1. A negative dimensional space as described by V Maslov considers gaps in a solid as negative dimensional, presumably a solid missing a line is -1 dimensional and a solid missing a plane is -2 dimensional. This seems to be incorrect as it is confusing measure with dimension. If anything one might think of the holes as having negative measure (relative to the solid).

2. A less scientific answer is to say something has negative dimension if it appears to fade away as you look more closely. For example getting closer to a cloud, or to a face painted in pointilism, or perhaps on very close inspection a smile disappears and becomes just connected cells. In other words, for something to reduce in intensity as you look with more detail, you need that something to be a concept rather than just a raw set of points.

3. Benoit Mandelbrot had a definition of negative dimensions when you are describing the intersection of different shapes under a small amount of noise. The intersection of a point and a volume is 0 dimensional, a point and a plane is -1 dimensional, a point and a line is -2 dimensional and between a point and a point is -3D. The idea is that as you look more closely the likelihood of intersection within a reducing size ball reduces at different rates for these cases. It is a nice idea which removes a special case in a certain set operator equation, but it assumes noise so is a somewhat specific definition. It also creates some odd cases such as a 1D plane (if you a intersecting it with a plane) and intersecting more than two sets gives unbounded negative dimension. In fact it is a specific type of idea 2, as the dimension is being defined for a concept (the intersection of sets) rather than just sets.

Instead I propose that we can produce negative dimensional geometry, in this case negative dimension curves, by using the fact that square distance of time quantities is negative. Here we assume the set will embed in the standard n+1 Minkowski space-time. The proper length of time-like curves can decrease with increasing detail level if they tend toward light-speed motions:

Reducing dimension time-like curves

This particular family of curves is the 1+1D Minkowski equivalent of a Koch curve in 2D Euclidean space. In fact, we can think of the Koch curve, the Weierstrass function and this curve to be the same type, only differing by whether the second dimension squared is positive, zero or negative respectively. And accordingly the larger angles increase, make no difference and decrease the dimension respectively, from a 1D line.

The same curves and same dimensions also exist on the complex plane, provided that it is with respect to the real-length of the curve, defined as:

The same curves and same dimensions also exist on the complex plane, provided that it is with respect to the real-length of the curve, defined as:

In 2+1 space-time the results represent a point moving around in 2D and are perhaps more visually interesting. Here I follow the replace pattern of the Koch curve in 2D space, but the bend angle doesn't increase the curve length by a fixed percentage (like the Koch curve), instead it is achieved by boosting the motion in order to decrease the proper time by a fixed percentage. As a result the dimension decreases and the measure is of the proper time, meaning we quantify the period of the motion as the time according to the point, in seconds^D where D is the dimension.

This effect is a bit like the internal scattering of light in some medium. Lower dimensions are like there are a greater range of particle sizes, with few that cause large reflections and many that cause tiny course changes. D=1 is like all particle sizes (or scattering angles) are the same.