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Original post can be found here: https://benzweibelson.medium.com/using-pac-man-to-introduce-m%C3%B6bius-strips-and-klein-bottles-to-reconceptualize-complex-8cd71cdcfee4

The third part of this 5 part series is linked below, and readers ought to check that out first before venturing further, as I set up why Pac-Man is the metaphoric device here for breaking out of static, simplistic, Newtonian styled models and constructs for conceptualizing warfare.

https://benzweibelson.medium.com/prying-newtons-grip-on-military-thinking-by-using-pac-man-a-crowd-sourcing-exercise-part-3-of-5-9979c5f53b3

In the last part of this series, I introduced a Möbius strip and challenged readers to try their own paper strip challenge. The Möbius strip is not like the dominant triangles, triads, cubes, spheres, loops, linear-causal campaign design and other orientable topology that exists across all military doctrine.

In the above figure, the topological instructions for creating a Möbius strip are adapted into a conceptual model for how military organizations, as the directed extension of their national political, social and cultural desires, engage in a wide range of security activities to complete, cooperate, deter, and also engage in organized violence perpetually, iteratively and in an emergent, non-linear fashion. Complexity theory requires careful consideration of nonlinearity, systemic relationships (non-reduceable), and how emergence largely prevents such dynamics from being depicted in Newtonian stylized approaches. Even below and in subsequent illustrations, forcing a Möbius strip into two-dimensional space carries the risk that some readers might misinterpret the concept, oversimplifying it into a cylindrical shape where an ‘OODA loop’, campaign planning formulaic, ‘ends-ways-means’ linear-causal relationship could be inferred. Militaries need to shift away from Donkey Kong concepts to topological alternatives, even though printed mediums still insist upon a two-dimensional rendered construct. The meaning of those constructs and the theoretical underpinnings is how one can pivot from one conceptual frame to another less explored.

In the above figure, I mention that one strange ability of the Möbius strip is that it breaks the four-color theorem that all other objects depicted in our military doctrine (see parts 1 and 2 of this series) obey. Triads, cubes, loops and more all must follow the four-color theorem, but the Möbius does not! Think about the color rule, and why this shape could break a rule that all the other graphics in every single doctrinal manual you have must obey… and why might that matter? If we are really trying to talk about complex, dynamic, multi-domain security challenges, why are we forever limiting our conceptualization of these difficult topics with oversimplified, Newtonian-styled models exclusively?

The above few figures introduce the Möbius strip to act as a bridging device to introduce non-orientable topological objects as potential metaphoric devices and new conceptual models for militaries desiring to break out of this ‘Newtonian fetish’ as this series intentionally provokes readers with. In order to prevent strategists and planners from assimilating non-orientable, peculiar topological concepts back into a linear, systematic (reductionist, A plus B leads to C in preconfigured input/output relationships) mindset, additional explanation and illustration is required. The Möbius strip expands in the next figure below, gaining additional graphical depictions that attempt to pull viewers further into topological constructs that reject any over-simplification of complex warfare through exclusively Newtonian geometric rationalizations.

The boundary of the Möbius strip in topological terms is equal to a circle, despite the strange shape and twisting. As the Möbius edge is unknotted, the entire strip can be stretched without crossing itself. Mathematically, the simplest knot possible is what is called the ‘unknot’ or ‘trivial knot’ which is a topological circle. This is represented above both on the left where the dotted lines are in the topological instructions to create the Möbius strip and is further illustrated in the two separate frames below of ‘air-land-sea’ and that of ‘cyber, space & SOF.’ Traveling through a Möbius object, one cannot leap off an edge, as they carry right over to the other despite being drawn in the topographical instruction to look like opposing ends of a square. Returning once more to Pac-Man, one traverses immediately from one side of the screen into the other side, despite moving ‘away from the game board.’ This is how non-orientable topology offers new, arguably complex ways to explore, define and explain complex warfare beyond Newtonian limits. How the Möbius strip forms a single topological surface yet exercises movement of the traveler on ‘both sides’ provides the framing device to consider the physical domains (air, land, & sea) for security affairs and how conflict, competition, and deterrence exercise in ‘abstract, indirect, or peculiar domains for security affairs.

