Interface Math
The Subtle Forces Beneath Modern Design and Why They Matter More Than We Realize
by Trudy Hall
I began developing my approach to design when I was a student learning the formal vocabulary of digital composition. I spent hours studying navigation, alignment and the visual logic that holds an interface together. The training helped me understand how information sits on a screen, although it did not prepare me for the deeper shifts that occurred as I moved from one digital environment to another. Some interfaces created a sense of grounded comprehension that allowed thought to develop with ease. Others produced a strain that I struggled to express. That contrast revealed something design education rarely names. Digital systems shape the conditions in which people think long before they notice that anything has happened. In this essay, I draw from research emerging from laboratories and institutes around the world to show how these forces operate beneath everyday experience.
I understand ideas most clearly through visual structure, so I turned to a series of symbols to study what I was sensing in a visual and organized way. My interest in geometry and algebra began long before my interest in design. I loved the way clean forms revealed hidden relationships. That instinct carried into my work. When a system felt disorienting or unexpectedly comforting, I looked for the underlying logic that produced the effect. Over time, these observations formed the design framework I developed after college — a method for examining how digital ecosystems shape the internal ground on which meaning forms.
The first pattern involved containment, or the degree to which an interface provides stable orientation. Cognitive scientists at Princeton describe how structured surroundings reduce the amount of information the mind must actively process. A clear, consistent information architecture allows a user to form an internal map of the system. A navigation bar that stays visible during a long scroll preserves orientation, giving the user a reference point as the page unfolds. When containment weakens, the burden shifts to the user. Many retail sites still force shoppers to reconstruct their entire path just to make a minor filter change. In those moments, the system withdraws support at the exact moment a person needs it most.
The second pattern centered on agency, or the user’s sense of authorship over their own movement. Research from the MIT Media Lab shows that people feel most capable when the interface reflects intention rather than redirecting it, yet digital systems often complicate that feeling. A streaming platform may advertise an abundance of choice while quietly shaping the outcome through placement, timing and visual weight. The countdown to the next episode creates a strong default even when the user has not consciously agreed to continue. The structure presents possibility while minimizing the moment in which a different action could occur. Agency measures whether a system supports genuine choice rather than a simulation of it.
A third pattern emerged through rhythm, the cadence at which information is delivered and processed. Researchers at Stanford’s Persuasive Technology Lab describe comprehension as a sequence that depends on periods of intake followed by periods of integration. Many interfaces ignore this cognitive rhythm. Onboarding flows bombard new users with instructional pop-ups before the core function is even understood. News feeds refresh themselves in the middle of a paragraph. A user must reorient through no fault of their own. These interruptions fracture the continuity needed for understanding. Rhythm evaluates how well an interface aligns with the pace at which comprehension forms rather than the pace at which content can be pushed.
A fourth pattern involved divergence, or a system’s ability to support nuance. Many interfaces condense complexity for efficiency. A wellness application that collapses varied physiological signals into a single score offers quick communication at the cost of precision. Research from Columbia and the University of Edinburgh shows that layered information structures lead to deeper understanding. Interaction designers refer to this as progressive disclosure. When systems allow context to unfold gradually, users build more complete internal models. Divergence measures interpretive elasticity — the balance between overwhelming users with uncontrolled choice and flattening experience through excessive simplification.
A final pattern, honesty, became central because digital systems communicate constantly. Every notification, delay or confirmation shapes a user’s sense of what a platform is doing. Research from the Oxford Internet Institute shows that trust depends on feedback that is proportional and accurate. A system that reflects real events builds confidence. A system that simulates urgency to increase engagement reshapes perception in ways that leave a user uncertain about what is real. Interfaces that refrain from coercive banners or constant cookie prompts often feel more trustworthy for this reason. They echo the experience of walking into a store without a hard-sell clerk hovering nearby, allowing the user to explore without being steered toward choices they did not initiate. Honesty evaluates whether a system conveys truthful information or pressure disguised as feedback.
As I studied platforms across different domains, I began to see a structure that resembled the logic of a spider’s web, particularly on social platforms where engagement depends on sustained capture. These systems draw users in through signals designed to attract focus, then hold them through infinite scroll, which removes the natural stopping points that help people regulate their own movement. Interaction loops consume time through gestures that feel intentional while offering little closure. Notifications, fresh content indicators, and algorithms trigger the initial pull. The feed itself produces the paralysis — not through force, but through the absence of boundary. Hours pass inside small interactions that appear chosen but are shaped by the environment’s architecture. This model clarifies how digital systems subtly influence user action.
The framework I developed eventually became a method for evaluating not only usability but also the internal climate produced by a digital system. An interface with weak containment forces users to stabilize what the structure fails to hold. An interface that undermines agency pushes people toward predetermined outcomes. An interface with disrupted rhythm accelerates past the point where comprehension can take root. An interface with limited divergence reduces complex thought. An interface with compromised honesty reshapes perception. These forces influence not only how people move through digital systems but also how they make sense of them.
As the work matured, I began describing the relationship among these patterns as a conceptual equation — not mathematical, but structural. Each force reinforces or erodes the others. Containment stabilizes the ground on which understanding forms. Agency supports the sense of deliberate movement. Rhythm aligns interaction with the pace of human comprehension. Divergence creates space for nuance to emerge. Honesty determines whether the system reflects reality instead of manipulating its interpretation. When these forces remain in balance, an interface strengthens a person’s ability to think, interpret, and respond. When they fall out of balance, the system may still function, but often at the expense of the user’s clarity.
What began as doodles in a datebook became my symbolic method for understanding the lived experience of digital life. The work taught me that interfaces do more than display information. They influence the internal terrain through which people make sense of the world. My system offers new vocabulary and a blueprint for bringing those forces into view. When design aligns with cognition and truth, digital systems can support the very capacities that brought them into being. Even small adjustments can restore balance, and even subtle changes in structure can reveal how experience is made.
