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Buffon's Needle Experiment


2023-05-28 |    0

Materials needed:

  • Several matchsticks or toothpicks of equal length
  • A large sheet of paper or a flat surface
  • Two matchsticks or toothpicks
  • A ruler
  • Pencil or pen

Instructions:

  1. Find a flat surface to work on, like a table or the floor.
  2. Take the large sheet of paper and place it on the surface.
  3. Take two matchsticks or toothpicks.
  4. Use the ruler to draw a straight reference line horizontally across the paper. This will be your baseline.
  5. Place one matchstick at one end of the reference line.
  6. Align the second matchstick next to the first, ensuring they are touching end to end.
  7. Without moving the matchsticks, slide the ruler until its edge touches the other end of the second matchstick.
  8. Hold the ruler in place and remove the matchsticks.
  9. Mark the position where the second matchstick ended using a pencil or pen.
  10. Repeat steps 5-9 to measure and mark additional points along the baseline, ensuring the same measured distance between each pair of adjacent points.
  11. Connect the marked points using the ruler to draw parallel lines on the paper, extending beyond the area where the matchsticks will be placed.
  12. Scatter the matchsticks randomly on the paper, ensuring they fall within the gaps between the lines.
  13. Start counting how many matchsticks cross one of the lines on the paper.
  14. Once you've counted a good number of crossings, write down the total number of matchsticks you placed and the number that crossed a line.
  15. Use the following formula to estimate π:

    π ≈ (total number of matchsticks) / (number of crossings
 
Please watch this video to see the full experiment.
 
Explanation:

 

Consider a unit circle, which is a circle with a radius of 1 unit, centered at the origin of a coordinate system.

Now, imagine randomly dropping matchsticks onto a sheet of paper with evenly spaced parallel lines drawn. The distance between each pair of adjacent lines equals the length of the matchstick.

Let's focus on the matchsticks that intersect the lines and their relationship to the unit circle:

  1. Matchsticks that intersect the lines outside the unit circle:

    • These matchsticks do not provide any information for estimating pi.
  2. Matchsticks that intersect the lines within the unit circle:

    • These matchsticks are valuable for estimating π.

    You can also try the interactive simulation found here.
     

Now, let's delve into the explanation using the unit circle:

  • Each matchstick that intersects the lines can be considered a random point falling within the area bounded by the lines.
  • The probability of a matchstick intersecting the lines is related to the ratio of the area of the unit circle to the total area bounded by the lines.
  • The area of the unit circle is π (pi) since it is defined as the ratio of a circle's circumference to its diameter.
  • The total area bounded by the lines can be calculated by multiplying the length between the lines (equal to the length of the matchstick) by the length of the segment along which matchsticks are dropped.
  • By conducting the Buffon's Needle Experiment and estimating the ratio of the number of matchsticks that intersect the lines (N_crossings) to the total number of matchsticks dropped (N_total), we can approximate the ratio of the area of the unit circle to the total area bounded by the lines.

Therefore, the estimation of pi using the unit circle can be expressed as:

π ≈ (N_total) / (N_crossings)

In this explanation, we utilize the properties of the unit circle and the matchsticks' intersections with the lines to estimate the value of π.

By increasing the number of trials and matchsticks dropped, the estimation becomes more accurate, approaching the true value of π.

 
 
 
 

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