A "crooked triangle" often refers to a Reuleaux triangle, a distinctive geometric shape that is curved yet maintains a constant width. While not a traditional polygon with straight sides, its three rounded "corners" give it a triangular appearance, making it a fitting answer for what one might call a "crooked" or non-standard triangle.
Understanding the Reuleaux Triangle
The Reuleaux triangle [ʁœlo] is a fascinating mathematical shape, best described as a curved triangle with constant width. It is the simplest and most well-known curve of constant width other than the circle. This means that no matter how you orient it, the distance between any two parallel tangent lines is always the same. This unique property sets it apart from conventional triangles.
Key Characteristics
Despite its name, a Reuleaux triangle is not a true polygon. Its defining features include:
- Constant Width: This is its most remarkable property. If you measure the distance across the shape in any direction, it will always be the same. Imagine rolling it between two parallel lines – it would maintain contact with both lines simultaneously, just like a circle.
- Not a Polygon: Unlike standard triangles which have straight edges and sharp vertices, a Reuleaux triangle is bounded by three circular arcs. These arcs are centered at the opposite "vertices."
- Three Lobes/Corners: While its sides are curved, it distinctly features three points that can be thought of as rounded corners, giving it its triangular resemblance.
Why it's "Crooked"
The term "crooked" aptly describes a Reuleaux triangle because it deviates significantly from the straight-sided, Euclidean triangle we typically envision. Its curved boundaries and unique constant width property make it an unconventional, yet mathematically precise, variation of a triangular form. It appears "bent" or "non-straight" compared to its traditional counterpart.
Applications and Examples
The peculiar properties of the Reuleaux triangle lend themselves to several clever and practical applications across various fields:
- Drilling Square Holes: One of the most famous applications is in the design of drill bits that can drill nearly square holes. While they don't produce perfectly sharp corners, they come remarkably close.
- Coin Shapes: Some countries, like the United Kingdom (e.g., the 20p and 50p coins), use Reuleaux polygons (including triangles) for their coinage. These coins are easily identifiable, fit vending machine slots designed for circles of the same width, and are less likely to roll away.
- Pencil Shapes: Some pencils are manufactured with a Reuleaux triangle cross-section, providing a comfortable grip and preventing them from rolling off flat surfaces.
- Manhole Covers: Although not strictly Reuleaux triangles, curves of constant width are often used for manhole covers because they cannot fall through their own holes, regardless of orientation.
Reuleaux Triangle vs. Equilateral Triangle
Here's a comparison to highlight the differences between a Reuleaux triangle and a standard equilateral triangle:
| Feature | Reuleaux Triangle | Equilateral Triangle |
|---|---|---|
| Sides | Three curved arcs | Three straight lines |
| Vertices | Three rounded "corners" | Three sharp vertices (60° angles) |
| Width | Constant width in all orientations | Varies depending on measurement orientation |
| Shape Type | Curve of constant width | Polygon |
| Area | Smaller than an equilateral triangle of the same width | Larger than a Reuleaux triangle of the same width |
