Page 30 - Abstract book - TJSSF-2020

Thailand – Japan Student Science Fair 2020 (TJ-SSF 2020)
“Seeding Innovations through Fostering Thailand – Japan Youth Friendship”

Buffon’s leaf problem
Approximating pi using fallen leaves

Takano haruka
1
Advisors: Kida Hideaki , and Matsuda Keisuke
1
2

Hiroshima University High School, Hiroshima, Japan
1
Graduate School of Frontier Biosciences, Osaka University, Osaka, Japan
2
Osaka University Hospital, Osaka, Japan
2

Abstract

“Buffon’s needle problem” is one of the most famous math problems. Buffon proved that when needles
of length l are randomly thrown on parallel lines placed at regular intervals of d, the probability of the
needles to cross any of the lines is equal to 2l/πd. The theory can approximate the value of π by counting
the number of needles that cross the parallel lines. Based on this theory, we have been trying to build
a theory which can approximate the value of π by fallen leaves on a brick pavement. To reach our goal,
there are two tasks. One is that leaves have diversity in their shapes. To find how our theory can deal
with the diverse shapes, we have formulated an equation to approximate the value of π by throwing
needles of varied lengths, and we confirmed the validity of the equation based on experiments and
simulation programs. The other is that a leaf is a more complex figure than a needle is. To find the
probability for leaves to cross a line when thrown on parallel lines, we proved that the convex hull of a
concave polygon crossing a line is the necessary condition for the concave polygon to cross a line and
found the probability for regular n polygons to cross a line based on a previous study.

Keywords: Buffon’s needle problem, probability, approximate, π, fallen leaves, convex hull, concave
polygon

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