Bonfire Space: Lights That Cannot Reach Each Other

Bonfires on a simple map share two directions. In an orthogonal high-dimensional space, a third fire can lie beyond both.

Imagine an endless night sky scattered with bonfires. Each one burns, but its light never reaches another. I call this kind of high-dimensional representation space bonfire space.

A shared map in low dimensions

Start with three bonfires on a flat, two-dimensional map. Measure their positions from the same origin: A is one step east, B is one step north, and C is one step east and one step north. In coordinates, A = (1, 0), B = (0, 1), and C = (1, 1), so C = A + B.

Two independent directions cover this whole map. Once a third fire’s coordinates are given, you can describe how to reach it using those same directions. Knowing A and B alone does not tell you where an unknown C is; they give you a shared language for locating it.

Figure 1: Left: three bonfires on a 2D map, where C = A + B. Right: in the orthogonal high-dimensional model, C occupies a direction outside the plane spanned by A and B.

Independent directions in bonfire space

A bonfire’s “location” is a direction in representation space, rather than a place you can mark with three ordinary spatial coordinates. A high-dimensional space has room for many independent directions. Each bonfire shines along its own.

Figure 2: Like isolated stars: knowing A and B gives you no star map to C.

Here, “far apart” means orthogonal: one bonfire’s direction has no projection onto another’s. They share no direction that could serve as a guide from one to the other.

Try to read one through the other by taking their inner product, and the result is zero. This is the “nothingness” of bonfire space: no shared signal. Even combining the two directions leaves you within the space they span. You cannot reach a third bonfire whose direction is orthogonal to both.

Figure 3: Left: orthogonal directions have an inner product of zero. Right: combinations of A and B stay in their plane and cannot reach C. Only three directions are shown for illustration.

The difference is that two independent directions span the entire 2D map, while a third bonfire in this model lies beyond the first two directions. High dimensionality makes room for many such directions; mutual orthogonality is the assumption that keeps the fires separate.

Every bonfire burns. None gives directions to the others.