Meditations On Geometry is a collection of research articles and related material whose central theme is the study of Mathematics and Geometry.

The current issue - Volume 01 - is a series of articles investigating the parameter space of the Unit Quaternion. Through the lens of the Quaternion it is possible to understand the multidimensional nature of reality, as the 3 spatial dimensions that we know very well are in fact rooted in 4 spatial dimensions. 

These articles are available for anyone looking to satisfy their curiosity about this most intriguing geometric object. For anyone familiar with the mathematical algebra of the Quaternion currently under development, and interested in pursuing their own research in this area - or even contributing to this body of work - I would like to hear from you.


Brian O'Sullivan

Having completed my PhD in December 2012, after what felt like a long 6 year struggle with the Quantum Theory, I was able to recognise that the mathematics utilized by the theory lacks a concrete geometric foundation. During that period I had studied the geometric models which were popular at the time which I found entirely unsatisfactory. Thereafter, I pursued my own research interests in an effort to uncover a self consistent mathematical algebra which has the potential to answer some of the many questions raised by the Quantum Theory.

These efforts have lead to me to study both the Unit Quaternion and Differential Geometry, and ultimately to create this website to share my research and related ideas with any and all people interested in the pursuit of a complete Physical theory. You will find links to all my current research articles (from 2014 - present) on this website. Links to my pre-2014 articles are below.


" ... arithmetic, geometry and other subjects of this kind, which deal only with the simplest and most general things, regardless of whether they exist in nature or not, contain something certain and indubitable. For whether I am awake or asleep, two and three added together are five and a square has no more than four sides."
Descartes - Meditations On First Philosophy (1641)

Geometry and arithmetic are disciplines of thought that form a basic element of reality. Music is to our heart's ear, as geometry is to our mind's eye. In this realm we can listen to and commune with nature.

I have always enjoyed the quiet solace found in the study of these thought forms. There is beauty in their elegance, and there is a simple pleasure that can be derived from these disciplines as they are both a science and an art.

  • The science involves asking a question and then solving it. The solution is either right or wrong, and this is the elegant beauty of science. Solving a problem is a journey, and until the destination is reached the process continues.
  • The art is more subtle - it is the How. The art is found in both the posing of the question, and the presentation of the solution. This is the yin and the yang. One cannot exist without the other and their aesthetic forms are interdependent. 

In my experience the process always follows a familiar path. Whether the question asked is 'What is 1056 divided by 52?' or 'What is the geometric phase of the qubit?', my initial position is typically utter confusion and bewilderment. Once the dust settles and research begins the problem soon unravels. What is useful is gathered and what is not is discarded. Perspective then develops until the point is reached when the solution is absolutely crystal clear, and I am shocked that it took me so long to find it. It is so obvious, how did I miss it!?

The title of the website ''Meditations On Geometry'' is chosen to reflect that which I do. I meditate on geometry. There is no goal other than playing with, visualizing and surfing the geometry, from inception to conclusion. The meditation begins with posing a question, meditating on the meaning of that question and listening to the geometry while the process unfolds.

"... for geometry, you know, is the gate of science, and the gate is so low and small that one can only enter it as a little child."

William K. Clifford


Research Articles 2009-2013 (click the title, or arXiv link to download the pdf)


Coherent transport by adiabatic passage on atom chips. Physical Review A 88 053618 (2013). arXiv:1309.2451

Tadhg Morgan, Lee O'Riordan, Neil Crowley, Brian O'Sullivan, Thomas Busch.