Triangles, Fractales and Spaghetti

TL;DR

This paper explores the probability of forming triangles from broken spaghetti segments, revealing a fractal pattern in the sample space and analyzing the likelihood of creating nearly equilateral triangles.

Contribution

It introduces a fractal-based geometric framework to analyze the broken spaghetti problem and computes probabilities for specific triangle configurations.

Findings

Identifies a fractal pattern underlying the problem

Calculates probability of forming a triangle from random breaks

Estimates probability of obtaining near-equilateral triangles

Abstract

There is well-known problem of geometric probability which can be quote as the Broken Spaghetti Problem. It addresses the following question: A stick of spaghetti breaks into three parts and all points of the stick have the same probability to be a breaking point. What is the probability that the three sticks, putting together, form a triangle? In these notes, we describe a hidden geometric pattern behind the symmetric version of this problem, namely a fractal that parametrizes the sample space of this problem. Using that fractal, we address the question about the probability to obtain a $\delta$-equilateral triangle.