(I have to admit that this post is unfinished. It has been sitting there in draft for a while and I decided now to publish it anyway mainly for the image that it includes. If I find the time it will be expanded. In the meantime, enjoy the inspiration.)

In an artistic mood I created this collage¹ with the Ouroborus in the right hand coming down from the upper right corner – as if descending from a higher sphere – and another hand holding a torus-like tube coming from the left bottom corner, the human realm. Both hands meet in the center of the picture. The observant reader will, of course, recognize the hands as detail in Michelangelo’s “The Creation of Adam” (1512)², that are part of his famous frescos in the Sistine Chapel, just turned upside-down to fit with the two objects placed in them. Click to flip the image and you will recognize them – but make sure the objects don’t fall out! :-P

In the following I will explain what they signify and how I got inspired to make this little piece of artwork. (You may want to toggle dark mode for a better viewing experience.)

Hand implements Link to heading

Two objects have been placed in the hands that correspond: one symbolic for the principle of circularity (the Ouroborus), the other referring to a topology of the Nervous System (the Torus). Consider their meaning by contemplating how they may be related.

While writing this essay I realized that others have published similar interpretations³ of Michelangelo’s painting. Is that what I had in mind when placing the Ouroborus and the Torus in these hands?

The Ouroborus Link to heading

The alchemist notion of transformation is captured in the symbolism of a snake-like creature biting its tail. Why? Because there is an intuition that circularity brings about appearances. Circularity may be viewed as a universal principle. If a token is recycled in a system capable of cognitions it either emerges as an eigen-value, a kind of perceptual invariant, or it will pass momentarily and vanish from perception.

The Ouroborus⁴ (which literally means “tail-eater” in Greek) is a symbol already found in ancient Egypt, originally attributed to renewal and regeneration. It has been appropriated by cybernetics, the theory of autopoesis, and as the form of self-reference.

The Torus Topology Link to heading

The torus has a doubly-closed surface. One around the circumference and one around the tube. The double-closure indicates that two structures are interacting here. In the case of an organism these may be understood as the sensory and motor surfaces of the nervous system. One modulates the other and both form a system capable of perceiving.

Let’s have a look at the torus in mathematical terms. The torus is also called ring torus or torus of revolution, as it revolves a circle around a central axis. Its algebraic surface in 3-space, $\R^3$, is

$$ (x^2+y^2+z^2+R^2-r^2)^2 = 4R^2(x^2+y^2) $$

where $R$ is the major radius and $r$ is the minor radius⁵.

Or, for a vector $\mathbf{p}$ pointing from origin parametrized with $\theta, \phi\ \in [0,2\pi)$ where $\theta$ is the angle around the tube of the torus, and $\phi$ the angle along its circumference,

$$ \mathbf{p}(\theta,\phi) = \begin{bmatrix} (R+r\cos\theta)\cos\phi \\ (R+r\cos\theta)\sin\phi \\ r\sin\theta \end{bmatrix} $$

describes the points on the torus surface⁶.

The torus is degenerate if the major radius tends to zero ($R \rightarrow 0$) and becomes a two-sheeted sphere with radius $r$, the non-zero minor radius. If on the other hand the revolving minor radius tends to zero ($r \rightarrow 0$), it becomes a circle with $R$, the major radius. And if both radii approach zero – a point in the origin.