Dissertation: The Illusion of Polarity-How Ternary Systems Reveal the Constructed Nature of Duality
Introduction
Human perception and scientific convention have long relied on polarity-the division of phenomena into opposites such as positive/negative, left/right, or up/down. Cartesian coordinates, with their four quadrants, are a classic example of this dualistic thinking. However, when we move to systems involving three axes-such as ternary or barycentric coordinate systems-the concept of polarity dissolves, revealing that what we often call "polarity" is a man-made construct, not a fundamental property of reality.
I. The Cartesian System and the Birth of Polarity
The Cartesian coordinate system divides space into four quadrants, each defined by the sign of its x and y values. This creates a clear sense of "positive" and "negative" directions-an intuitive but ultimately arbitrary framework. The axes themselves are chosen by convention, and their "polarity" is a product of how we define them, not an intrinsic feature of the underlying space.
II. Ternary Systems: Three Axes, No True Polarity
A. Structure of the Ternary Plot
A ternary plot (or triangle plot) is a graphical representation of three variables that sum to a constant. Each corner of the triangle represents 100% of one component, and any point within the triangle represents a unique ratio of the three components. The axes are not orthogonal but are instead arranged at 120° angles, and movement along one axis inherently affects the others due to the constraint that all three must sum to the same total.
B. No Polarity-Only Proportion
In a ternary system, there is no "negative" or "positive" direction. Instead, each axis represents a proportion of the whole, and all values are necessarily non-negative. The idea of "opposite" directions does not apply; rather, each component is balanced against the others, and the entire system is defined by relative ratios, not absolute positions or polarities.
III. Mapping Cartesian Points to Ternary Space
When we attempt to map points from a Cartesian system-where negative values are possible-onto a ternary system, we find that negative values have no direct analogue. Instead, all points are represented by their absolute contributions to each axis. This process "folds" negative quadrants into positive space, demonstrating that the distinction between positive and negative is not fundamental, but rather a byproduct of our chosen coordinate system.
IV. The Constructed Nature of Polarity
A. Polarity as a Human Convention
Polarity arises from the way we choose to divide and measure space. In binary systems, we are forced to choose "sides," but in ternary (and higher-order) systems, the need for polarity disappears. The axes become mutually dependent; increasing one necessarily decreases the others, but there is no "negative" or "opposite"-only a shift in balance.
B. Implications for Science and Philosophy
This has profound implications: what we perceive as opposites may simply be artifacts of our models, not features of reality. In chemistry, physics, and even social systems, ternary and higher-order relationships are often more accurate representations. The illusion of polarity is created by our frameworks, not by nature itself.
V. Conclusion
By adopting ternary and barycentric coordinate systems, we reveal that polarity is not a property of the universe, but a lens through which we view it. When we move beyond dualistic thinking, we see that all values are relative, all axes are interdependent, and the concept of "opposite" is a convenient fiction. This insight not only deepens our understanding of mathematics and science but also invites us to reconsider the binaries that shape our worldview.
In summary:
Ternary plots and systems show that polarity is not inherent to reality; it is a human-made construct, imposed by the limitations and choices of our coordinate systems. By embracing models that do not require polarity, we can more accurately describe and understand the complex, interdependent nature of the world.