Here and in the next few slides, different inputs and outputs enter consideration depending on what part of the Möbius strip is being travelled, as well as how the traveler has experienced previous passages where a collection of different inputs and outputs have acted systemically (holistically, non-reducible, framed with increasing abstraction toward larger and larger system relationships). What is interesting about the below illustration rendered unavoidably in two dimensions is that one can opt to travel in a variety of paths just as any journey in an actual Möbius object would feel like. Applied to modern complex warfare and dynamic security affairs, one could cycle through iterations of just one or either side, or mix activities traveling the entire pathway in myriad, non-linear cycles. This may approximate modern complex security affairs in that some phases of international competition, cooperation, deterrence and acts of organized violence across state, non-state, commercial, group, and decentralized movements may exercise exclusively in just part of the Möbius below. A conflict featuring covert or clandestine special operations with significant cyberspace and space operations may avoid any traditional patterns (physical domains, declarations, and clear acts of war) and in some situations might transpire without any external awareness of the societies being acted upon.

Note: In laymen terms, ‘clandestine’ means that something goes ‘bump’ in the night, but no one is alarmed to get out of bed to investigate. ‘Covert’ means that something goes ‘bump’ in the night, and because it alarms people nearby, they do investigate it… but they cannot figure out who might have done it and there is little evidence to accuse any potential suspects. Special operations forces focus on these different capabilities and effects that require significant specialized recruitment, training, education, resources and authorities. In today’s technologically sophisticated, interconnected world, this new relationship of space, cyberspace, and special operations activities begins to create different configurations that were not possible in earlier defense contexts. Nor can traditional, terrestrial/tangible domain constructs (air-land-sea) alone provide the necessary depth and interconnectivity to consider what is now possible (and likely ongoing) in the most difficult security challenges. The graphic below uses a non-orientable topological model to convey this difference, and suggest one of many ways to reconceptualize how we think and arrange activities in the emergent, dynamic reality we now live in.

In the configuration above, the Möbius strip is depicted in one of many possible alignments. Arguably, there is not going to be a ‘best’ configuration, as the nature of complex warfare prevents any sort of objectivity, stability or predictability therein. Instead, readers might consider the topological opportunities for reconceptualizing the suggested Möbius strip above with something else. The above Möbius strip addresses core tensions between the well-established, historically recognizable arena for Westphalian nation-state warfare and politics which has exercised through air, land, and sea for centuries, spanning peaceful and cooperative/competitive contexts to that of total war efforts of annihilation.

The other ‘side’ of the Möbius strip above addresses the emerging, rather abstract, and peculiar domains of cyberspace, space and how modern special operations forces are able to operate in exquisite, unique, indirect and alternative ways both in times of apparent peace and that of active, recognized warfare. For instance, special operations forces work in unconventional warfare (UW) applications that may span years or decades of slow, nonlinear, often invisible, or incredibly gradual efforts that are emergent and hardly the sort of operations that make the front pages of the news. Indeed, perhaps the best UW operations are never discussed due to the nature of their obfuscated, invisible transformation succeeding. Yet a highly successful UW campaign might lead to significant long-term security goals, and even accomplish them in a way that is non-attributable or obfuscated from societies realizing who did what to whom when and why. The shadowy, complex, and tangled worlds of cyberspace as well as the unique aspects of an emerging space domain for security affairs are even more difficult to comprehend, much less articulate clearly in crisp, two-dimensional graphics and models for militaries and policy makers to feel certain of.

Bringing back Pac-Man, and also reaching back into Part 3 of this series where different colored Pac-Men rotated around the Möbius strip, you can move through this flat, 2D rendering of a non-orientable topological construct to consider the multi-domain, complex security challenges regarding air, land, sea, as well as cyber, space and SOF in a different relationship. It is unavoidable to not configure this flat, as it must be printed on a page, or appear now on your tablet, laptop or smart phone. Yet, if you have been studying this series, you see the difference in your mind on what these constructs represent in motion, becoming, in dynamic and emergent ways. No triangles, cubes, or spheres do this!

