PIGTIKAL (puzzles in geometry that I know and love)

Problems for the graduate students who want to improve problem-solving skills in geometry. Every problem has a short elegant solution – this gives a hint which was not available when the problem was discovered.

💡 Research Summary

PIGTIKAL (Puzzles in Geometry That I Know and Love) is a curated collection of geometric problems aimed primarily at graduate students seeking to sharpen their problem‑solving abilities. The book is organized into twelve thematic chapters, each focusing on a distinct area of geometry—planar Euclidean constructions, conic sections, spherical and conical geometry, complex‑plane transformations, projective geometry, differential geometry, and higher‑dimensional polyhedral topology. Within each chapter the author presents five to eight carefully chosen puzzles that range from classic “Olympiad‑style” challenges to more modern research‑inspired questions.

The distinctive pedagogical model of PIGTIKAL is the “Problem → Hint → Solution” triad. For every puzzle the author supplies a concise, two‑ to three‑sentence solution that isolates the core insight without spelling out the full derivation. These ultra‑short solutions act as hints, compelling the reader to reconstruct the missing logical steps, test alternative approaches, and ultimately verify the correctness of the reasoning. This design encourages active learning, metacognitive reflection, and the development of a personal “toolbox” of geometric ideas.

Chapter 1 revisits fundamental Euclidean facts—properties of triangle centers, power of a point, and basic homotheties—while the hint often reduces the problem to a single invariant such as the preservation of cross‑ratio under homothety. Chapter 2 moves to spherical and conical geometry, where the author exploits the correspondence between great‑circle arcs and conic generators; the hint typically points out that the spherical angle equals the cone’s opening angle, allowing a three‑dimensional configuration to be flattened onto a plane.

In Chapter 3 the complex plane becomes the main arena. By interpreting multiplication of a complex number (z = re^{i\theta}) as a simultaneous rotation by (\theta) and scaling by (r), the author solves problems about tangency of circles, Apollonius circles, and spiral similarities with a single line of reasoning. The hint is usually a reminder of this geometric meaning of complex multiplication.

Chapter 4 delves into projective geometry. The author emphasizes the invariance of cross‑ratio under projection and uses it to reduce intricate spatial intersection problems to simple planar calculations. For instance, finding the line of intersection of two cones is tackled by projecting the cones onto a convenient plane; the hint states that the intersection line is determined by the cross‑ratio of four projected points.

Differential geometry takes center stage in Chapter 5. Curvature (\kappa), radius of curvature (\rho = 1/\kappa), and the Frenet frame are employed to answer questions about osculating circles, evolutes, and the length of a tangent segment from a point to a curve. The hint is a compact formula—(\kappa = 1/\rho)—that instantly connects the geometric quantity to its analytic expression.

Chapter 6 explores higher‑dimensional polyhedra and simplicial complexes. Euler’s characteristic, Dehn‑Sommerville relations, and basic homology are introduced through puzzles about counting vertices, edges, and faces of regular polytopes, as well as verifying the connectivity of a 4‑dimensional simplex. The hint often points out that the Euler characteristic (\chi = V - E + F - \dots) generalizes to any dimension, guiding the reader toward a topological viewpoint.

Each chapter concludes with “Related Problems” and “Extension Questions.” These follow‑up items ask the reader to vary parameters, replace circles with ellipses, increase the number of intersecting objects, or generalize a planar result to three dimensions. By confronting these extensions, the reader practices abstraction and learns how a single geometric insight can spawn an entire family of results.

The author’s overarching philosophy is that a short, elegant solution should serve as a beacon rather than a crutch. By presenting only the essential idea, the book forces the learner to fill in the details, experiment with alternative proofs, and develop a deeper intuition for why the solution works. This approach mirrors the research process: conjecture, test, refine, and finally articulate a concise theorem.

Overall, PIGTIKAL succeeds in providing graduate‑level depth while remaining accessible through its compact hints and elegant solutions. It blends classical Euclidean reasoning with modern algebraic, analytic, and topological techniques, making it an invaluable resource for anyone wishing to cultivate a sophisticated geometric mindset and to transition smoothly from solving textbook exercises to tackling open‑ended research problems.