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Counting Intersections in the Manhattan Grid: Formulas and Patterns

This study examines the intersections within a Manhattan two-way grid and formulates methods for counting intersections both inside a given radius and on the boundary. Utilizing geometric insights, we explore the rhombus-pattern area created by boundaries and derive the formulas for N(r) and B(r). By analyzing congruent right triangles, we find the number of intersection points inside a radius and on the boundary, revealing patterns in their progression. We also investigate the behavior of these counts as radius increases, linking intersection counts with areas in a rigorous manner.

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Counting Intersections in the Manhattan Grid: Formulas and Patterns

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  1. Manhattan Intersections Number within a Boundary and Number on the Boundary The N(r) and B(r) Formulas Jeff Rose - MAE 6127

  2. This is the Manhattan two-way grid with the distance from the origin to each intersection shown. Our task is to count the number of intersections within the boundary Here are the first few counts: Jeff Rose - MAE 6127

  3. Each radius forms a rhombus pattern that is rotated 45° about the origin with respect to the horizontal. From a previous solution, the area of this rhombus is 2r 2. For example, if r = 3, then A = 2(3)2 = 2(9) = 18 Jeff Rose - MAE 6127

  4. However, when looking for a pattern in the intersection count problem, I find it easier to look at it as four congruent right triangles with one point leftover at the origin. Each triangle has one row with r intersections and subsequent rows with one fewer intersection all the way to the last row - which has 1 intersection Jeff Rose - MAE 6127

  5. With compliments to Gauss, the sum of this arithmetic sequence is That means the number of intersections in each triangle is r + (r - 1) + (r - 2) + ... 3 + 2 + 1 Jeff Rose - MAE 6127

  6. So, the number of intersection points enclosed by radius r, N(r), is four times the number in each triangle - plus 1 for the origin Jeff Rose - MAE 6127

  7. To count the number of intersections that are exactly r distance from the origin, we go back to the rotated rhombus. Here’s a chart that shows the boundary intersections: Jeff Rose - MAE 6127

  8. The formula for B(r) is pretty straight forward from the chart. When r = 0, the number of boundary intersections is 1 - otherwise, it’s 4 * r B(0) = 1 B(r) = 4r Jeff Rose - MAE 6127

  9. The last part of the problem is to investigate what happens to N(r) when r gets very large. This chart shows some values of N(r) and A(r), as well as the ratio of the two: As r gets larger, the ratio of N(r) to A(r) goes to 1. Only the highest order term of the polynomials will matter for large values of r. Both formulas have 2r2 as the highest order term - which means r   N(r) = A(r) Jeff Rose - MAE 6127

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