A small geometry experiment / Excelsior

Constant width,
moving center.

A Reuleaux triangle fits between the same two parallel lines at every orientation. Turn it and watch the amber dot: the board stays at the same height while the shape’s center moves.

A Reuleaux triangle between two horizontal lines one unit apartThe amber center changes height as the rounded triangle turns, while the shape remains in contact with both lines. A dashed circle shows a circular roller of the same width. Upper boardFlat floor center d
Width between the lines1.000 d
Center above the floor0.423 d

Orientation study: horizontal travel is omitted. The dashed circle has the same width. The amber dot marks the symmetry center of the rounded triangle. Use the slider’s arrow keys for one-degree steps.

Build it from three arcs

Start with an equilateral triangle whose side is d. At each vertex, draw the shorter circular arc of radius d joining the other two vertices. The enclosed rounded triangle is a Reuleaux triangle.

Its width is the distance between parallel supporting lines. That distance remains d as the shape turns. Extrude the shape into a roller, put it on a flat floor and rest a horizontal board on top: the board can stay level as the roller turns beneath it.

The axle follows a different path

A dot fixed at the center rises and falls. For this shape, its height above the floor ranges from (1 − 1/√3)d to d/√3: about 0.423d to 0.577d. A wheel attached to an axle there would give a bumpy ride, even though the board stays level.

There is another coincidence with a circle of diameter d. Each curved side is a 60° arc of radius d, so the three arcs have total length 3 × (πd/3) = πd. Width and perimeter agree; the shapes are still different.

Geometry reference: Francis Su and collaborators, Reuleaux Wheel, Math Fun Facts. Diagram and center-height calculation by Excelsior. The extrema use the circular arcs analytically; the displayed outline is sampled for drawing. This is a geometric model, not a physical experiment.