截图来自Mathematics Analysis and Approaches
Pi (π) is more than just a number for circles.
In 1777, the French naturalist Georges-Louis Leclerc, Comte de Buffon, posed a question in his essay Essai d’Arithmétique Morale:
If you randomly toss a needle onto a sheet of paper ruled with equally spaced parallel lines, and the length of the needle ll is less than the distance between the lines dd, what is the probability that the needle crosses a line?
He derived the answer: P=2l/πd
Buffon had both the means and the curiosity to test his theory. Legend has it he personally tossed thousands of needles to verify the result.
We can simplify the setup: let the needle length be exactly half the line spacing — that is, l=d/2.
Then the formula becomes: P=1/π
Which means:
total tossesintersections≈πintersectionstotal tosses≈π
For safety, we replaced needles with pick-up sticks.
Grade 10 students took turns conducting the experiment on the first and second floors of the library.
Our first 767 tosses, collected by Grade 10 students on the second floor of the library, gave us a π value of 3.10526 — slightly below the true value.
Then we moved the experiment to the classroom. Grade 11 and Grade 12 students took over, tossing sticks freely from floor level.
The new data swung wildly — some rounds far too high (3.84), others surprisingly low (2.66). But those very fluctuations pulled our π value upward, from below 3.14 to above it.
Grades 11 and 12 together made 1281 tosses with 396 intersections, giving a standalone π value of 3.235 — clearly on the high side.
But when all the data were combined (2288 tosses, 733 intersections), the final π value settled at 3.12142 — closer to the true value 3.14159.
The highs and lows pulled against each other, and the larger sample brought π back home.
- θ: the angle between the needle and the parallel lines, ranging from [0, π]
- D: the distance from the midpoint of the needle to the nearest line, ranging from [0, d/2]
When the midpoint’s distance to the line is less than or equal to this projection, one end of the needle reaches the line.
图片来源《潜藏在圆与球之中无限延续的数-圆周率》
截图来自Mathematics Analysis and Approaches
It took us three days to complete 2,288 tosses.
We then invited our computer science teacher, Ms. Salar, to help us create a simulation — a digital version of Buffon’s Needle experiment.
With a single click, 5,000 tosses were completed in an instant.
We ran it up to 150,000 tosses.
On the screen, the estimated value of π fluctuated with each batch of tosses, gradually converging and slowly approaching 3.14.
2,288 tosses — a human inquiry into mathematics.
150,000 tosses — computing power bearing witness to the same truth.
Our sincere thanks go to the library for their generous support. Below are some books on π for those who wish to explore further — all available at the library.
