This research paper started with the discovery of an interesting hypothesis of Kepler with a rich history, which started with an ordinary snowflake and took 400 years to prove. Although this hypothesis has a rich history, there is very little information on this topic and even less illustrations. Finding this problem we had the idea of creating various 3D models on this topic to improve and understand the essence of the hypothesis. These 3D models built from the data on Kepler's hypothesis could make a major breakthrough in the development of various areas of our lives. The models will be freely available, this will help a lot of people for their research and discoveries. Each model has its own meaning and evidence in Kepler's hypothesis. This hypothesis already covers physics, chemistry, logistics and also involves computer graphics. With our 3D models, scientists and enthusiasts are free to experiment and explore new research horizons in a virtual 3D world.

Figure 1. Our 3D models

In 1611 the German scientist Johannes Kepler in his work ‘On the Hexagonal Snowflake’ for the first time formulated a hypothesis that greatly influenced the future of the mathematical sciences, this hypothesis states that the densest packing in three-dimensional space is achieved in a hexagonal or cubic lattice. This idea originated with Kepler after realising that snowflakes have a perfect hexagonal shape, after snowflakes fell on his coat alone during a New Year's Eve party. Johannes Kepler often asked the question, ‘Why six? What was the physical reason for this?’, these questions were a challenge for Kepler to solve this question.However, it took more than four centuries of humanity to prove the hypothesis, thanks to the efforts of mathematicians and their calculations this hypothesis succeeded thanks to the development of computer technology. Thanks to this attempt, the hypothesis was put forward, which became the basis for one of the most important questions in geometry: how densely can identical spheres be packed in three-dimensional space? The hypothesis states that the best way of packing is to arrange the spheres in the form of hexagonal dense packing (HDP) or cubic dense packing (CDP), as in stacks of oranges.[1]

 Figure 2. Title page of the first edition of I. Kepler's work On Hexagonal Snowflakes, 1611.Credits:http://www.joostwitte.nl/M_Galilei/Johannes_kepler_snowflake.pdf

This problem has acquired great importance not only in mathematics, but also in physics, chemistry, materials science and other fields of science. Full proof of the hypothesis became possible only at the end of XX century due to the development of computer technologies. And its simplified form of proof was presented only in the beginning of XXI century.  This article considers the history of the hypothesis, the key stages of its proof and its impact on modern science.

1)Kepler's idea:In his work of 1611 Kepler paid attention to hexagonal structures of snowflakes and suggested that the packing density of spheres in nature is maximum at hexagonal arrangement. Kepler took for the further analyses and researches known and usual at first sight to us things such as bee honeycomb which are constructed on hexagon and also was taken a pomegranate and the question was asked why it has a rhomboidal form? But this hypothesis was not proved by Kepler that passed on to the next great minds.[1]

Figure 3 . 6-sided snowflake. Credits:  keplers-new-years-gift-on-the-six-cornered-snowflake

2)The achievements of Gauss (1831):Gauss proved that among all lattice packings of spheres, the f.c.c. arrangement provides the highest density. This was a crucial step in confirming Kepler's conjecture under the condition that spheres are arranged in a regular lattice.

Limitations: Although Gauss established the maximum density for regular arrangements, he acknowledged the complexity of proving the conjecture for irregular or non-periodic packings. He indicated that while irregular arrangements might achieve higher densities over small volumes, extending these arrangements to larger volumes would ultimately reduce their density.[2]

3)The work of Thomas Hales (1998-2014):Thomas Hales completed the proof of Kepler's hypothesis in 1998, using computational methods.The proof of Kepler's hypothesis Hales was engaged in from 1992 to 1998. But due to the underdevelopment of technology the proof was accurately proved and verified by computer systems only in 2014-2015.[3;4]

Kepler Conjecture: The conjecture states that the maximum density for packing congruent spheres is achieved by the face-centered cubic arrangement, with a density of 

π/√18≈0.74, meaning that approximately 25.95% of the volume remains unfilled.[5]

 Figure 4. Credits: mathematical-mysteries-keplers-conjecture     

Figure5.Credits:http://www.joostwitte.nl/M_Galilei/Johannes_kepler_snowflake.pdf

• Logistics and Shipping: The principles derived from the Kepler Conjecture can optimize the packing of spherical objects, such as fruits or balls, in shipping and storage. Efficient packing minimizes wasted space, leading to cost savings in transportation and storage.
•  Data Compression: The algorithms developed during the proof of the Kepler Conjecture have applications in digital data compression. By understanding how to pack data more densely, these methods improve storage efficiency and speed up data transmission processes.
•  Material Science: Insights from the conjecture help researchers understand atomic arrangements in crystalline structures. This knowledge is crucial for developing new materials with specific properties, as the arrangement of atoms significantly influences material behavior.
•  Traffic Management: Techniques related to linear optimization used in proving the conjecture are applicable in traffic management systems. These methods can optimize routes and reduce congestion, contributing to more efficient urban planning.
•  Coding Theory: The conjecture's principles are analogous to problems in coding theory, where optimal arrangements of codes can enhance error correction and data integrity during transmission6.
•  Computer Graphics: In computer graphics, understanding sphere packing can improve rendering techniques by optimizing how objects are placed within a three-dimensional space, enhancing visual realism and performance.

•  Interdisciplinary Research: The quest to prove the Kepler Conjecture has spurred advancements in computational mathematics and algorithm development. This research fosters collaboration among mathematicians, physicists, computer scientists, and engineers, enhancing problem-solving capabilities across various scientific domains.
•  Mathematical Exploration: The methods developed for proving the conjecture have broader implications for solving other complex mathematical problems, demonstrating the interconnectedness of mathematical theories and their applications.

Kepler's hypothesis is a unique and rare example of how a simple observation of nature and fun can lead to the formation of a fundamental mathematical problem. Its proof, which took a great deal of time and was proved only with the help of computer technology of our time, not only confirmed the genius of Kepler's intuition, but also marked a new era and new opportunities not only in mathematics but also in people's lives, where traditional methods are complemented by computational ones.

Modern applications of the hypothesis cover a wide range of sciences, including physics, chemistry, biology, logistics and even computer graphics. It remains a symbol of the link between nature, mathematics and the new technologies of the new age, showing that new solutions and breakthroughs cannot be initiated and solved without the past.

1.  http://www.joostwitte.nl/M_Galilei/Johannes_kepler_snowflake.pdf
2.  https://www.ias.ac.in/article/fulltext/reso/002/06/0060-0067
3. https://annals.math.princeton.edu/wp-content/uploads/annals-v162-n3-p01.pdf
4. https://phys.org/news/2017-06-mathematicians-formal-proof-kepler-conjecture.html
5. https://mathworld.wolfram.com/KeplerConjecture.html
6. https://ars.electronica.art/aeblog/en/2017/05/31/die-keplersche-vermutung/