New Reuleaux Triangle Magic  Summary and Q&A
TL;DR
Shapes of constant width, such as the Reuleaux triangle, have fascinating properties and applications, including creating a unique ferris wheel and solving common engineering challenges.
Questions & Answers
Q: What is the defining property of shapes of constant width?
Shapes of constant width have the same width no matter which direction they are measured, making them unique and different from regular polygons or circles.
Q: How can shapes of constant width be applied in practical situations, such as coins and manhole covers?
Noncircular coins of constant width are cheaper to produce and can be easily identified by coinoperated machines. Manhole covers of constant width, like the Reuleaux triangle, ensure safety by never falling into smaller holes.
Q: What challenges arise when constructing shapes of constant width with evensided polygons?
When constructing shapes of constant width with an evensided polygon, the resulting shape will not have a constant width as it will be wider in some directions and narrower in others.
Q: What is Barbier's theorem, and how does it relate to shapes of constant width?
Barbier's theorem states that all shapes of constant width have the same perimeter as a circle, which is a surprising and fascinating property of these shapes.
Summary & Key Takeaways

Shapes of constant width, like the Reuleaux triangle, have the same width no matter which direction they are measured, making them perfect for creating mesmerizing structures like a ferris wheel.

These shapes can be constructed by drawing circles centered around the corners of certain polygons, such as equilateral triangles or regular pentagons.

Shapes of constant width have practical applications, such as creating coins that are easier and cheaper to produce, and ensuring manhole covers cannot fall into their corresponding holes.