Shapes of space

There is a branch of mathematics, topology, which can be described as the study of the possible “shapes of space”, referring to the 3-dimensional space as we experience it in the world, as well as 4-dimensional space-time and other more abstract spaces (such as those of higher dimensions). It is the study of structure at a more basic level than geometry because it contains no notion of distance or angles. It is sometimes dubbed “rubber sheet geometry”, for reasons which will be clear shortly.

Topologically speaking, a brick is the same as a ball and a coffee cup is the same as a doughnut. In both cases each could be morphed into its counterpart smoothly with only stretching, squeezing, folding or twisting allowed, but not tearing, breaking or hole-filling (assuming the objects were made of pliable enough material), or in the jargon of mathematics “remaining invariant under homeomorphic deformation”. Topology is also concerned with the definition of regions and their connectedness (within and between), for example the notions of “inside” and “outside”.

There is one aspect of topology that I think is relevant to philosophy, particularly Merleau-Ponty’s phenomenology. That is the concepts of intrinsic and extrinsic points of view. The intrinsic viewpoint is from the perspective of the mathematical object itself (or a being living on it). For example, for a 2-dimensional being living on a torus (the surface of a doughnut) the universe is just the surface. Nothing else exists. That is the intrinsic viewpoint. This being can carry out experiments which will confirm that its universe conforms to a Euclidean geometry (for example the angles of triangles add to 180° and the circumference of a circle = 2πR).

It would also be possible in principle for the being on the torus to verify that travel in a straight line in either of two orthogonal directions would result in return to the same place, and that some loops cannot be contracted to a point (the loops that from the extrinsic viewpoint are snagged around the inner “hole”). From the intrinsic viewpoint there is no “hole”. The world is complete in itself. From the extrinsic viewpoint on the other hand, the torus is situated in an ambient 3-dimensional space. Travel paths that from the intrinsic viewpoint are geodesics (shortest possible routes or straight lines) are from the extrinsic viewpoint curves (such as circles).

The analogous situation arises for a being on a sphere (the 2-dimensional surface of a ball). From the intrinsic point of view the surface is the whole universe. The angles of triangles will add up to more than 180° and the circumference of circles will be less than 2πR, hence the geometry is non-Euclidean.

The point is that due to a theorem Theorema Egregium (Remarkable Theorem) proved by Gauss, the curvature of the “world” (curvature being a purely theoretical notion since as far as the resident being is concerned the world is perfectly flat and curvature has no physical meaning – curved through what?) can be measured purely on the surface (“in the world”) alone, which will determine the appropriate geometry of “the world” (Euclidean or non-Euclidean) without recourse to a mystical “3rd dimension”.

I am thinking that perhaps topology might be fruitful in somehow shedding light on those terms, such as connectedness, boundary, inside, outside etc. which also crop up in philosophy and related disciplines, and I am not the first to have this idea.

Philosophers such as René Descartes and Jacques Lacan, for example were very interested in topology. Descartes was a pioneer in the field, discovering what became known as Euler’s characteristic more than 100 years before eminent mathematician Leonhard Euler did so in 1752. Lacan is another kettle of fish entirely, however. But while someone like physicist Alan Sokal (perpetrator of the “Sokal hoax” resulting from the publishing of this paper) would no doubt mock Lacan’s use of examples from the field of topology to illustrate points in the fields of philosophy and psychoanalysis, I have a more charitable view. At the very least, the contemplation of such wonderful objects as the Möbius strip, the Klein bottle and Borromean Rings (employed by Lacan as described here and here ) should be encouraged to help us think more imaginatively and less dogmatically about concepts such as connectivity and inside/outside-ness.


Topology abounds in even more astoundingly intuition-defying objects, for example fractal objects such as the Koch snowflake (which has finite area but infinite length perimeter) and Alexander’s Horned Sphere.

“Although they separate space into two regions, those regions are so twisted and knotted that they are not homeomorphic to the inside and outside of a normal sphere”.

Rather, the inside is, but the outside is not, because it is not simply connected (some loops cannot be contracted to a point).

My point with these curious topological oddities is this. If the (apparently) two sides of a surface can in fact sometimes be just one side and if sometimes the regions of inside and outside are unable to be cleanly distinguished, then perhaps the apparent Cartesian dualism split (“mind is interior and matter is exterior and never the twain shall meet”) might turn out not to be so hopelessly clearcut, with tendrils of each realm ambiguously and inextricably embracing the other, taking the form of a topological entity. I am attracted to the phenomenology of Merleau-Ponty for just that reason. And, after all, the fractal objects of pure mathematical thought have turned out to be highly useful in modelling natural systems, so why not consider topology as one more tool for modelling ontology?

I know this post has been pretty far out there in the speculative stratosphere for me, but it was good to indulge my passion for topology and mix it up with my new found interest in phenomenology. I expect the next few posts will be more down to earth.


About middleeuropeanmelancholy

64 year old Australian born male. Into travel, poetry, philosophy, music, popular physics, mathematics (especially topology)...
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2 Responses to Shapes of space

  1. DylanTrigg says:

    Have you had a chance to read Steven Rosen’s “Topologies of the Flesh”? I think you’d enjoy it:

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