What Are the Ideal Proportions for a Cylinder to Act as a Three-Sided Die?

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SUMMARY

The discussion focuses on determining the ideal proportions of a cylinder that can function as a three-sided die, landing on its side or ends with equal probability. Participants emphasize the importance of analyzing the forces acting on the cylinder when it is tipped at various angles. The key takeaway is that the relationship between the cylinder's length and the angle at which it tips is crucial for achieving the desired random landing behavior. Drawing diagrams to visualize these forces is recommended for better understanding.

PREREQUISITES
  • Understanding of basic physics principles, particularly forces and angles.
  • Familiarity with geometric shapes, specifically cylinders.
  • Knowledge of probability concepts related to random outcomes.
  • Ability to create and interpret diagrams illustrating physical scenarios.
NEXT STEPS
  • Research the physics of tipping objects and equilibrium conditions.
  • Explore geometric properties of cylinders and their impact on stability.
  • Learn about probability distributions related to random landing outcomes.
  • Study practical applications of three-sided dice in gaming and simulations.
USEFUL FOR

This discussion is beneficial for physicists, mathematicians, game designers, and anyone interested in the mechanics of randomization and object stability.

Helios
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There is a cylinder of certain proportions, that when dropped, lands either on its side or one of its two ends with equal probability. A coin is too short of a cylinder. A solid pipe is too long of a cylinder. Somewhere in between is a cylinder that acts like a three-sided die. I've imagined this problem but don't know how to solve it.
 
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Draw a picture. (Always good advice)

You have a cylinder that touches the ground at one point on the rim, tipped at some angle. Draw the forces acting on the cylinder and ask for what angles does the cylinder tip over to the side and for what angles does it tip back to land on an end. You now have a relation between the length of the cylinder and the random angle it lands at.
 

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