For a circle with radius ( R ) and ( n ) equally spaced nodes
The task is to find the values of ( n ) that allow the distance ( D ) between adjacent nodes to fit an integer number of times across the diameter of the circle. The distance ( D ) between adjacent nodes is given by:
Conclusion
The instances where the distance ( D ) between nodes fits as an integer multiple across the diameter of the circle are:
- ( n = 2 ): The distance ( D ) fits exactly 1 time across the diameter, dividing the circle into two equal parts.
- ( n = 6 ): The distance ( D ) fits exactly 2 times across the diameter, dividing the circle into six equal parts.
- ( n = \infty ): A more abstract concept, where the distance ( D ) fits an infinite number of times across the diameter, representing a continuum or infinite division.
Condition
The condition we're looking for is when the distance ( D ) between nodes fits as an integer multiple across the diameter. We have:
Simplifying, we get:
The sine function has a range between -1 and 1, so the only integer values of ( k ) that would make the equation valid are 1 and 2.
For ( k = 1 ), we get ( \sin\left(\frac{\pi}{n}\right) = 1 ), which leads to ( n = 2 ). For ( k = 2 ), we get ( \sin\left(\frac{\pi}{n}\right) = \frac{1}{2} ), which leads to ( n = 6 ). No other integer values for ( k ) will satisfy the equation, hence these are the only integer solutions for ( n ), besides the limit case of ( n = \infty ).
The String Universe-Circle Theorem
Definitions and Notations
- Let ( \mathcal{U} ) be a one-dimensional string of infinite length.
- Let ( R ) be a positive real number representing the radius of a circle.
- Define ( D ) as a segment length in ( \mathcal{U} ).
- Let ( k ) be an integer, and let ( 2R ) be partitioned into ( k ) equidistant segments each of length ( D ) such that ( k \cdot D = 2R ).
- Let ( n ) be the number of such segments ( D ) that are chords of the circle with radius ( R ).
Theorem
For a circle to emerge from the partitioning of ( \mathcal{U} ) in the manner described, ( n ) must be ( 2 ), ( 6 ), or ( \infty ).
Conditions
- For ( n = 2 ): ( D = 2R ) and ( k = 1 ).
- For ( n = 6 ): ( D = R ) and ( k = 2 ), corresponding to the inscribed hexagon.
- For ( n = \infty ): ( D = 0 ) and ( k = \infty ).
Interpretation
The circle's unique properties of uniformity and symmetry are emergent phenomena arising from the specific geometric arrangement of equidistant segments within ( \mathcal{U} ), satisfying ( k \cdot D = 2R ) for ( n = 2, 6, \text{ or } \infty ).