Generating the Greater Reality
of One-Sided Surfaces

I will keep adding extensively to this site. And I am writing a book that explains in simple terms the process of generating and mathematically categorizing one-sided (non-orientable) surfaces. It also shows a surprising beauty that lies at the core of these surfaces.

For now let me share a few thoughts about the Metamobius algorithm. It is a powerful process that generates and categorizes one-sided (non-orientable) surfaces. It also generates a whole world of two sided (orientable) connected surfaces, but the focus here is on one-sidedness.

The greater fundamental reality of one-sidedness (beyond the Mobius strip) has never before been realized because to do so from the perspective of the surfaces themselves, such as boundary condition, is virtually impossible given their complexity and endless variety. The Metamobius algorithm solves this problem with a unique method of first capturing the space or perspective around which the surface is defined. It then generates and topologically categorizes the surfaces. 

I’ve used the Metamobius algorithm to create thousands of amazing one-sided surfaces. When I created the first one I had only a vague idea of what it might look like. When it emerged it was absolutely beautiful to me and it took my breath away. Soon after creating the algorithm, I also created a way within it to interconnect these beautiful surfaces to create fascinating and beautiful landscapes of one-sidedness.

This site shows only a few basic surfaces which you can see in the Gallery, but the site will go through a complete overhaul in the next while with many more surfaces and a much more in-depth explanation of the process. 

Over the last few years I have been fortunate to have made presentations of the Metamobius algorithm in Waterloo, Ontario where I live - a few to a local high school, and over the last few years, four major presentations to international math conferences at the University of Waterloo. This is thanks to a friend who championed the algorithm from the beginning, and to the organizer of the conferences who recognized its importance. 

At the presentations I not only explain the process of creating and categorizing Metamobius surfaces, called metamobs, I also present very cool models of some of the simpler surfaces. Actually I am often asked, “Can metamobs be made in real life”? Yes, of course. Just as the Mobius strip can be made, all metamobs can be made by any process that you would use to create any other kind of surface. The only catch is that the surfaces range from simple to extremely complex, and they get complex very quickly after the simpler ones. So to make most of the metamobs will take some serious thought and creativity. I am hoping to be able to mathematically generate them at some point, and then even the most complex ones - the Metamobius landscapes in particular - will be generated graphically and 3-D printed.

I spent about 30 years trying to create the Metamobius process. When it finally began to take shape I faced the challenge of also creating a naming convention for the surfaces - hence their technical names as indicated in the Gallery. These names reflect the path within the Metamobius algorithm used to generate each surface, and the resulting topological description or category.

They won't convey much meaning here now but that will soon change with the new website. I will also supply a thorough explanation in the upcoming book on the process.

For instance, the surface shown on this page is a Pc T3+S2(alt) m2 = N b2 x-5 n5 t24. 

Yikes!!! That's way too much of a mouthful, but the names contain all the information needed to create and categorize the surfaces. Their names are equations. The left side is the generator. It reflects the space substructure (the sub) around which the surface is derived, and the method applied to the sub to generate the surface. The right side is the descriptor which represents the resulting topological  description or category.

I learned how to derive the right side with the great help of a topology advisor and mentor, and for whose help I will always remain grateful. 

I am now building a data set based on a mathematical approach involving space substructure, minimum triangulation, Euler characteristic, genus, and boundary condition.   

I had originally called each new surface the name of a family member or friend, but with so many surfaces this practice became impossible - I have a small family and not that many friends! 

I hope you enjoy viewing the few that are posted here. They are unique to the world and represent the most important advancement in one-sided surfaces since the discovery of the Mobius strip itself 160 years ago. There are many more to come with increasingly more emphasis on their artistic beauty.

A note on the term Metamobius - In 2010, when I first started creating these surfaces, I hadn't yet named the algorithm. I emailed my niece about my discovery, explained a bit of the process, and included an example surface. Within a short time she had emailed me back suggesting the name Metamobius. I adopted it immediately.   

Ted Gibbons