June 2024 - Topology and Core Value Systems!

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 these branches of geometry?

In this final blog piece in the series correlating mathematics with business processes, I want to explore Topology. What can we infer from this type of geometry?

Most mathematicians associate the emergence of topology as a distinct field, with the 1895 publication of Analysis Situs by the Frenchman Henri Poincaré. There are many others though, who believe that several topological ideas had found their way into mathematics during the previous century and a half.

Topology is the part of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending. This branch of mathematics studies properties of spaces that are invariant under any continuous deformation. It is sometimes called "rubber-sheet geometry" because the objects can be stretched and contracted like rubber, but cannot be broken. As we will see later, it is this fundamental principle defined as extrinsic topological equivalence, which offers a link to the world of business.

When a continuous deformation from one object to another can be performed in a particular ambient space, the two objects are said to be isotopic with respect to that space. For example, consider two circles inside a larger circle with a point inside them. In a two-dimensional ambient space these two smaller circles cannot be continuously deformed into each other to merge the points inside them; it would require cutting the circles open to allow this possibility. However, if three-dimensional space serves as the ambient space, a continuous deformation can be performed, as in a sphere, or a doughnut! Thus, these two are isotopic with respect to three-dimensional space, but they are not isotopic with respect to two-dimensional space.

The notion of objects being isotopic with respect to a larger ambient space provides a definition of extrinsic topological equivalence, in the sense that the space in which the objects are embedded plays a role. The example above motivates some interesting and entertaining extensions. One might imagine a pebble trapped inside a spherical shell. In three-dimensional space the pebble cannot be removed without cutting a hole through the shell, but by adding an abstract fourth dimension it can be removed without any such surgery.

Now, let’s try and relate these concepts of isotopism, ambient spaces and extrinsic topological equivalence to the world of business.

Businesses operate within an ecosystem. An ecosystem is the ambient space. The boundaries of the ecosystem are defined by the nature of the business as well as its scope (or extent). As can be inferred, higher dimensions of the ecosystem will happen when the nature of the business expands or its scope widens to include new customer segments or supply chain partners.

Some core and intrinsic value systems govern the way businesses aim to deliver whatever their customers want. In a phase where the operations are stable around the current ecosystem (akin to the lower 2D in mathematics), these businesses face a dilemma about breaching some of core values (even as other core values remain intact) if they want to access some attractive opportunity. Such combinations of core values are said to be non-isotopic in the existing ecosystem. An example of non-isotopic core value dilemma may be environmental-protection and job/skills-creation for a coal mining company, which may result in a stumbling block for business growth and expansion.

However, if the ecosystem upgrades to a higher order where the technology supply chain enables improved environmental performance and the skills needed in such an ecosystem can facilitate additional jobs, then in this higher order of ecosystem, the same combination of core values can become isotopic.

Extrinsic topological equivalence, as a corollary, is all about businesses embedding themselves continuously into higher orders of ecosystems (continuous improvement), so that all their core values continue to remain un-breached, even as they pursue growth.