Activity 1-11: Curves of Constant Width

www.carom-maths.co.uk Activity 1-11: Curves of Constant Width

How would you measure the width of a closed curve? Suppose you took a rectangle that is 2 cm by 1 cm: what is the maximum width it could have? The minimum?

Is it possible to have a curve that is a constant width, whichever direction this is measured in? A circle clearly counts, but is there anything else? There is a curve called a Reuleaux Triangle that also qualifies.

Draw an equilateral triangle ABC, and then add arc BC drawn with centre A, arc AC drawn with centre B, and arc AB drawn with centre C. There is nothing special about using a triangle here. You could do the same with a regular pentagon, or any regular n-agon for n odd. In fact, our fifty-pence piece and our twenty-pence piece are Reuleaux septagons. (What practical advantage might there be in this?)

Barbier’s Theorem states that every curve of constant width w has a perimeter of πw. Now there is a useful formula called the Isoperimetric Inequality: If any curve has perimeter L and area A, then 4πA ≤ L2. Equality holds if and only if the curve is a circle. (Check that equality holds for a circle!)

So if our curve has perimeter πw, and area A, then A ≤ πw2/4. Equality is achieved by the circle. It can be proved that the area of the Reuleaux Triangle is a minimum for a curve of constant width w. The area here is (-√3)w2/2 = 0.7047...w2.

There are in fact infinitely many curves of constant width that we can create. Curve of Constant Width link http://www.cut-the-knot.org/Curriculum/Geometry/CWStar.shtml

What about three dimensions? Is there a surface of constant width, other than the sphere?

We could create a Reuleaux tetrahedron, by drawing four spheres with centres at the corners of a regular tetrahedron. It seems intuitively obvious that this will be a solid of constant width, but this is NOT the case!

If the tetrahedron’s edge-length is 1, and we measure the solid’s width from mid-arc to opposite mid-arc, we get the value 1.0249...

Meissner & Schiller (1912) showed how to modify the Reuleaux tetrahedron to form two surfaces of constant width that are called Meissner tetrahedra. Throw away the dark shaded sharp sections above on three edges that meet at a vertex, and replace them with rounded sections.

Campi, Colesanti & Gronchi (1996) showed that the minimum volume surface of revolution with constant width is the surface of revolution of a Reuleaux triangle through one of its symmetry axes. Bonnesen & Fenchel (1934) conjectured that Meissner tetrahedra are the minimum-volume three-dimensional shapes of constant width, a conjecture which is still open...

With thanks to:ChrisSangwin. Wikipedia, for another excellent article. Carom is written by Jonny Griffiths, hello@jonny-griffiths.net

Activity 1-11: Curves of Constant Width