Modelling surface diagrams

Question

While working on a project in higher category theory, a branch of mathematics, I was lead to draw so-called "surface diagrams" similiar to those that can find in the following papers; page 22-23, page 13 or in this blogpost.

You can think of them as objects in three-dimensional space made up of various coloured sheets of paper (of different shape, sometimes with their boundary partially jagged) glued together at parts of their boundary. Onto these "surfaces" one draws coloured "lines" (1-cells).

A self-drawn example is this one:

Reading from bottom to top (and right to left):

• 1st slice: two sheets (one jagged, the other not) are merged into one;
• 2nd slice: two sheets are merged into one at the first black line. Then the resulting sheet and the blue sheet are merged at the red line. Next the resulting sheet is merged at the black line with the sheet closest to the viewer;
• 3rd slice: Two sheets are merged at the black line;
• Note that the blue patch is not a hole, but a coloured sheet.

I am trying to model these three-dimensional objects in Blender — as a total beginner in Blender.

• It is possible to model these surfaces with Blender, right? Or are there better suited alternatives?
• How would you approach modelling such objects? Different suggestions welcome. Ideally, after having drawn one such surface one should be able to glue sheets of paper onto its side easily and stack two such surfaces on top of each other. I have unsuccessfully played around with Blender, trying the shrinkwrap modifier and various other modifiers to bend objects the way needed for my surface diagrams.

Answer 1

Is this the kind of result you are trying to achieve? It's pretty easy to extrude new faces (sheets) from a Blender plane that acts as the main sheet.

Create a plane (⇧ Shift + A Add > Plane), rotate it vertically (R, X 90) then enter Edit Mode Tab).

You subdivide the plane (or add new loops with CTRL+R), pick edge select (2), select edges, press E to extrude and you will pull another sheet from the main plane.

Then you switch to vertex select (1), pick vertices and drag them around (with G) until you get the result you want. Or, in edge select, you can press R and rotate the edges. You can always add more edges with CTRL+R or by subdividing the entire model (⇧ Shift + A, Right Click, Subdivide). To add an edge between two vertices or make a face out of 3 vertices, select them and press F.
To delete a vertex, edge or face, select it and press X, then pick what feature you want to delete.

These are the basic tools for a beginner to start editing meshes.

You will need some patience until you get the desired concept. Use G and R to move and rotate vertices in combination with X, Y, Z to specify the axis for better control of the transformations.

Answer 2

(Using Blender 3.6.8)

Approach

Use NUBRS surfaces and Bezier curves for smoothness, and Geometry Nodes for non-destructive workflow.

Procedure

Add Surface/Nurbs Curve to model each branch:

Extrude NURBS curves to make NUBRS surfaces:

Hooks are useful to constrain adjacent control points to move together:

Add Curve/Bezier to model each morphism (?), adjusting control points position in vicinity of the supporting surface:

A GeometryNodes modifier is added to each curve, to project it on its supporting surface and to make it thicker:

Add Mesh/Ico Sphere to model nodes:

Resources

• Blender file:
• Replicating Shrinkwrap modifier: see Raycast Node: Use target normals as rays instead of source normals (replicate Shrinkwrap Target Normal Project mode)
• Colouring face both sides: see How to apply a different color to the face of part of an object
• Rendering jagged border: see How to create a procedural chevron like texture/material?

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More advanced setups

Blender file:

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