why study discrete mathematics, The kissing problem
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why study discrete mathematics
The kissing problem
Why study discrete mathematics?
Discrete mathematics is a fundamental pillar of computer science, providing the mathematical underpinnings for algorithms, data structures, and cryptography. It also boasts diverse real-world applications, from network analysis to logistics. Studying discrete mathematics sharpens your problem-solving and logical reasoning skills, opening doors to lucrative careers in software engineering, data science, and cybersecurity. Furthermore, understanding discrete mathematics is essential for navigating today's digital world, as it underpins technologies and data-driven decision-making. Additionally, it's an intellectually stimulating subject with rich and diverse concepts that challenge and engage the mind. Whether you're pursuing a technical career or simply seeking to understand the world around you better, discrete mathematics is a valuable and rewarding field of study.
The kissing problem
"How many non-overlapping unit spheres can simultaneously touch ("kiss") another unit sphere of the same size in a given dimension?"
Imagine you have a central sphere, and you want to surround it with other spheres of the same size, such that they all touch the central one but don't overlap with each other. The maximum number of such surrounding spheres is the "kissing number" for that particular dimension.
In one dimension (on a line), the kissing number is 2 (one sphere on each side).
In two dimensions (on a plane), the kissing number is 6 (think of arranging coins around a central coin).
The three-dimensional case (our usual 3D space) is more complex and was the subject of a debate between Newton and Gregory. It was finally proven that the kissing number in 3D is 12.
The problem gets increasingly difficult as we move to higher dimensions. Determining the exact kissing number is only known for a few specific dimensions. There are general bounds and estimates, but finding precise values remains an active area of mathematical research.
Why is it called the "kissing problem"?
The term "kissing" is used because the spheres are tangent to each other, just like two objects touching lightly.
Why is the kissing problem important?
Sphere packing: It's directly related to the problem of how dense spheres can be packed together, which has applications in coding theory, crystallography, and even understanding the structure of the universe.
Error-correcting codes: The arrangement of spheres in the kissing problem can be used to design efficient codes that can detect and correct errors in data transmission.
Mathematical beauty: Beyond its applications, the problem itself is a fascinating mathematical puzzle that has captivated mathematicians for centuries.