# 3D Delaunay Lofts

Hi, Everyone

I am interested in the Delaunay Lofts in voronoi tessellation.
However, I can’t build this three-layer geometry model in the grasshopper.
The Delaunay Lofts from the bottom to the top is(1) a regular hexagonal; (2) a square, and; (3) another regular hexagonal tiling.

Reference: Sai Ganesh Subramanian et al, 2019, Delaunay Lofts: A biologically inspired approach for modeling space filling modular structures.

Fenzo

It’s been a little while since I’ve worked with lofts in grasshopper, but a big part of getting consistent behavior out of them revolves around equal numbers of vertices and ensuring that curve directions are the same. The loft is trying to match points, and much like a grasshopper solution where you’re adding with two lists of numbers, where list A has 6 entries and list B has 4, you’ll have a A1+B1, A2+B2, A3+B3, A4+B4, A5+B4, A6+B4 situation. The loft will produce odd results because it’s trying to match the leftover vertices to create the form.
In the bottom of that last image it looks like two of the hexagon vertices converge on a single one of the rectangle’s. So in some ways this is an A1+B1, A2+B1, A3+B2, A4+B3, A5+B3, A6+B4 matchup, producing those lovely triangle shapes in the lower corners.
(Please excuse if this is bad practice) I don’t have time at the moment, but I’d recommend making the “rectangle” in your loft a polygon with 6 vertices, but with 2 zero-length edges, so the loft functions as expected. Try doing it manually, with rhino geometry and lofts, first.

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There’s two quick options I came up within the attached definition that could have what you’re looking for, although I’m sure there’s way more elegant ways to do this.

Both are explained within the file but a quick summary below of the two options:

• Option A: goes through a similar workflow to what @Aramon went into detail in his post (and my usual preferred way). I tend to do a lot of form finding exercises and this is a good general approach that you can use on many environments. Getting familiar with how the “data” relates to the form helps a lot.

• Option B: uses a convex hull approach which does not have the need to filter and allocate the sequence/order of the vertices, only downside, you won’t be able to define your vertex connections.

I’d say the post above by @Aramon is spot on regards to a good general workflow to achieve what you want if you’d like to give it a go yourself, I would not worry too much about loft being the only solution.

Form Finding_Hexagonal Scutoid_AC.gh (31.5 KB)
Form Finding_Hexagonal Scutoid_Model.3dm (734.3 KB)

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it’s an interesting topic!

found this video that describes a bit better the logic behing the paper: https://www.youtube.com/watch?v=SA-Mq9lvv7s

I guess the initial and final exagon structure is just a byproduct of how points are distributed on the plane

start, mid, and final sets of points, connected with their respective axis

slice those axis with a range of planes along Z:

and voronoi each of these set of points (I have hidden perimetral cells as they deform due to Voronoi boundary being a rectangle):

which looks pretty similar to what’s described in the video:

of course, the more divisions (slicing planes) the more definition the shape gets:

here is a very bad and non-optimized sketch:

Delaunay_Lofts_Sketch.gh (15.0 KB)

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Thank you @Alebrew and @aramon. You guys both gave me a clear way to build this interesting model. Now I have some ideas to create different geometry interlocking, such as adding the Metaball for the organism shape would be a fun puzzle. Cheers!

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@inno So happy that you found the video! Yes, the paper shows the geometry transformation from the bottom to the top is amazing and exciting. I think this slice allows us to have something more customized shape on it. Thank you for making this sketch!

I think the most interesting -and probably useful- part is at minute 4:07, where they give some hints about using the wallpaper geometric patterns to generate space-filling shapes as interpolations of regular tilings

whenever you move any step forward in that direction and are willing to share, I would enjoy much reading about those

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