![Tetrahedron centroid gimp](https://upload.wikimedia.org/wikipedia/commons/a/a3/Tetrahedron_centroid_gimp.png)
Commandino's theorem, named after
Federico Commandino
Federico Commandino (1509 – 5 September 1575) was an Italian humanist and mathematician.
Born in Urbino, he studied at Padua and at Ferrara, where he received his doctorate in medicine. He was most famous for his central role as translator o ...
(1509–1575), states that the four
median
In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic fe ...
s of a
tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
are concurrent at a point ''S'', which divides them in a 3:1 ratio. In a tetrahedron a median is a line segment that connects a vertex with the
centroid
In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ob ...
of the opposite
face
The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may aff ...
– that is, the centroid of the opposite triangle. The point ''S'' is also the centroid of the tetrahedron.
History
The theorem is attributed to Commandino, who stated, in his work ''De Centro Gravitatis Solidorum'' (The Center of Gravity of Solids, 1565), that the four medians of the tetrahedron are concurrent. However, according to the 19th century scholar Guillaume Libri,
Francesco Maurolico (1494–1575) claimed to have found the result earlier. Libri nevertheless thought that it had been known even earlier to
Leonardo da Vinci
Leonardo di ser Piero da Vinci (15 April 14522 May 1519) was an Italian polymath of the High Renaissance who was active as a painter, Drawing, draughtsman, engineer, scientist, theorist, sculptor, and architect. While his fame initially res ...
, who seemed to have used it in his work.
Julian Coolidge
Julian Lowell Coolidge (September 28, 1873 – March 5, 1954) was an American mathematician, historian and a professor and chairman of the Harvard University Mathematics Department.
Biography
Born in Brookline, Massachusetts, he graduated from Har ...
shared that assessment but pointed out that he couldn't find any explicit description or mathematical treatment of the theorem in da Vinci's works. Other scholars have speculated that the result may have already been known to Greek mathematicians during antiquity.
Generalizations
Commandino's theorem has a direct analog for
simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
es of any
dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
:
: Let
be a
-simplex of some dimension
in
and let
be its vertices. Furthermore, let
, be the medians of
, the lines joining each vertex
with the centroid of the opposite
-dimensional
facet
Facets () are flat faces on geometric shapes. The organization of naturally occurring facets was key to early developments in crystallography, since they reflect the underlying symmetry of the crystal structure. Gemstones commonly have facets cut ...
. Then, these lines intersect each other in a point
, in a ratio of
.
Full generality
The former analog is easy to prove via the following, more general result, which is analogous to the way
levers
A lever is a simple machine consisting of a beam or rigid rod pivoted at a fixed hinge, or ''fulcrum''. A lever is a rigid body capable of rotating on a point on itself. On the basis of the locations of fulcrum, load and effort, the lever is di ...
in physics work:
: Let
and
be
natural numbers
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''cardinal n ...
, so that in an
-
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
,
pairwise different
points are given.
: Let
be the centroid of the points
, let
be the centroid of the points
, and let
be the centroid of all of these
points.
: Then, one has
::
: In particular, the centroid
lies on the line
and divides it in a ratio of
.
Reusch's theorem
The previous theorem has further interesting consequences other than the aforementioned generalization of Commandino's theorem. It can be used to prove the following theorem about the centroid of a tetrahedron, first described in the ''Mathematische Unterhaltungen'' by the German
physicist
A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe.
Physicists generally are interested in the root or ultimate caus ...
:
[Friedrich Joseph Pythagoras Riecke (Hrsg.): ''Mathematische Unterhaltungen. Zweites Heft.'' 1973, S. 100, 128][In den ''Mathematische Unterhaltungen'' (Zweites Heft, S. 128) wird auf die S. 36 von Reuschs Abhandlun]
''Der Spitzbogen''
verwiesen.
: ''One may find the centroid of a tetrahedron by taking the
midpoints of two pairs of two of its opposite
edges and connecting the corresponding midpoints through their respective midline. The intersection point of both midlines will be the centroid of the tetrahedron.''
Since a tetrahedron has six edges in three opposite pairs, one obtains the following corollary:
:''In a tetrahedron, the three midlines corresponding to opposite edge midpoints are
concurrent
Concurrent means happening at the same time. Concurrency, concurrent, or concurrence may refer to:
Law
* Concurrence, in jurisprudence, the need to prove both ''actus reus'' and ''mens rea''
* Concurring opinion (also called a "concurrence"), a ...
, and their intersection point is the centroid of the tetrahedron.''
Varignon's theorem
A specific case of Reusch's theorem where all four vertices of a tetrahedron are
coplanar
In geometry, a set of points in space are coplanar if there exists a geometric plane that contains them all. For example, three points are always coplanar, and if the points are distinct and non-collinear, the plane they determine is unique. Howe ...
and lie on a single plane, thereby degenerating into a
quadrilateral
In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...
, Varignon's theorem, named after
Pierre Varignon, states the following:
[Coxeter, op. cit., S. 242][''DUDEN: Rechnen und Mathematik.'' 1985, S. 652]
:''Let a quadrilateral in
be given. Then the two midlines connecting opposite edge midpoints intersect in the centroid of the quadrilateral and are divided in half by it.''
References
External links
* {{MathWorld, title=Commandino's Theorem, urlname=CommandinosTheorem
''A Couple of Nice Extensions of the Median Properties''
Theorems in geometry
Euclidean geometry