Unexplored Alleys in Non-Euclidean Geometries: New Discoveries in Tiling Patterns

While the realms of mathematics, particularly in areas such as algebra, geometry, and arithmetic, seem to have been extensively explored, there are still corners that remain uncharted. In this article, we delve into the fascinating world of non-Euclidean geometries, specifically hyperbolic and spherical geometries, where new tiling patterns and functions are being discovered. These findings represent a significant breakthrough in the geometric understanding of complex shapes and their relationships.

Understanding Euclidean vs. Non-Euclidean Geometries

Euclidean geometry is the familiar geometry of flat surfaces, where the angles of a regular polygon are directly related to the number of its sides. For instance, equilateral triangles have an angle of 60 degrees, squares have 90 degrees, and so on.

In non-Euclidean geometries, such as spherical and hyperbolic geometries, the relationships between angles and sides are more complex. This is due to the curvature of the space. For example, consider a right-angled pentagon in a hyperbolic space. All such pentagons, despite varying in shape, share the same edge length, which is a counterintuitive result (see illustration below).

Edge Functions and Their Challenges

Building upon the concept of edge functions, which determine the edge lengths necessary for certain polygon groups to fit around a vertex, we can explore the intriguing question of whether these edge functions yield the same result for multiple sets of polygons. This is a complex problem that requires detailed numerical computation in most cases.

For instance, consider the following tiling pattern:

35618 3669. This notation represents the arrangement of polygons around a vertex, where 3 stands for an equilateral triangle, 5 for a pentagon, 6 for a hexagon, and 18 for an 18-gon. Similarly, 45512 represents a different arrangement. The fact that both quartets can fit around a vertex at the same edge length is a remarkable discovery.

Navigating the Unexplored Ground in Hyperbolic Tiling

The examples above are part of the uncharted territory in hyperbolic tiling, an area rife with potential for new discoveries. Hyperbolic tilings, where surfaces are curved in a way that makes the sum of angles on a polygon less than 360 degrees, offer a vast universe for exploration.

Steps in Exploring New Geometric Concepts

Inventing something new, especially in mathematics, begins at an undergraduate or graduate level. It requires a deep understanding of existing theories and a willingness to hypothesize and generate new conjectures. The critical step is research and verification; to ensure that your work is novel and not a repeat of previous findings.

A detailed process might include:

Learning the foundational knowledge of the area you wish to explore. Generating hypotheses and conjectures based on your knowledge and observations. Validating these conjectures by checking against existing literature and mathematical proofs. Exploring further through detailed research and potentially with the help of an advisor. Continuing the process until you have a comprehensive understanding or your advisor validates the merit of your research.

Conclusion

While the world of mathematics may seem complete, there are always new avenues to explore, particularly in obscure fields like hyperbolic and spherical geometries. The discovery of new tiling patterns and functions in these geometries represents a significant leap in our understanding of geometric relationships and opens up new possibilities for mathematical exploration.

As we continue to venture into these unexplored alleys, we not only enrich our understanding of mathematics but also pave the way for new innovations and applications.