May 2024 - Projective Geometry and Branding!

Geometry, the branch of mathematics concerned with the shape of individual objects, spatial relationships among various objects, and the properties of surrounding space, is one of the oldest branches of mathematics; it has evolved from the study of flat surfaces (plane geometry) and rigid three-dimensional objects (solid geometry) to analysing the most abstract thoughts and images which might be represented and developed in geometric terms. The main branches of geometry are Euclidean Geometry, Analytic Geometry, Projective Geometry, Differential Geometry, Non-Euclidean Geometries, and Topology.

What can entrepreneurs learn from each of these six branches of geometry?

In my last four blogposts, I had endeavored to explore the lessons that entrepreneurs and business leaders could infer from Euclidean and Analytic Geometry. Some readers have continued to send me interesting feedback and have encouraged me to continue this adventure of correlating mathematics with business processes.

Emboldened by their comments, I want to explore Projective Geometry next. What can we infer from this type of geometry?

In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations.

It has its origins in the early Italian Renaissance, particularly in the architectural drawings of Filippo Brunelleschi and Leon Battista Alberti, who invented the method of perspective drawing.

The French mathematicians Girard Desargues and Blaise Pascal took the first significant steps by examining what properties of figures were preserved (or invariant) under perspective mappings. In general, by ignoring geometric measurements such as distances and angles, projective geometry enables a clearer understanding of some more generic properties of geometric objects.

Projective Geometry deals with the relationships between geometric figures and the images, or mappings, that result from projecting them onto another surface. Common examples of projections are the shadows cast by opaque objects and motion pictures displayed on a screen.

As the images below can will help illustrate, some of the fundamental theorems of Euclidean geometry such as those of similarity, preservation of angles and intersection of parallel lines seem to lose sanctity when projections on to different planes of higher dimensions are made.

                                                 

 

                                                             

However, Desargues clearly showed that while some properties such as distances and angles are not preserved in projections, there will be other properties, such as collinearity (three points on a line in real plane will also be in a line on the projected plane), that will continue to be invariant irrespective of whether our focus is on the reality plane or on the projected plane.

This concept of a dichotomy between Variant and Invariant properties offers the parallels for understanding a fundamental aspect about entrepreneurial practices. This relates to Branding.

As commonly understood, a brand is a name, term, design, symbol, or any other feature that distinguishes one seller's product or service from those of others. Branding is a process of creating a distinct identity for a business in the minds of your target audience and the general population.

                                          

The process of branding is evidently one that involves projecting some fundamental truths about the value propositions that the enterprise offers to consumers of its products and services.

The overarching objective is that the consumer is able, to distinctively perceive and remain satisfied that her/his needs and wants are being met at a value-point that is a “wow”!

And, for success in such an objective, projecting and capturing the mindshare of positive perceptions of the consumer demands a clinical assessment of what “brand-truths” will remain invariant during the branding process and which ones will lose their validity.

Understanding the critical nature of the invariant-propositions and the invalid-propositions and working to managing these during the branding process is akin to what projection-painters and machinists do when converting 2-dimensional drawings into three-dimensional visual masterpieces and precision machine parts.

Understanding the laws governing “real-world” of designing and producing a product or service and knowing how they will work when projected to the “esoteric and fussy world” that exists in the consumers’ mind needs a lot of mathematical jugglery - of the projective geometry kind!