Since no one answered the question directly, Iâ€™ll take my captain obviousâ€™ hat and explain how this is done.

He makes it sound like this scene is the graph of a complex mathematical equation, and implies there was no program involved to further imply this idea. He is lying; or, to be more precise, heâ€™s misleading people into a lie he never states explicitly. This is a simple volumetric ray tracer written in shader language where models are defined by implicit equations.

What he means by â€śno programâ€ť is â€śI wrote a small programâ€ť, and what he means by â€śno 3d POLY modelsâ€ť is â€śI used a different representation than the one used in most 3d programsâ€ť.

The fact he wrote a small program is obvious so Iâ€™m just going to skip that. The only thing to mention here is that the program he wrote is running not on your processor, but on your GPU as a shader. This is probably why he suggests there are â€śno programs involvedâ€ť, besides the fact the wrote the program himself. By definition, a shader is a small program whose main function is to â€śshadeâ€ť, or to calculate lighting, but modern cards are able to run arbitrary code. That is what is known as a compute shader, or a general purpose shader program.

As for the models themselves, theyâ€™re volumetric models derived from implicit equations. This is why he claims this is a â€śmathematical sceneâ€ť, but itâ€™s no more a mathematical representation of a 3d world than a Blender scene is. Itâ€™s just a different representation.

To understand what an implicit volumetric definition is, let us analyze a simple 2d circle. The equation could be:

x^2 + y^2 - c^2 = 0

This is an implicit circle of radius c. Any solution to this equation is a point on the circleâ€™s perimeter. This is what the graph of the equation really is: the set of all points that solve it. Hence, to â€śrenderâ€ť this circleâ€™s outline, you could â€ścast 2d raysâ€ť, testing all (x, y) coordinates along this ray against the equation to see if (x, y) should be drawn. If the equation is true, then (x, y) lie on the circleâ€™s perimeter. Because you â€śmarchâ€ť along the ray, this is called ray marching.

The reason this is called a â€śvolumetricâ€ť representation is because we can tell whatâ€™s inside and outside the circle based on a Signed Distance Funcion(SDF), hence we are aware of the volume, not just the surface of the object. The SDF is a signed function that evaluates to less than 0 if a given point P lies inside a certain volume, greater than 0 if it lies outside said volume, and 0 if it lies exactly on the surface.

Let D(P, Q) be the shortest distance function between points P and Q, and let O be the origin of a circle. For a circle, the SDF would then be the function that â€śchecksâ€ť whether the distance from P to O is less than the radius r of the circle. In mathematical language:

F(P, r) = D(P, O) - r

This is what he means by â€śthere no poly modelsâ€ť. Models were built from similar fundamental implicit equations, and more complicated ones are derived from said primitive equations. Theyâ€™re then rendered through this process. Thereâ€™s also a technique for describing implicit volumetric models using a hierarchical tree structure with boolean operations whose concepts he may or may not have applied mentally when building the scene. He also wrote definitions for simple math and linear algebra functions to help with coordinate space transformations, and performs most computations inline(so if youâ€™re not familiar with them, it looks like magic).

How does this differ from the usual representation? Most 3d programs care only for the surface of the model and have no interest for the volume(ever noticed 3d models are â€śhollowâ€ť, not solid?). To describe the surface we approximate it through a finite set of polygons, and the surface itself is calculated from interpolation of surface information during a several steps process called rasterization.

In the end of the day this is still the same basic math any graphics programmer knows by heart, so heâ€™s no more a math genius than the Blender devs are. I will say though that even if the math is nothing special, putting something into practice something is. Thatâ€™s worth more than theoretical knowledge in this field.