The Reuleaux tetrahedron is the intersection of four balls of

radius In classical geometry, a radius (: radii or radiuses) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The radius of a regular polygon is th ...

''s'' centered at the vertices of a regular

tetrahedron In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...

with side length ''s''. The spherical surface of the ball centered on each vertex passes through the other three vertices, which also form vertices of the Reuleaux tetrahedron. Thus the center of each ball is on the surfaces of the other three balls. The Reuleaux tetrahedron has the same face structure as a regular tetrahedron, but with curved faces: four vertices, and four curved faces, connected by six circular-arc edges. This shape is defined and named by analogy to the Reuleaux triangle, a two-dimensional

curve of constant width In geometry, a curve of constant width is a simple closed curve in the plane (geometry), plane whose width (the distance between parallel supporting lines) is the same in all directions. The shape bounded by a curve of constant width is a body of ...

; both shapes are named after

Franz Reuleaux Franz Reuleaux (; ; 30 September 1829 – 20 August 1905) was a German mechanical engineer and a lecturer at ''Technische Hochschule Berlin'' (today Technische Universität Berlin), later appointed as the president of the academy. He was often c ...

, a 19th-century German engineer who did pioneering work on ways that machines translate one type of motion into another. One can find repeated claims in the mathematical literature that the Reuleaux tetrahedron is analogously a surface of constant width, but it is not true: the two midpoints of opposite edge arcs are separated by a larger distance, :

\left(\sqrt3 - \frac2 \right) \cdot s \approx 1.0249\,s.

Volume and surface area

The

volume Volume is a measure of regions in three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch) ...

of a Reuleaux tetrahedron is :

\frac\big(3\sqrt2 - 49\pi + 162\tan^\sqrt\big) =
 \frac\left(32\pi - 81\cos^\left(\tfrac 1 3\right) + 3\sqrt\right) \approx 0.422\,s^3.

The

surface area The surface area (symbol ''A'') of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the d ...

is :

s^2 \approx 2.975\,s^2.

Meissner bodies

Ernst Meissner and Friedrich Schilling. showed how to modify the Reuleaux tetrahedron to form a surface of constant width, by replacing three of its edge arcs by curved patches formed as the surfaces of rotation of a circular arc. According to which three edge arcs are replaced (three that have a common vertex or three that form a triangle) there result two noncongruent shapes that are sometimes called Meissner bodies or Meissner tetrahedra. Bonnesen and Fenchel. conjectured that Meissner tetrahedra are the minimum-volume three-dimensional shapes of constant width, a conjecture which is still open. In 2011 Anciaux and Guilfoyle. proved that the minimizer must consist of pieces of spheres and tubes over curves, which, being true for the Meissner tetrahedra, supports the conjecture. In connection with this problem, Campi, Colesanti and Gronchi. 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.

Man Ray Man Ray (born Emmanuel Radnitzky; August 27, 1890 – November 18, 1976) was an American naturalized French visual artist who spent most of his career in Paris. He was a significant contributor to the Dada and Surrealism, Surrealist movements, ...

's painting ''Hamlet'' was based on a photograph he took of a Meissner tetrahedron, which he thought of as resembling both Yorick's skull and Ophelia's breast from

Shakespeare William Shakespeare ( 23 April 1564 – 23 April 1616) was an English playwright, poet and actor. He is widely regarded as the greatest writer in the English language and the world's pre-eminent dramatist. He is often called England's natio ...

's ''

Hamlet ''The Tragedy of Hamlet, Prince of Denmark'', often shortened to ''Hamlet'' (), is a Shakespearean tragedy, tragedy written by William Shakespeare sometime between 1599 and 1601. It is Shakespeare's longest play. Set in Denmark, the play (the ...

''.

References

External links

* * There are also films and eve
interactive pictures
of both Meissner bodies. * {{cite web , author = Roberts, Patrick , title = Spheroform with Tetrahedral Symmetry , url = http://www.xtalgrafix.com/Spheroform2.htm Includes 3D pictures and link t
mathematical paper
showing proof of constant width. Euclidean solid geometry Geometric shapes Constant width