How would you create these types of abstract data visualization / mathematical forms graphics?

Question

See attached for images of the graphics I'm looking to emulate.

I have 3 questions:

1. What software would you need to make graphics in these styles? Could they be done in illustrator? Or would you need to use something like Blender / Cinema4D?

2. For the graphics in the grey background (image 2, 3), there's a very subtle grid-like pattern on the visualizations - how would you achieve that? How would you achieve the grid for the graphics with the black background (image 1)?

3. What would this style of illustration be called?

Answer 1

This is only a partial answer. It's about the surfaces on the grey background.

Assuming you do not expect anything which is accompanied by an exact math formula those surfaces or at least something resembling can be produced in Illustrator. An example:

Draw a rectangle. Make a copy of it:

In the left one of the copies is filled with a colorful gradient. In the middle there's gradient from solid white to black. In the right the BW gradient version is rasterized and re-vectorized by applying Object > Create Object Mosaic. The rasterized version is deleted.

The object mosaic has got an envelope distort with a simple mesh:

This phase is difficult. It needs extreme patience to fold a surface in an interesting way. I had only 2x2 mesh, but you need more if you are going to make complex foldings. See NOTE1.

Make a copy of the distorted mosaic and release the envelope distortion. You get a free mesh which can be applied directly to the colorful gradient. Use Object > Envelope Distort > Make with Top Object:

The released mesh is the light grey shape on the top. It's applied to the colorful gradient in the right. As you see the colors are not bent, but it's good enough here for me. Another more complex coloring method is to drag swatches to the mesh nodes. You can insert nodes with the mesh tool simply by clicking the mesh, if needed. An example (no extra nodes inserted):

Another way to get bent colorful gradient is to rasterize the gradient before applying the envelope distortion. An example:

The price is to have a raster image in the work. It doesn't stand scaling to a bigger size and the edges are clearly less perfect than in the full vector version. Many of us avoid raster parts in their work just for these reasons. The drawbacks are not harmful if you have rasterizing resolution high enough for the final usage.

You can combine the colorful shape and the distorted grey mosaic to get the subtle grid. Align the mosaic and the colorful shape, bring the mosaic to top and give to it blending mode Hard Light:

I inserted a grey BG to prevent the white corner fading and the black corner looking too dark. You can use limited scale in the original BW gradient, say only between 20% brightness and 80% brightness or reduce the opacity to prevent too white and too dark.

NOTE1: It really needs some thinking and practicing to use envelope distortion properly to fold a shape over and behind itself. The nodes have certain back-front layering order and you cannot change it. In the next example you cannot change which is in front and which is in back:

Answer 2

Some of these you could 'fake' in Illustrator, but for most of them, some kind of 3D software is required. You can see the polygons (squares that make up the mesh) on some of those 3D shapes, which indicates they were made in 3D software. All of these can be made in Blender, which is free.

Answer 3

1. Nearly any application would suffice. A painting, application, a vector application, a 3d graphics package, a 2d-3d CAD package, a mathematics software, programming language or a graphing tool would be able to do this. Or any combination of these.

   What you would choose depends entirely of what you know how to use, how much exploration you intend to do, and how many different works of this kind you intend to do and what is your end use case.

   My personal short list of applications, in order of testing, for this use case would be:

   1. Mathematica
   2. Matplotlib
   3. Postscript + Illustrator
   4. Tikz
   5. Maya
   6. Illustrator
   7. Blender

2. By applying flat shading on each square?

3. 3D Plot.

But seriously I can not stress this enough: You need to use a application that suits your knowledge. It does not really matter if a 3D application could do this in 15 minutes if your 50 hours away from using a 3D application in the first place. Likewise you could probably do this in a math package in 15 minutes too but again if you don't have the mathematical background or don't know the functions of what you want to plot again of no use for you today.