Topology

Topology is the branch of mathematics concerned with the geometric deformations of objects. According to its rules, a certain type of flat square - in which opposite edges have been mathematically linked - is equivalent to a holed-doughnut, or torus, because one can easily be turned into the other. (New Scientist Magazine)

Torus

A manifold is a topological space that is locally Euclidean (i.e., around every point, there is a neighbourhood that is topologically the same as the open unit ball). To illustrate this idea, consider the ancient belief that the Earth was flat as contrasted with the modern evidence that it is round. The discrepancy arises essentially from the fact that on the small scales that we see, the Earth does indeed look flat. In general, any object that is nearly "flat" on small scales is a manifold, and so manifolds constitute a generalization of objects we could live on in which we would encounter the round/flat Earth problem, as first codified by Poincaré.

More concisely, any object that can be "charted" is a manifold.

One of the goals of topology is to find ways of distinguishing manifolds. For instance, a circle is topologically the same as any closed loop, no matter how different these two manifolds may appear. Similarly, the surface of a coffee mug with a handle is topologically the same as the surface of the donut, and this type of surface is called a (one-handled) torus. (Wolfram Mathworld).

Tactile understanding

I wanted to gain tactile experience of the ring torus. Instead of making it from a square paper, I found an alternative via the internet, a paper ‘slice’ template. I hand cut and slot each slice together. Mathematics is the abstract science of number and space. By making this space physical emphasises the importance of haptic (tactile) feedback in understanding objects in space (children predominately learn this way). The object itself is quite ‘dumb’, crudely assembled but expresses the idea that we are approximations (as with topology) and this ‘dumbness’ may allow you to see yourself in the object, in a more ‘naked’ way.

Each part containing the whole and vice versa

The following information is based on an Interview with Dr David Peat about 'Wholeness and the Implicate Order'. The torus is just one shape out of many from a flat square; you could construct an origami crane, for example. The theory of the Implicate Order developed by the physicist David Bohm can be illustrated as such; the paper contains all potential transformations into many objects, the methods are the same, but the precise sequence has to be followed to create the objects. But all sequences are simultaneously present in the Implicate Order. The mind of the person folding selects one and follows necessary steps to transform it into the physical realm. Therefore, the Implicate Order is seen as holding potential information for all processes of transformation in both physical and mental realms; everything is enfolded into everything else.

The one ‘slice’ I left out is a play on the notion of each part containing the whole and vice versa.

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