Startup Geometry Part One

I promised some more thoughts around my favourite shape in the world. I am super appreciative of the responses obtained from Barbara Tellalian and others.

Today's two thoughts are also dedicated to 12 special individuals, along with my father:

• Erik Modahl
• Jason Kraus
• Shweta Agrawal
• Mark Zheng
• Hao He, CFA, MBA
• Parna Sarkar-Basu
• Christopher R. Dube
• Coach AK Ikwuakor
• Daniel Hulter
• Roxana Grunenwald
• David Zhou
• Greg Isenberg.

Our Startup Spheres Are More Oblongated Than Exact...

In classical mechanics, orbits are rarely perfect circles; they are ellipses, stretched by gravitational influences and the momentum of the moving body. Likewise, in startups, the idea of a predictable trajectory—one where you follow a set plan, hit expected milestones, and land smoothly at success—is a comforting illusion. In reality, founders experience something closer to an oblate spheroid, stretched and compressed by external forces beyond their control.

These distortions arise from three fundamental forces in the startup universe:

• Luck – The unpredictable gravitational slingshot that can propel a company forward or send it spiraling off course. A random investor introduction, an unexpected viral moment, or even an economic downturn can stretch or shrink a startup’s orbit in ways no one could have foreseen.
• Timing – Like planetary alignment, market readiness determines whether an idea takes off or crashes. Founders often believe their efforts alone dictate success, but much like a spacecraft waiting for the right launch window, external timing is crucial. The best founders recognize when their market is contracting or expanding and adjust their trajectory accordingly.
• Execution – This is the force that stabilizes the otherwise chaotic nature of startup life. While luck and timing distort the shape of the journey, execution—persistence, strategic adaptability, and resilience—determines whether the founder stays in motion or burns up on reentry.

From a psychological perspective, humans naturally seek predictability. The brain relies on cognitive biases like pattern recognition and linear extrapolation, which is why so many founders assume that early traction will continue in a straight line.

However, research in behavioral economics shows that success often follows an S-curve, where initial growth is slow, then explosive, and eventually stabilizes. The inability to recognize this natural pattern is why many founders overestimate early success and underestimate the long, slow struggle of scaling.

The successful founder, then, is not the one who assumes a perfect sphere but the one who recognizes the oblong nature of their trajectory and adjusts accordingly. They build flexible plans, expect deviation, and accept that even their most carefully plotted course may be stretched or compressed by forces outside their control.

The Best Founders Are Dodecahedrons

A dodecahedron is a polyhedron with 12 flat faces, 20 vertices, and 30 edges. It is one of the five Platonic solids, which means it maintains perfect internal symmetry while still being complex and multi-dimensional. This makes it an ideal metaphor for a great founder—someone who is not just a single-faced specialist but a multi-faceted operator who can engage with different challenges from different angles.

Why 12 Faces?

In the startup world, a founder must embody multiple roles to survive and thrive. Each face of the dodecahedron represents a distinct but interdependent competency:

• Visionary – Seeing the future before it exists.
• Operator – Turning ideas into structured execution.
• Salesperson – Convincing customers, investors, and talent.
• Strategist – Navigating competition and positioning the company.
• Technologist – Understanding the underlying mechanics of their product.
• Fundraiser – Persuading investors while maintaining leverage.
• Negotiator – Managing partnerships, deals, and contracts.
• Recruiter – Attracting top talent in an increasingly competitive world.
• Storyteller – Crafting the narrative that makes the company more than just a business.
• Crisis Manager – Making high-stakes decisions under pressure.
• Learner – Continuously evolving to stay ahead.
• Resilient Human – Withstanding failure, rejection, and personal sacrifice.

A founder who excels in only one or two of these areas is like a cube—functional but predictable, easy to tip over in a volatile environment. A sphere might be smooth and polished, but it lacks the defined edges necessary to gain traction. The dodecahedron, by contrast, maintains internal stability while offering multiple contact points with the world.

Mathematical Properties of a Dodecahedron and Their Relevance to Founders

Let's examine a few key facets of this remarkable shape...

Structural Integrity – Unlike simpler geometric shapes, the dodecahedron distributes stress evenly across its 12 faces. This is analogous to how a strong founder distributes their skills, preventing any single weakness from collapsing the whole structure.

Non-Uniform Contact with the Environment – If you roll a dodecahedron on a surface, it lands on different faces unpredictably. This is how founders experience the startup journey—constantly shifting between roles depending on what the moment demands, and not becoming "stuck" to one preferential posture.

Golden Ratio Connections – The dodecahedron is deeply connected to the golden ratio, a mathematical constant found in nature, art, and complex systems. Some studies suggest that humans subconsciously find structures based on the golden ratio aesthetically pleasing and functionally effective. Similarly, great founders create companies that feel intuitively "right"—products that fit market needs seamlessly, teams that function harmoniously, and cultures that attract the best talent.

Psychological Perspective: Why Founders Must Be Multi-Faceted

The most successful founders operate in a state of cognitive flexibility, as Chris Dube once told me: "effortless action and a mind disciplined enough to allow itself to be open governs when positive outcomes, and special moments in a startup's journey occur with great profundity" utilizing the ability to shift thinking between different concepts and to consider multiple perspectives simultaneously.

Neuroscience shows that individuals with high cognitive flexibility are better at problem-solving, more creative, and more resilient in the face of uncertainty—all traits necessary for startup success.

Rigid thinkers (who embody only one or two faces of the dodecahedron) often struggle when their initial strategy fails because they lack the adaptability to shift perspectives. In contrast, multi-faceted founders pivot with purpose, knowing that their success depends on the ability to reframe problems rather than being stuck in one-dimensional thinking.

The Founder’s Geometry

Let's conclude where we began -- by making the argument for this way of thinking, having noticed other nuances in the innovation universe around us...

• The naive founder believes they are a sphere, rolling smoothly along a predefined track.
• The rigid founder behaves like a cube—stable but inflexible, unable to adapt when things go off-plan.
• The chaotic founder is a jagged shape, bouncing unpredictably with no real control.
• The best founder is a dodecahedron—balanced, complex, multi-faceted, and capable of engaging with the world in ways that maximize their odds of success.

The key lesson?

Founders who recognize the oblong nature of their path and who develop the multi-faceted depth of a dodecahedron are the ones who not only survive but thrive in the unpredictable universe of startups.