Explaining Shapes Of Constant Width


Shapes of constant width are so fun to play with. The motion initially looks impossible, but once you understand how the simple reuleaux triangle is constructed, you can easily see why it behaves the way it is.

Here's a snippet showing what's going on

reuleaux triangle shape of constant width explained

During the short research I made on these shapes I discovered you can "combine" two shapes of constant width to obtain a new one. The way you combine these shapes is by a mathematical operation called a Minkowski sum, which has a pretty cool geometric representation, as seen in these animated gifs

minkowski sum of shapes of constant width
minkowski sum rounded reuleaux triangle

All shapes of the same width also has the same circumference. The proof for this relies on the fact that the Minkowski sum of a shape of constant width and its flipped version, always results in a circle with diameter twice the width of the original shapes, as can be seen here:

minkowski sum to proof that all shapes of the same width have the same circumference - reuleaux pentagon

minkowski sum to proof that all shapes of the same width have the same circumference - right triangle

minkowski sum to proof that all shapes of the same width have the same circumference - reuleaux triangle

I was also curious if you could construct shapes of constant width without using circular arcs. Apparently you can! You might have noticed that sliding the pentagon around the reuleaux triangle resulted in a new shape (of constant width by definition) that has no circular arcs. Another example for creating "non-circular" shape of constant is this shape which I made in the build video.

shape of constant width with no circular arcs

The construction of this shape is described in the book "How Round Is Your Circle", which has more information about these shapes in general.

The top part of this shape is half of an ellipse with semi major axis half the desired width , or in other words

The bottom part is generated by sliding one end of a straight line the length of the desired width so it's perpendicular to the ellipse. The other side traces the bottom of our new shape of constant width.

In order to draw it, I derived the analytical curve defined by this construction, here is its parametric representation

The parameter t is the ellipses x coordinate.

If you want to make this unique shape yourself and don't care much for the math, you can simply scale the templates as desired.

The models for everything 3D printed in this video can be found on my thingiverse page

reuleaux tetrahedron meissner tetrahedron

Partial list of sources:

[1] minkowski sum of shapes, barbier’s theorem: Soifer A. Geometric etudes in combinatorial mathematics (2ed., Springer, 2010)(ISBN 0387754695)(O)(300s) - page 155 and on

[2] meissner tetrahedron: http://www.softouch.on.ca/kb/data/Scan-130117-0001.pdf

[3] special shapes/solids of constant width: https://arxiv.org/pdf/1504.06733.pdf

[4] special shape: Bryant J., Sangwin C. How Round Is Your Circle page 195

[5] meisner: https://www.swisseduc.ch/mathematik/geometrie/gleichdick/docs/meissner_mysterious%20_bodies.pdf

[6] https://www.swisseduc.ch/mathematik/geometrie/gleichdick/meissner-en.html

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