Blowing up
Sunday, August 10th, 2025
Blowing up is a concept from algebraic geometry. Blowing up turns a neighbourhood in a Euclidean space into a more complex manifold (called the ‘blow up’ of the original neighbourhood). The transformation preserves the Euclidean geometry in most parts of the neighbourhood, but expands a particular subspace to have extra dimensions. This can help to resolve singularities, with applications in singular learning theory.
The simplest non-trivial example is blowing up a point in a 2-dimensional neighbourhood (say, a disk). Unfortunately, even this example is pretty hard to visualise—at least for someone like me who doesn’t think natively in terms of projective geometry and quotients.
Most attempts to depict this blow up focus on showing a single part of the resulting 2-dimensional manifold at a time. However, it’s actually possible to represent the entirety of the manifold in three dimensions. In fact, it has the topology of the well-known Möbius strip.
Below is a Three.js visualisation of the operation, and a brief discussion of some of its properties.
Visualisation
We start with a two-dimensional disk. We want to blow up the central point into a one-dimensional space representing all of the different lines along which we could approach that point. We want to somehow do this without disturbing the topology of the rest of the disk.
How is this possible? Drag the slider to see:
Click and drag on the manifold to rotate it:
Observe how the resulting manifold looks like a Möbius strip. The central point is replaced by a circle (black line). A Möbius strip is the perfect manifold to preserve the topology of the rest of the disk, because it still allows us to walk ‘around’ the ‘centre’ by completing two revolutions around the strip. The coloured boundary in the visualisation represents one such circuit.
Exercises
I think this visualisation could be improved in several ways.
The main focus of this visualisation is the start point (the disk) and the end point (the blow up). The transition between them is imperfect, with some creasing of the manifold and some parts of the space passing through others. I don’t think it’s possible to continuously deform the original neighbourhood into the blown up manifold without these kinds of artefacts, because the two surfaces have different numbers of holes. However, it may be possible to interpolate between the two manifolds in a way that is more visually appealing.
It would be cool if there were a version of this construction that showed the original neighbourhood and the blow up side by side (spinning, say), and made it so that if you highlight a point (or line) on one of the surfaces, it shows the corresponding point (or line) on the other. For points, this would more clearly show that the white sections are homeomorphic. For lines, this would more clearly show the motivation for the blow up, which is to expand the central point into a full projective space (here a circle) to allow tracking different lines of approach as different points.
The blow up is a step that is used during the resolution of singularities. If there were a curve in the neighbourhood with a singularity at the point, then blowing up the point would normally lead to a curve with a less severe singularity in the blown up space. It would be cool to extend this visualisation to contain an example curve with a singularity, before and after the blow up.
This blow up can be defined in several steps. First, take the product of the neighbourhood of the plane (a disc) and one-dimensional projective space (a circle). This product can be visualised as a solid donut. Next, in each copy of the disc along projective space (each slice of the donut), discard everything except the subset of points along a line through the origin with the same direction as represented by that part of projective space. (This is how I finally realised that the resulting space was the above Möbius strip.) It would be cool to create a visualisation that shows this construction, step by step.