Euler Vase - Possible Openscad?

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Euler Vase - Possible Openscad?

unkerjay

Euler Vase
http://www.instructables.com/id/Euler-Vase/

It looks like a polygon with truncations and snubs.  Flattened on the bottom and the top
and a boolean (difference) cylinder.

This can be done in Blender using Math Solids under Extra Objects and the Boolean Modifier.

Is it possible to do this programmatically using OpenSCAD?
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Re: Euler Vase - Possible Openscad?

MichaelAtOz
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Yes, you could build all the polygons yourself, but I'd map the points of the vertices and hull it. Like so:
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Re: Euler Vase - Possible Openscad?

leebc
In reply to this post by unkerjay
Thus unkerjay hast written on Mon, Mar 13, 2017 at 06:09:00PM -0700, and, according to prophecy, it shall come to pass that:
> Euler Vase
> http://www.instructables.com/id/Euler-Vase/
>
> It looks like a polygon with truncations and snubs.  Flattened on the bottom
> and the top and a boolean (difference) cylinder.
>
> Is it possible to do this programmatically using OpenSCAD?

I would make this by starting with a rectangular cube, and differencing
off a bunch of other cubes that have been rotate() { translate() { rotate() } }
at various  distances and angles.

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Re: Euler Vase - Possible Openscad?

Ronaldo
In reply to this post by unkerjay
The vase is a simple polyhedron so it is certainly possible. Doing it randomly is easy.

n  = 50; // number of facets
d  = 40; // approx. center diameter
h  = 80; // vase height
r  = 80; // trimming radius
hd = 15; // hole diameter

difference(){
    intersection() {
        cube([d,d,h], center=true);
        intersection_for(i=[0:n-1]) {
            a = rands(-1,1,2);
            b = asin(h/r/2)/2;
            rotate(b*a[0],[-sin(360*a[1]),cos(360*a[1]),0])
            rotate([0,0,360*a[1]])
                translate([r-d/2,0,0])
                    cube(2*r, center=true);
        }
    }
    // the hole
    translate([0,0,-h/2+1])
        cylinder(r=hd/2,h=h);
}
Although the result may be disappointing :)

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Re: Euler Vase - Possible Openscad?

unkerjay
Ronaldo, I think yours comes closest to being programmatically easier to
modify.

But, I've also noticed that, in changing the variables, there are some combinations
that work better than others and possibly either some constraints or if/then might
be useful.

It looks like a matter of "shaving" (slicing?) if you will, substance from a polygon, in that respect
I think, you're on to something, leebc, the trick would be in adjusting the angles (or number)
of the cubes from one variation to the next.  Maybe a loop to adjust the number of cubes and
allow for random placement (probably with constraints) as some placements would work better
than others.

MichaelatOz, yours looks interesting.  I think in terms of the original example, perhaps
best to start with a rectangular cube more tall than wide, then you'd still have to boolean
(difference) the cylinder for the hole.

He references the math of the process in Step 5 if that helps:

"Step 5: The Math of Euler Characteristic (optional)

The title "Euler Vase" refers to the mathematical relationship between Vertices, Edges, and Faces. Regardless of how you choose to shape the planes of the vase, the Euler characteristic, X, will always be the same.

X = V - E + F = 2 where V is the number of vertices, E is the number of edges, and F is the number of faces. For this vase, X = 2, the same Euler Characteristic. You can count the edges and faces you just sanded to verify this for yourself. Use pieces of tape to mark which faces/edges/vertices you count as you go, record the number, then remove the tape and start again. You can pretend the hole isn't there and count the top as one face, or you can subdivide the cylindrical portion into faces and edges and vertices and count them. However, if you make your vase into a "doughnut" by drilling all the way through, then the Euler characteristic becomes zero! Zero is the Euler characteristic for all toruses, such as doughnuts and coffee mugs."

It could likely be done more inexpensively with wood, but, then, you'd need all the requisite tools as well
as the wood itself.

Just seems like, once it's figured out, it could be more readily designed and modified programmatically either via Python (as in the Math Objects settings (advantage there, it's possible to save the variations or using Geodesic Dome settings and saving the variations in separate layers, Nodes in Blender (Sverchok - addon) or via code in OpenSCAD.

As has been noticed in Blender, Extra Objects (Curves or Mesh) are one off settings.  Move or Scale your object and the initial settings window disappears.

That's why, I think OpenSCAD (or Blender scripting or nodes) might be a better approach.

An interesting exercise.  And different from the vases commonly seen from OpenSCAD.
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Re: Euler Vase - Possible Openscad?

Ronaldo
2017-03-14 9:49 GMT-03:00 unkerjay <[hidden email]>:

An interesting exercise.  And different from the vases commonly seen from
OpenSCAD.

I disagree. The kind of vase we can get from the proposed method is a polyhedron as any other vase created with OpenSCAD. Or better, it is a small subset of it: it is a convex polyhedron.

I see two interest in this kind of construction for someone starting his studies in geometry: to understand a well known property of convex sets and recognize the conditions under which the Euler characteristic holds. The woodwork process is just a sequence of planar cuts trimming the initial block. Geometrically this may be regarded as a set intersection of an intermediate solid with a half-space. If the initial block is convex, the final solid will be convex because half-spaces are convex. The behind concept here is the well proved fact the intersection of convex sets is a convex set. The hole will destroy the convexity.

I have not read the instructables in detail to see how it applies the Euler characteristic. But some caution should be taken here and not only with the genus. If we start with a rectangular block, we will count V = 8, E = 12 and F = 6, so X = 2 as expected. Now suppose that we make a non-trespassing hole with a square section in the block top face. New vertices, edges and faces will be added: dV = 8, dE = 12, dF = 5 and the total characteristic will be 1! Has Euler equation failed? No, because it is valid only for polyhedron with faces without holes. A small trick allows that faces with holes be considered: add two edges (not intersecting any other) to the face connecting two vertices of its inner ring (or rings) to two vertices in the outer ring. We have added now 2 edges and 1 face and the total characteristic is again 2.

When there is a genus, the Euler characteristic changes to 2 - 2*G where G is the number of genus.

In a polyhedron, the facets are planar. What would happen with the Euler characteristic if instead of trimming the initial block by half-spaces we use non-planar half-spaces? Nothing changes, the Euler equation still holds even if the facets are non-planar. It is important to understand that Euler equation relates to topology and not only geometry. It is a relation valid for any 2-manifold subdivided in vertices, edges and faces.

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Re: Euler Vase - Possible Openscad?

unkerjay
Most of what I've seen, openscad related is of vases built around a hollow.

Euler vases differ from that in that, as shown, they are polygons into which
holes are added.

I know that they are "polygons" and in that respect, not particularly exceptional.

In terms of what I've seen the vast majority, different, interesting.

From an analytical, mathematical perspective, I'm out of my league.

From an aesthetic perspective, relative to the lion's share of vases I've seen
designed by others using OpenSCAD - different.

Not necessarily remarkable or radically different.  Perhaps mundane from the
vantage point of possibility of designability.

But, first I've seen of it.  Might not be new.  New to me.
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Re: Euler Vase - Possible Openscad?

unkerjay
And, not just a platonic solid.  That would truly be mundane.

This method adds character, if you will, to the shape.

A difference, so to speak, between a block of ice (or stone, or wood)
and a sculpture.  The selective, subjective removal.

A dodecahedron with a hole in it is still a dodecahedron with a hole in
it.

A round, or ovular bowl or a tall rectangular box with a hole in it not particularly
distinguished, notable or noteworthy.  

This is something more.