Star Construction of Shapes of Constant Width

The Reuleaux triangle is the simplest (after the circle) example of shapes of constant width. The applet below shows how to construct other, less regular, shapes of constant width starting with star polygons.


This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


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Start with an equilateral, but not necessary equiangular star. Proceed as in the case of the Reuleaux triangle. Specifically, use vertices of the star as centers to draw circular arcs of the radius equal to the side of the star. The arcs should connect pairs of adjacent vertices.

If we think of the arcs as bridging between the sides (or their extensions), we could draw arcs of a radius augmented by some positive quantity a. This creates gaps at the vertices of the star that could be filled with arcs of radius a.

Vertices of the star are draggable. The foregoing constructions will produce shapes of constant with as long as all sides cross each other and the number of vertices is odd.

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The number of vertices in the star construction of shape of constant width is always odd.

Indeed, mark a vertex and the opposite arc. Count the number of vertices and arcs between the marked vertex and the marked arc. To the left of the marked vertex there will be, say, LV vertices and LA arcs. To the right of it there will be RV vertices and RA arcs. Since every arc follows a vertex and vice versa, LV = LA. Similarly, RV = RA. Since every arc lies opposite a vertex and vice versa, LV = RA and RV = LA. Therefore, all four numbers are equal to, say, N. Adding the marked vertex, we see that the total number of vertices equals 2N + 1.

There is a different approach to constructing the shapes of constant width. Known as the Crossed-Lines method, it is more general in that it uses a greater variety of radii than the star construction, which uses only two.

References

  1. M. Gardner, The Unexpected Hanging and Other Mathematical Diversions, The University of Chicago Press, 1991
  2. H. Rademacher and O.Toeplitz, The Enjoyment of Mathematics, Dover Publications, 1990

Related material
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Various Geometric Constructions

  • How to Construct Tangents from a Point to a Circle
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  • Constructions Related To An Inaccessible Point
  • Inscribing a regular pentagon in a circle - and proving it
  • The Many Ways to Construct a Triangle and additional triangle facts
  • Easy Construction of Bicentric Quadrilateral
  • Easy Construction of Bicentric Quadrilateral II
  • Four Construction Problems
  • Geometric Construction with the Compass Alone
  • Construction of n-gon from the midpoints of its sides
  • Short Construction of the Geometric Mean
  • Construction of a Polygon from Rotations and their Centers
  • Squares Inscribed In a Triangle I
  • Construction of a Cyclic Quadrilateral
  • Circle of Apollonius
  • Six Circles with Concurrent Pairwise Radical Axes
  • Trisect Segment: 2 Circles, 4 Lines
  • Tangent to Circle in Three Steps
  • Regular Pentagon Construction by K. Knop
  • Shapes of Constant Width

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