Through the analytical lens and abstract thinking, mathematicians have continually contributed to the understanding of geometry, providing comprehensive insights into the structures that are all too familiar in our daily lives. The pursuit of identifying the best versions of iconic shapes has long intrigued scholars and enthusiasts alike, leading to a series of fascinating discoveries that unpack the intrinsic properties of these geometrical forms.

Historically, the quest for defining ‘best’ has been rooted in the principles of symmetry, simplicity, and optimality. The circle, with its perfectly equidistant points from a central locus, represents a kind of geometric nirvana, symmetrical and simple. Similarly, the square signifies optimal subdivision of space into equal areas with minimal perimeter.

Recent endeavors by mathematicians have delved deeper into classic forms, exploring their various manifestations across different dimensions and contexts. The groundbreaking research was driven by mathematical optimization and computer algorithms designed to test and compare different shapes under manifold criteria.

One landmark finding concerned the Platonic solids—a set known since antiquity—which mathematicians re-evaluated. Through computational analysis, they confirmed that these shapes indeed represent an ideal form considering factors like symmetry, surface area to volume ratio, and stability.

Another intriguing area of study has been on tessellations: patterns that fill a plane with no overlaps and no gaps. Mathematicians have unearthed new tessellation shapes that minimize boundaries—a characteristic that could inform efficient spatial planning in various scientific fields.

In higher dimensions, research on hypercubes and other complex polytopes has unveiled configurations with unprecedented efficiency properties. These shapes could have significant implications for data storage and retrieval systems in computing architectures.

More esoteric advances involve curved spaces and non-Euclidean geometries. The studying of optimized shapes within these realms has not only deepened understanding but has also revealed inherent beauty in forms unfathomable within traditional Euclidean constraints.

Mathematicians believe that their work decoding these ideal shape structures holds potential for multiple applications—a ranging spectrum from architectural design to nanotechnology. As such, this melding of aesthetic allure with practical function captures the quintessence of mathematical elegance—an art form sown from the seeds of logical rigor and empirical investigation.

Where this journey into geometrical perfection will lead remains an ever-evolving narrative. However, its impact resonates far beyond the confines of mathematics; it inspires a deeper appreciation for the subtle contours governing our physical reality.