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Geometric Constructions through Ordered Sets

This paper proposes a novel approach to geometric construction problems by modeling the given figure as an element within an ordered set generated through a logical chain of steps, demonstrating the method's efficacy through three classical examples involving proportional segments, Gothic details, and proportional angles.

Original authors: Liudmyla Morozova

Published 2026-08-18

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CC BY 4.0

Original authors: Liudmyla Morozova

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Geometry is often taught as a set of rigid rules for drawing lines and shapes with a straightedge and a compass. In this classical world, a problem is usually a puzzle with a single, static answer: draw a line here, cut a circle there, and the solution appears. For centuries, mathematicians have accepted that some problems are simply impossible to solve with these tools, such as turning a circle into a square of the same area. However, there is a different way to think about these challenges. Instead of hunting for a single, isolated figure, one can imagine a process that generates a whole family of related shapes. This approach treats geometry not as a search for a destination, but as a journey through a sequence of steps where each new shape grows naturally from the one before it. By focusing on the logical chain that connects these shapes, rather than just the final result, it becomes possible to see the hidden structure of a problem and find solutions that might otherwise remain invisible.

A researcher named Liudmyla Morozova has developed a method based on this idea of ordered sets, where a geometric figure is viewed as just one element in a larger, connected sequence. Her work explores how this perspective can solve three distinct types of construction problems that are usually handled with standard, and sometimes cumbersome, techniques. The first problem involves dividing a line segment into five equal parts. The traditional way to do this requires drawing a separate, auxiliary line outside the main figure, marking off equal distances, and then drawing a series of parallel lines to transfer those measurements back to the original segment. This method works, but it feels somewhat artificial because the solution exists outside the figure itself. Morozova's approach changes the game by building a chain of circles that are all tangent to a single point. As these circles grow larger, they create a natural sequence of diameters. By drawing a line through this sequence, the circles automatically divide the line into the correct proportions. The solution emerges from the internal logic of the shape itself, without needing to step outside the figure or draw parallel lines. The result is a set of segments that are ordered by length, where the ratios between them are preserved no matter how the line is tilted.

The second example takes this concept further by tackling a problem found in Gothic architecture: fitting a small circle perfectly inside a curved, triangular space formed by two large arcs and a straight line. Usually, solving this requires complex algebraic calculations to find the exact center of the small circle. Morozova's method, however, treats the problem as a story of transformation. She begins with a simplified, "collapsed" version of the shape where the curves have flattened into a single line. From this starting point, she imagines the shape unfolding step by step. As the shape expands, the relationship between the parts remains constant, guiding the construction of the final solution. Instead of calculating a center point, the method identifies the exact spot where the small circle touches the straight line, allowing the rest of the circle to fall into place naturally. This process reveals that the original problem is not just about one specific drawing, but is the starting point of a whole family of related configurations. The solution is found by following the logical path of this unfolding sequence.

The third and most complex application of this idea extends the concept of proportion to angles. Dividing an angle into a specific ratio is notoriously difficult with a straightedge and compass, and many such divisions are impossible to do directly. Morozova's approach reverses the usual direction of thought. Instead of trying to cut a given angle into pieces, she starts by creating a chain of equal circles that form a series of increasing angles. This creates a framework where angles are linked in a sequence, like a ladder of steps. She then treats the size of the circles as a variable that can change smoothly. As the circles get smaller, the angles they form change continuously, creating a bridge between different sizes. By placing the specific angle to be divided onto one of these steps, the method determines a precise size for the circles that fits the angle perfectly. This choice then selects a specific path through the chain of shapes. Moving to a different step in this same path reveals the proportional angle needed. The construction is completed by transferring this result back to the original framework. This technique shows that even when a direct division is impossible, the relationship between angles can be understood and constructed by moving through a continuous family of shapes.

The paper concludes by acknowledging the limits of this method. While the approach of ordered sets offers a powerful new way to organize the logic of geometric construction, it does not break the fundamental rules of what can be built with a straightedge and compass. It cannot solve problems that are mathematically impossible, such as creating a line equal to the circumference of a circle, because those problems rely on numbers that cannot be constructed with these tools. However, within the realm of what is possible, this method reorganizes how we see the problem. It shifts the focus from a single, static answer to a flowing sequence of related figures. This sequence acts as a kind of information flow, where the relationships between shapes are preserved and redistributed as the construction progresses. By treating geometry as a generative process rather than a static puzzle, the work reveals a deeper structural principle: that the path to a solution is often more revealing than the solution itself.

