Surface of constant width
Encyclopedia
In geometry
, a surface of constant width is a convex
form whose width, measured by the distance between two opposite parallel
planes touching its boundary, is the same regardless of the direction of those two parallel planes. One defines the width of the surface in a given direction to be the perpendicular distance between the parallels perpendicular
to that direction. Thus, a surface of constant width is the three-dimensional analogue of a curve of constant width
, a two-dimensional shape with a constant distance between pairs of parallel tangent lines.
More generally, any compact
convex body D has one pair of parallel supporting planes in a given direction. A supporting plane is a plane that intersects the boundary of D but not the interior of D. One defines the width of the body as before. If the width of D is the same in all directions, then one says that the body is of constant width and calls its boundary a surface of constant width, and the body itself is referred to as a spheroform.
A sphere
is obviously a surface of constant width. Contrary to common belief the Reuleaux tetrahedron
is not a surface of constant width. However, there are two different ways of smoothing subsets of the edges of the Reuleaux tetrahedron to form Meissner tetrahedra, surfaces of constant width which were conjectured by to have the minimum volume among all shapes with the same constant width; this conjecture remains unsolved. Among all surfaces of revolution
with the same constant width, the one with minimum volume is the shape swept out by a Reuleaux triangle
rotating about one of its axes of symmetry .
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, a surface of constant width is a convex
Convex set
In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...
form whose width, measured by the distance between two opposite parallel
planes touching its boundary, is the same regardless of the direction of those two parallel planes. One defines the width of the surface in a given direction to be the perpendicular distance between the parallels perpendicular
Perpendicular
In geometry, two lines or planes are considered perpendicular to each other if they form congruent adjacent angles . The term may be used as a noun or adjective...
to that direction. Thus, a surface of constant width is the three-dimensional analogue of a curve of constant width
Curve of constant width
In geometry, a curve of constant width is a convex planar shape whose width, defined as the perpendicular distance between two distinct parallel lines each intersecting its boundary in a single point, is the same regardless of the direction of those two parallel lines.More generally, any compact...
, a two-dimensional shape with a constant distance between pairs of parallel tangent lines.
More generally, any compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
convex body D has one pair of parallel supporting planes in a given direction. A supporting plane is a plane that intersects the boundary of D but not the interior of D. One defines the width of the body as before. If the width of D is the same in all directions, then one says that the body is of constant width and calls its boundary a surface of constant width, and the body itself is referred to as a spheroform.
A sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
is obviously a surface of constant width. Contrary to common belief the Reuleaux tetrahedron
Reuleaux tetrahedron
The Reuleaux tetrahedron is the intersection of four spheres of radius s centered at the vertices of a regular tetrahedron with side length s. The sphere through each vertex passes through the other three vertices, which also form vertices of the Reuleaux tetrahedron...
is not a surface of constant width. However, there are two different ways of smoothing subsets of the edges of the Reuleaux tetrahedron to form Meissner tetrahedra, surfaces of constant width which were conjectured by to have the minimum volume among all shapes with the same constant width; this conjecture remains unsolved. Among all surfaces of revolution
Surface of revolution
A surface of revolution is a surface in Euclidean space created by rotating a curve around a straight line in its plane ....
with the same constant width, the one with minimum volume is the shape swept out by a Reuleaux triangle
Reuleaux triangle
A Reuleaux triangle is, apart from the trivial case of the circle, the simplest and best known Reuleaux polygon, a curve of constant width. The separation of two parallel lines tangent to the curve is independent of their orientation...
rotating about one of its axes of symmetry .