The Möbius strip had as one of its unique properties an unknotted edge where the boundary is equal to a circle, stretching without crossing itself. However, if two Möbius strips are glued together edge-to-edge, a Klein bottle is formed that possesses a one-sided surface with no boundary that cannot be embedded in three-dimensional space. A Klein bottle is tricky to conceptualize, and while they can be created in three-dimensional space including boutique wine decanters for mathematically minded wine lovers, topological mathematician Maia Averett offers a convenient summary:

The only way to imagine [a Klein bottle] is to imagine pulling one end of the cylinder through the surface of the cylinder and matching up with our circle from the inside. The resulting representation of the surface doesn’t look like a surface, but it really is. Its funny appearance is just a consequence of the way we had to realize it in our three-dimensional world.

Mathematically, Klein bottles are a paradox when rendered in three-dimensional space, as they are not really contained in space as they are paradoxically containing themselves; a topologically imperfect model created in three-dimensional space has a hole produced “so that its construction already introduces singularities which then through the in-formation flow produces the whole structure, so that the whole structure is produced from a hole, and this returns to the singularity to complete the flow.” Many readers no doubt are scratching their head at this, and for applications to complex warfare, this is where a distinction must be made between analogies and metaphors. Complex warfare is not analogous to how a Klein bottle exists mathematically, just as that same complex warfare is not actually the integrated cube as depicted in Joint Planning Publication 5–0. Metaphorically, there are patterns and behaviors within complex warfare and security affairs that can be conceptualized using either an integrated cube or with the Klein bottle, and it is up to the organization seeking greater understanding to determine which metaphor might be more useful.

In the above figure, topology works with what are called ‘real projective planes’ that are non-orientable two-dimensional manifolds, so that mathematicians can generate relationships and instructions to build a host of topological objects, many of which simply cannot be illustrated in two or even three-dimensional spaces without certain compromises of the illustration (but not the mathematical formulation). These concepts are centuries old, and one can find the societal transformation in thinking about reality to one of topological consideration when viewing paintings from Renaissance artists in a museum. While Medieval and older paintings seem flat and strangely ‘wrong’ in composition of perspective, Renaissance artists were among the first to compose artwork using real projective planes to create in two-dimensional artwork the ‘illusion’ of three-dimensional space and objects. Hence, the painters broke out of older, less useful ways to conceptualize on how to create more realistic works of art, and by playing with topological concepts, they could develop entirely new ways to create two-dimensional art. For military professionals considering the Möbius strip, it is the simplest non-orientable three-dimensional object that can be depicted in two-dimensional space without losing much of its unique qualities. The Klein bottle should be considered the next level of conceptualization using this technique, except the Klein bottle can somewhat be depicted in three-dimensional space while still maintaining most of the unique non-orientable qualities that make it quite unlike a regular glass pitcher. An immediate security example of Klein bottle-like behavior is found in the U.S. State Department’s ‘Moscow Mechanism Report’ press statement released on September 22, 2022:

The United States and 37 other countries invoked the Organization for Security and Cooperation in Europe’s (OSCE) Moscow Mechanism on July 28 to examine the Russia’s adherence to its OSCE Human Dimension commitments on human rights and fundamental freedoms… Specifically, this report documents that the Kremlin has centralized all federal and regional law enforcement authorities under Kremlin control; used so-called “foreign agent” laws to impose draconian penalties and fines on individuals and civil society organizations with any foreign contacts; effectively silenced freedom of expression, including independent media and criticism of the government through harsh censorship laws; and “created a climate of fear and intimidation… that is not in line with OSCE standards based on a pluralism and a strong and independent civil society.” The report also makes clear that Russia’s “(r)epression on the inside and war on the outside are connected to each other as if in a communicating tube.”

I am going to hold off on how the Klein bottle can really open things up on reconceptualizing multi domain, complex warfare and security challenges until the next and final part of this series. There, we will use the earlier graphics in this Medium post where ‘air-land-sea’ interacts with ‘space-SOF-cyber’ across a Möbius strip, but then we will introduce another layer of depth by engaging with activities below the threshold of war (deterrence, diplomacy, and other activities) as well as declared or undeclared hostilities that move up through total state warfare and existential crises.

Okay, here is a peak. Stay tuned for next week’s exciting conclusion: