- #1
Loren Booda
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What is the sequence described by the counts of integer Cartesian coordinates (x, y) within circles of successive whole number radii centered at the origin?
Sequence of circumscribed Cartesian coordinates
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- #2
arildno
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Hmm..it was trickier than I thought to get out an explicit formula.
Nice problem!
Nice problem!
- #3
CRGreathouse
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It's the Gauss Circle Problem, with values http://www.research.att.com/~njas/sequences/A000328 or http://www.research.att.com/~njas/sequences/A051132 depending on how you handle the boundaries. I don't have an explicit formula for this myself, though the MathWorld article has a few sum representations.
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- #4
Loren Booda
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Thanks for your alertness, both.
1. What are Cartesian coordinates?
Cartesian coordinates are a system for representing points in a two-dimensional or three-dimensional space. They consist of two or three numerical values, known as coordinates, that together specify the precise location of a point in space.
2. What is the meaning of "circumscribed" in the term "Sequence of circumscribed Cartesian coordinates"?
In this context, "circumscribed" refers to a sequence of coordinates that are arranged or determined in a way that defines or encompasses a specific shape or area. In other words, the coordinates are arranged in a way that outlines or surrounds a particular region in space.
3. How is a sequence of circumscribed Cartesian coordinates used in science?
A sequence of circumscribed Cartesian coordinates is commonly used in various scientific fields, such as physics, mathematics, and engineering, to precisely locate and describe objects or phenomena in space. It allows scientists to perform calculations and analyses on the coordinates to understand the relationships and interactions between different points in space.
4. Can Cartesian coordinates be used in any dimension?
Yes, Cartesian coordinates can be used in any dimension, from one-dimensional lines to four-dimensional spacetime. However, the most commonly used Cartesian coordinates are two-dimensional (x, y) and three-dimensional (x, y, z) coordinates, which are used to represent points in a plane and in three-dimensional space, respectively.
5. How do I convert Cartesian coordinates to other coordinate systems?
To convert Cartesian coordinates to other coordinate systems, you need to use specific mathematical equations and formulas that take into account the different axes and units of the two systems. For example, to convert Cartesian coordinates to polar coordinates, you can use the equations r = √(x^2 + y^2) and θ = tan^-1(y/x). Different coordinate systems may require different conversion methods, so it's important to research and understand the specific equations for each conversion.
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