Technical Summary: Geometric Constructions through Ordered Sets

Author: Liudmyla Morozova
Subject: Geometric Construction, Ordered Sets, Discrete-Continuous Structures

Problem Statement

The paper addresses the methodology of solving classical geometric construction problems. Traditional approaches often rely on auxiliary figures (such as parallel lines in Thales' theorem) or algebraic reductions that locate specific points (e.g., centers of circles) outside the immediate logical flow of the given figure. The author posits that these methods can obscure the inherent logical structure of the solution, treating the target figure as an isolated entity rather than a product of a generative process. The core problem is to reformulate construction tasks so that the solution emerges naturally from an ordered sequence of related figures, thereby providing internal justification and logical completeness.

Methodology

The proposed methodology shifts the focus from constructing a single target figure to generating an ordered set or chain of figures. The process involves three key conceptual steps:

Sequence Generation: Constructing a sequence where each successive element is determined by its predecessors.
Embedding: Placing the given problem parameters (a specific segment or angle) into this pre-constructed sequence.
Extraction: Deriving the solution by moving within the sequence (e.g., changing a continuous parameter like radius) while preserving the structural ratios defined by the set.

This approach treats the construction as a "generative process" rather than a static assembly, utilizing the concept of an "information flow" that moves between an external discrete framework and an internal continuous realization.

Key Contributions and Results

The paper illustrates this methodology through three distinct geometric problems:

1. Proportional Segments

Traditional Approach: Dividing a segment into equal parts typically uses Thales' theorem, requiring an auxiliary ray and the construction of multiple parallel lines. This separates the solution from the original figure's internal structure.
Proposed Approach: The author constructs a chain of internally tangent circles centered on an auxiliary ray. The diameters of these circles form an ordered discrete sequence ( $1:2:3:4:5$ ). By drawing a chord from the origin through the intersection of these circles with a specific arc, the proportional divisions are transferred to the target segment.
Result: This method eliminates the need for parallel lines. The proportional relations arise intrinsically from the homothety of the circles, creating a continuous family of solutions where the ratios remain constant regardless of the chord's angle.

2. A Gothic Detail (Inscribing a Circle in a Curvilinear Triangle)

Traditional Approach: Based on Pólya's algebraic treatment, this involves finding the intersection of a symmetry axis and a parabola to locate the center of the inscribed circle.
Proposed Approach: The problem is reinterpreted as the "unfolding" of a degenerate triangle. Starting with a collapsed configuration (a single semicircle), the figure is transformed by displacing a circle. The solution is not found by locating a center, but by determining the point of tangency on a segment as part of a logically ordered sequence of configurations.
Result: The construction is generalized to include a "virtual point" that persists through the transformation. The specific solution is revealed as a particular stage in a continuous family of interrelated configurations, offering a geometric rather than algebraic derivation.

3. Proportional Angles

Traditional Approach: Dividing an arbitrary angle into a specific ratio often leads to separate, non-generalizable division problems, many of which are impossible with straightedge and compass.
Proposed Approach: The author constructs a "discrete-continuous structure."
- Discrete: A chain of equal circles generates a sequence of multiple angles ( $\alpha, 2\alpha, 3\alpha, \dots, n\alpha$ ).
- Continuous: The radius of these circles is treated as a variable parameter ( $r$ ). As $r$ varies from $R$ to $R/2$ , the angle on the $n$ -th link varies continuously from $n\alpha$ to $2n\alpha$ .
Result: To divide a given angle $n\beta$ , the author embeds it into the $n$ -th link of the sequence. This determines a specific radius $r$ , which defines an "inner chain" of circles. By moving to a different link within this same inner chain (preserving the radius), the corresponding proportional angle is obtained. This allows for the construction of proportional angles using only one parallel line, extending the concept of proportionality to angular magnitudes.

Significance and Claims

The paper claims that the primary significance of this work lies in the logical reorganization of geometric construction.

Internal Justification: By embedding the target figure into an ordered set, the solution is no longer an external imposition but a natural consequence of the structure's development.
Information Flow: The construction process is described as an information flow consisting of an external component (the abstract general idea, e.g., the sequence of multiples) and an internal component (the concrete realization via varying parameters).
Completeness of Choice: The methodology highlights how initial arbitrary choices (e.g., the direction of a ray or the value of an initial angle) lead to a "completeness of choice," where all equally possible variants are accounted for within the generated structure.
Discrete-Continuous Duality: The approach successfully unifies discrete sequences (the links of the chain) with continuous variation (the changing radius/angle), providing a robust framework for understanding proportional relationships.

Limitations

The author explicitly states that this approach does not expand the class of constructible magnitudes beyond what is possible with a straightedge and compass.

The method reorganizes the logical path of construction but cannot overcome inherent algebraic limitations.
Specifically, problems requiring the construction of transcendental numbers (such as squaring the circle or constructing a segment equal to a circle's circumference, $2\pi r$ ) remain unsolvable under this framework. The ordered structure can organize relations between constructible magnitudes but cannot render non-constructible magnitudes constructible.