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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-theknot.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:
Chris Sangwin.
Wikipedia, for another excellent article.
Carom is written by Jonny Griffiths, hello@jonny-griffiths.net