A wheel does not have to be a circle. Some strange shapes can roll smoothly between two parallel surfaces, and others can roll across a plane along winding paths. This article tours the Reuleaux triangle, the Meissner bodies, the sphericon, and the oloid. Along the way it uses Gaussian curvature to explain why some surfaces can be made from paper and rolled flat.
The Reuleaux Triangle
Start with an equilateral triangle ABC. Draw a circle centered at A through B and C, and likewise circles of the same size centered at B and C. The region inside all three circles is a Reuleaux triangle, a triangle-like shape with three circular arcs for sides [1]. It is named after the German engineer Franz Reuleaux.
Place two parallel supporting lines against the shape and roll it. It always fits snugly between them, and the lines stay the same distance apart. This makes it a body of constant width: its width is the same in every direction. A sheet resting on several such shapes would glide horizontally with no bumpiness.
Why It Can't Be a Wheel
Constant width is not the same as constant radius. Only the circle has constant radius, because a circle is exactly the set of points at a fixed distance from a center. A circular wheel's axle sits at that center and rides smoothly. A Reuleaux triangle has no such center, so an axle through it would not give a comfortable ride.
Meissner Bodies
The three-dimensional analogue starts with four congruent balls of radius S, each centered at a vertex of a regular tetrahedron with edge length S. Their intersection is the Reuleaux tetrahedron [2]. It is often claimed to have constant width, but it does not [3]. The regular tetrahedron is one of the Platonic solids, explored in this article on the five Platonic solids.
Here is a check. The two midpoints of a pair of opposite curved edges are separated by:
(√3 − (√2)/2) × S ≈ 1.025 S
The vertices, however, are only S apart, so the width varies from place to place.
The fix is to round off three of the curved edges. Meißner showed in 1911 how to modify the Reuleaux tetrahedron this way to obtain constant width [4]. The three edges can either meet at a common vertex or bound a common face, so there are two Meissner bodies [3]. Each has constant width, so it fits perfectly between two parallel supporting planes however it is rolled.
Curvature and Wrapping Paper
Now consider covering a surface with paper. A cylinder is easy to wrap, but a sphere is not: the paper creases and distorts. To make this precise, pick a point on a surface and zoom in. The surface looks like a flat plane, the tangent plane. A normal plane passes through the point at a right angle to the tangent plane, and it cuts the surface in a curve.
Different normal planes give curves with different curvature. On a cylinder, one normal plane gives a circle and another gives a straight line. These are the extremes, and in general the largest and smallest such curvatures at a point are the principal curvatures, κ₁ and κ₂.
Gaussian Curvature
Their product is the Gaussian curvature, named after Carl Friedrich Gauss:
K = κ₁ × κ₂
For a cylinder, K = 0 everywhere, which is why paper wraps around it neatly. A sphere of radius R has K = 1/R² everywhere, so paper cannot fit it. Zero Gaussian curvature forces at least one principal curvature to vanish [5]. Curvature of this kind also underlies the ideas in this tour of complex geometry shapes, such as the Klein bottle and other surfaces.
Developable Surfaces and the Sphericon
A surface with zero Gaussian curvature can be rolled out onto a plane region without stretching. Such a surface is called developable [6]. The sphericon is one example. A sphericon surface can be made from a paper template of four circular sectors joined edge to edge [7].
The paper template uses four circular sectors, each with angle 90√2°, approximately 127.28° [8]. To see the geometry, take a cone whose radius and height both equal r. Its slant height is √2r, so half its curved surface unrolls to a sector with angle π/√2 radians, or 90√2°. The sphericon rolls with a distinctive side-to-side wobble as successive parts of its conical surface contact the plane.
The name comes from Colin Roberts, who independently discovered the shape in the 1960s while carving wood [8].
The Oloid
To build the oloid we need convex sets. A set is convex if the segment between any two of its points lies in the set; a disc is convex. A non-convex set might have a dip, a gap, or several disconnected pieces. Filling in only what is necessary gives the smallest convex set containing the original, its convex hull.
Now take two perpendicular planes and draw a circle in each, so that each circle passes through the other's center. The oloid is the convex hull of these two circles. Paul Schatz discovered it in 1929 [9]. Its boundary includes developable surface patches whose Gaussian curvature is zero at regular points [10].
The oloid can roll across a plane despite being far from spherical. Its developable boundary brings successive parts of the surface into contact with the ground, illustrating how rolling behavior depends on the geometry of contact rather than roundness alone [9]. Solids built from profiles and rotations also matter in practice, as described in geometric shapes used in 3D printing.
The Takeaway
Roundness is not required for rolling. Constant-width bodies roll between parallel supports without bumps, and developable surfaces roll out flat. Even if you're not well-rounded, you can still roll.
Sources
- Meissner’s Mysterious Bodies Bernd Kawohl & Christof Weber June 19, 2011
- Reuleaux Tetrahedron — from Wolfram MathWorld
- Reuleaux tetrahedron
- Meissner Tetrahedra — from Wolfram MathWorld
- Developable Surfaces and the Geometry of Zero Curvature
- Discrete Geodesic Nets for Modeling Developable Surfaces
- Sphericon
- Sphericon — Wolfram MathWorld
- Computational Validation of the Oloid as a Local Optimum in the Developable Roller Family
- Rolloids Kenneth Brecher
Leave a Reply