More Efficient Geodesic Tessellations

Introduction

The goal here is a simplified presentation of the Rose Geodesic process which is a new, and efficient, method for geodesic tessellation. The intention here is not to present mathematical proof(s) but rather simply describe the process and present the simple, quite useable, results. To the best of my knowledge we have generally accepted Buckminster Fullers (BF) works as the best approach. I am also unaware of any other successful works along the lines of the Rose Geodesic methods. Here I would like to offer a concept which I have worked on, incrementally, for over 20 years. In essence I have followed the BF process until subdivision of platonic solid faces. There is great simplicity in the BF method which allowed ease of calculation. However, when studying the process and results, I was almost immediately drawn to a concept which potentially yielded a more efficient end result.

Having an idea is a great thing. Often these ideas improve one aspect of a situation while greatly compromising others. The basis for the Rose Geodesic method was conceptually developed in 2001 when I first began studying, and seeking improvements to, BF methods in finer detail. The initial step in refinement was clear but next steps still proved challenging. Many approaches were considered. Over time all final steps, except one, were cast away and the process was solidified. Despite conceptualizing these methods, recreationally, early in the new millennia I took me decades to find the time to devote to verifying, to my own satisfaction, its effectiveness. To my own surprise I have found a process more efficient than I had originally anticipated. Retrospectively, in my mind, the process is simple and elegant. My only regret with the Rose Geodesic method is not following my intuition more quickly and taking decades to share this process.

Rose Geodesic methods are characteristically unique from the BF method. BF subdivides equilateral triangular faces and then projects those points onto the spherical surface. This is a beautiful process due to the simplicity of the mathematical procedure and such a great technological advancement. Unfortunately the BF method results in complexity growing exponentially once exceeding the 3rd frequency subdivision. The advancement using the Rose Geodesic process is prolonging the subdivision of the face of the platonic solid until it is projected on the spherical surface. The key there is doing so in an elegant manner. Interestingly the complexity of the mathematical process isn’t unreasonable and becomes substantially less repetitive as the frequency of the tessellation increases. Ultimately the Rose Geodesic process yields a linear, 1:1, growth in unique components to frequency in subdivision. Additionally the Rose Geodesic process yields a greatly normalized unit length across the entire geodesic. Rose Geodesics are equally simple in general application, if not easier, for the end user. Ultimately I believe the Rose Geodesic method is a new process which is worthy of superseding the BF geodesic in most cases and may have significant impact in architecture, spherical modeling, viral analysis, and countless other applications.

John D. Rose