Bitcoin and Topology
Bitcoin and Topology
We’re thinking about Bitcoin wrong.
First principles.
Bitcoin isn’t a digital, decentralised currency.
It’s not a new asset class. It’s not digital gold, or a store of value, or a unit of account, or a new payment system, or a technological innovation. It isn’t philosophy or politics. It isn’t beauty or truth.
At least, not in its primary ontology.
Bitcoin is geometry.
Its financial properties and behaviours can be understood as manifestations of topology.
In its foundational modality, the way it manifests in the world, Bitcoin is a one-sided surface, a complex, non-orientable, non-linear, continuous, self-connected shape.
We’ll explain what we mean through reference to a Klein bottle.
A Klein bottle is created by taking a cylinder and connecting its two ends together, but with a twist through a fourth spatial dimension. The resulting shape has no ‘inside’ or ‘outside’, as traditionally understood. Instead, its single surface twists and turns through space, looping back on itself in a way that can’t be represented in three-dimensional, Euclidean space (R3) without intersecting itself.
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A Klein bottle requires four-dimensional space for a true representation where it doesn’t intersect itself.
A 3D sphere or a torus (think doughnuts) has two distinct sides (inside and outside). A Klein bottle only has one side. An ant crawling along the surface of a Klein bottle could cover the entire surface, seamlessly transitioning from what might appear as the ‘outside’ to the ‘inside’ without crossing an edge, and without ever leaving the surface.
The Klein bottle, in other words, manifests beyond the intuitive constraints of three-dimensional geometry. It represents the vastness of imagination. It maps the depth of our inquiry into the nature of space and dimensions.
Bitcoin conforms to the higher dimensional and surface behaviour of a Klein bottle.
This is easily proved.
1. Digital realm as fourth dimension
The Klein bottle exists comfortably without regard to conventional geometric limitations.
So does Bitcoin.
A Klein bottle doesn’t care about Euclidean space or logic.
Neither does Bitcoin.
Bitcoin’s existence challenges and extends beyond traditional three-dimensional financial constructs of currency, assets, and value storage. It introduces a new (fourth) spatial dimension of financial interaction,[1]incorporating decentralization, digital scarcity, and cryptographic security as intrinsic dimensions of its structure.
Like its other spatial properties, these characteristics are allocated across Bitcoin’s surface, and – through its distributed ledger technology (blockchain) – are accessible at all times and from all points on its surface.
The digital dimension that Bitcoin adds to traditional finance and asset management is complex, and neither entirely visible nor comprehensible from traditional (Euclidean) perspectives.
Grasping the impact and potential of Bitcoin from the perspective of the existing financial system entails navigating complexities of higher dimensions along with a non-trivial expansion of imagination.
2. Non-orientation and boundaries
Bitcoin is a self-connected, higher-dimensional shape, but integrates (in topological terms, forms a ‘connected sum’) with the fiat system through exchanges and transactions, while operating under entirely different principles.
In the same way as a Klein bottle challenges our notions of inside and outside, Bitcoin blurs the lines between different financial realms.
Bitcoin’s geometric properties and spatial dynamics account for its capacity to flow (move value) and operate seamlessly across national and economic boundaries, unhindered by the conventional barriers of cross-border payments.
Transactions on the blockchain traverse the surface of Bitcoin’s manifold, embodying a kind of topological fluidity.
The structure’s decentralized nature and the absence of a central authority controlling it is a function of its lack of orientation and boundary.
3. Local observations vs global reality
Sound predictions about reality are rarely made from observing only local regions of a plane (geometrical, aesthetic, psychological, emotional, existential). Knowing a small, localized region of a curve doesn’t necessarily reveal its global properties, such as overall shape, symmetry, or boundedness.
The challenge becomes even more profound with non-orientable surfaces like Bitcoin.
Individuals and institutions may perceive and interact with aspects of Bitcoin and the broader cryptocurrency ecosystem in vastly different ways, depending on their perspective and understanding of the ‘local region’ of the current context.
Each transaction using Bitcoin, no matter its size, reflects the network’s overall architecture (consistency of rules and protocols). However, the insights that understanding a single transaction can provide into the nature of the entire Bitcoin network are of necessity partial, and do not, for example, reveal much about overall health, scalability, and security challenges, or about the dynamics of node distribution and mining power concentration.
Just as it’s impossible to comprehend the entirety of a Klein bottle from a single, three-dimensional vantage point, it’s impossible to grasp the full complexity of Bitcoin without considering the additional dimension it introduces.
This doesn’t mean we shouldn’t try.
As we try, further rubrics will be added to the three listed above.
TL;DR
Bitcoin is a single-sided, non-orientable manifold. It is continuous, seamless, boundaryless. It does not represent personal autonomy, individual freedom, or self-sovereignity. Such claims make sense only within localized regions of the larger manifold.
Bitcoin manifests the principle of continuous, unified existence and interaction within the digital realm.
Recognising it as such will determine how we interact with it. How we interact with Bitcoin will determine the way we exist as individuals. How we exist as individuals will determine how we participate collectively in the world.
How we participate collectively in the world will determine the future.
[1] The fifth dimension of Bitcoin is time. Gigi has covered that.