Gyroid

dpont

Member
Hi,

There are various formulas of Gyroid,
here is a simple one, as a compound node,
for shape displacement (Null ->Shape ->Empty Volume)

Gyroid.jpg

View attachment SDF Gyroid_nodes.zip

Intersecting with a sphere,

CSG-Gyroid-Sphere.jpg

Also as a shader through a gradient, boring on a box, more interesting
on a curved mesh.

gyroid_shader.jpg gyroid_shader_bump.jpg

Denis.
 

dpont

Member
Of course you can use formulas for transformation,
but here a torus shape is easy to animate with the Twist node from DP Kit,
though the shape a precision of 0.05.

(animated gif)

torus_twisted.gif

Denis.
 

dpont

Member
Pitting the Gyroid, just adding a Math scalar Max node with Dot procedural,
before pluging in Displacement,

gyroid_dots.jpg

Next a Double Gyroid,

Double_Gyroid.jpg

View attachment 144651

Here a bit distorted with premilinary Gradient Scaling over distance,
Scaling goes over the sphere intersection,
(including artefacts caused by limited accuracy).

Double_Gyroid_gradscale.jpg

Animated,

Double_Gyroid.gif


(updated: missing connection in SDF Double Gyroid nodes)

Denis.
 

Attachments

  • SDF Double Gyroid_nodes.zip
    2.5 KB · Views: 214
Last edited:

dpont

Member
Triplic Periodic Functions

A shorter title for
Triply Periodic Bicontinuous Cubic Microdomain Morphologies
the math web page is here:
http://www.msri.org/publications/sgp/jim/papers/morphbysymmetry/text/final.html

Or even shortly known as minimal surfaces,
you can use basic functions, named F100, F110 etc,
(meaning you are using cosinus or sinus, simple coordinates or by a factor of 2 or 3),
with one or an addition of these functions with a scaling factor,
sometimes a constant offset...

Triply Periodic Functions.jpg


for the following équations, including the Gyroid,
(each reference is illustrated on the web page above)


Figure 1a. A level set approximation for the P surface:
F(100).

Figure 1b. A level set approximation to the I-WP surface:
F(110) + 0.25.

Figure 1c. Two-parameter experiment.
s*F(100) + (1 - s)*F(110) - t.

Figure 1d. An approximation of the O, CT-O surface.
0.6*F(110) - 0.4*F(111) - 1.

Figure 1e. An approximation to the Neovius’ surface C(P).
0.6*F(100) - 0.4*F(111) + 0.3.

Figure 1f. P+C(P) surface approximation.
0.35*F(111) + 0.2*F(100) + 0.2*F(222) + 0.1*F(200) + 0.05*F(300) +0.1*F(110).

Figure 1g. K surface approximation. This surface was also discovered by Karcher
0.3*F(100) + 0.3*F(110) - 0.4*F(200) + 0.2.

Figure 1h. F-RD surface approximation.
0.8*F(111) + 0.1*F(222) - 0.1*F(220).

Figure 2a. G (gyroid) level surface:
F(110).

Figure 2b. G' surface level set:
G' has several new features as compared to the gyroid, e.g. extra holes and tunnels.
by von Schnering and Nesper).
F(211) + 0.32.

Figure 2c. Two-parameter experiment
s*F211 + (1-s)*F220 - t.

Figure 2d. Double gyroid surface approximation.
0.8*F(211) - 0.2*F(220).

Figure 2e. An approximation to the L surface. The equation is:
0.5*F(211) - F(220) + 0.15.

Figure 3a. D surface level set approximation, shifted here by p /4 for a more familiar representation:
F(111).

Figure 3b. D' surface level set:
0.5*F(111) - 0.5*F(220) - 0.2.

Figure 3c. Double diamond surface
0.5*F(110) + 0.5*F(111).

Figure 4. The P2-DG surface.
It arises as a combination of terms of the double gyroid and P-surface level sets.
0.7*F(211) - 0.1*F(220) - 0.2*F(200).




View attachment Triplic Periodic Functions Nodes.zip


Hope there's not typo errors in my transcription...
As you can see in the above screenshot, the radius of the Shape is set to 4,
and you get only the shape starting, you may also scale
the Object Position before the connection.


Denis.
 
Last edited:

dpont

Member
Hi denis. Will any of the node setups you have attached here work in earlier versions of LW?

Shape and Signed Distance was introduced in LW 2019,
except my shading example in the first post, this is required for using those setups.

Denis.
 

lardbros

Not so newbie member
These are absolutely BRILLIANT Denis! :)

Thank you so much for sharing your efforts, it's so cool to see some of these really difficult mathematical shapes being done in LW.

I have a feeling it may be possible to mesh these using VDB? :)
 

daforum

da what?
Shape and Signed Distance was introduced in LW 2019,
except my shading example in the first post, this is required for using those setups.

Denis.

Thank you for the info Denis.
I love the examples in posts #4 and 5 and that's what I wanted to recreate.
 

dpont

Member
Truchet 3D

Beside the Gyroïd, often used as base for "cave travelling"
a similar infinite math shape generation,
the Truchet 3D, 3 unconnected tunnels
based on 3 quart part of torus,

Truchet3D.jpg

repeat and rotated randomnly,

(animated)
Truchet3D.gif



Denis.
 
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