub dodecahedron is certainly surprising. You might want to add a description of what the models are, how to make them, and of course a mention that you used Stella

The stripes are due to prisms on opposite faces whose lateral faces are not aligned. Would be easier to ensure opposite prisms have the same colour before taking the convex core rather than trying to fix it afterwards.

Also, for the N-prisms, you might want to make sure N is divisible by the rotational symmetry of the face on which it will be placed, eg divisible by 5 on a dodecahedron's face. That way overall symmetry should be maintained in the result. Maybe easiest to always use a value of N that divides all the common symmetries, ie 2, 3, 4, 5. Might even avoid the striping, but not sure. So N should be a factor of 60, eg N = 480, 540, or 600.

Rob.

[Ulrich]

.., and of course a mention that you used Stella

Okay. so you better use this link:

http://www.polyedergarten.de/e_cylintro.htm

I gave the snub dodecahedron a try with a 480-fold prism and it worked fine
(the 500-fold one failed here). The stripes were there too, but that's no
great problem.

Ulrich