Exploring Negative Dimensions in Geometry: A Novel Concept in Spatial Quantities

In the realm of geometry, dimensions have traditionally been considered non-negative integers. However, recent explorations delve into an intriguing concept: the idea of negative dimensions. This article delves into a geometric interpretation of negative dimensions, expanding our understanding of spatial quantities.

Understanding Dimensions

Different dimensions have distinct Euclidean spaces and associated shapes. For instance, a two-dimensional (2D) space is measured in square units (m2), a one-dimensional (1D) space in linear units (m), and a zero-dimensional (0D) space in point units (m0). Each of these dimensions corresponds to a specific type of shape: filled regions for 2D, lines for 1D, and a set of discrete points for 0D.

Negative Dimensions: An Abstract Concept

When we consider dimensions in spatial quantities, we can introduce the notion of negative dimensions. For a 0D entity, we measure in m0. For a 1D entity, we measure in m. What, then, would a -1D entity measure in? This can be understood as a density measure.

Interpreting Negative Dimensions

Shapes with negative dimensions can be seen as densities, such as the number of leaves per meter along a vine or the number of posts per meter along a fence. In both cases, the measure is of an unbounded set. Counting individual elements would fall under m0. Similarly, a -2D entity measures the number of flowers per square meter in a field or bumps per square meter on a wall. This is an unbounded set of points in the plane.

Intermediate Dimensions and Fractals

In addition to negative and positive dimensions, intermediate dimensions can be explored. Positive dimensions involve fractals, like the Koch curve, while negative dimensions involve unbounded point-wise equivalents. These intermediate dimensions blur the traditional boundaries between integer dimensions.

Irregular Negative Dimensional Shapes

The idea of negative dimensional shapes is not limited to regular patterns. Irregular shapes can also be considered, as long as they converge to a single density at large scales. This flexibility allows for a richer exploration of spatial relationships and properties.

Historical Context and Generalizations

Sergei Petlin.each studied a form of negative dimensions in his book Exploring Scale Symmetry, Chapter 9. He defined negative dimensions in terms of the intersection of shapes, such as the intersection of two lines in a three-dimensional space having a lower dimension. This led to defining intersections with lower-dimensional entities as dimensions with negative values. For example, the intersection of two lines in 3D space might have dimension -1, while the intersection of a line and a point might have dimension -2, and so forth.

Mandelbrot's Contributions

Beno?t Mandelbrot, a pioneer in fractal geometry, also discussed negative dimensions in the context of intersections. He noted that the intersection of two lines in 3D space had a lower dimension than the intersection of a line and a plane. Since the latter had dimension 0, he defined the intersection of two lines as having dimension -1. Similarly, the intersection of a line and a point had dimension -2, and the intersection of two points had dimension -3. While Mandelbrot's concept was developed for integer dimensions, extending it to non-integer values is a topic of ongoing research.

Conclusion

The concept of negative dimensions, while abstract, opens up new possibilities in geometry and spatial quantity measurement. From density measures to intermediate dimensions, these ideas challenge traditional notions of space and shape. Further exploration in this field could lead to new insights and applications in various scientific and mathematical domains.

Keywords: negative dimensions, geometry, spatial quantities