Commandino's theorem

Medians of a tetrahedron
with a focus on S.

${\displaystyle {\begin{aligned}&{\frac {|AS|}{|SS_{BCD}|}}={\frac {|BS|}{|SS_{ACD}|}}={\frac {|CS|}{|SS_{ABD}|}}\\=&{\frac {|DS|}{|SS_{ABC}|}}={\frac {3}{1}}\end{aligned}}}$

The set of Commandino is a proposition of the chamber geometry , which is based on the Italian mathematician Federigo Commandino back (1506-1575). He treats an elementary average property of the center lines (Engl. Medians ) of the general tetrahedron . The theorem is the three-dimensional analog of the average theorem about the bisectors in triangle geometry .

Formulation of the sentence

Given a tetrahedron . Each of the four corner points of is connected to the center of gravity of the opposite triangular surface by a straight line, namely by the corresponding center line   . ${\displaystyle {\mathcal {T}}\subset \mathbb {R} ^{3}}$ ${\displaystyle X=A,\,B,\,C,\,D}$ ${\displaystyle {\mathcal {T}}}$ ${\displaystyle {S'}_{X}}$ ${\displaystyle \Delta '_{X}}$ ${\displaystyle X}$ ${\displaystyle m_{X}\subset \mathbb {R} ^{3}}$

The following applies:

The intersection of     the four center lines consists of exactly one point. ${\displaystyle \textstyle \bigcap _{X=A,\,B,\,C,\,D}{m_{X}}}$

This is the center of gravity of the tetrahedron . ${\displaystyle S({\mathcal {T}})}$ ${\displaystyle {\mathcal {T}}}$

The division ratio in which the center of gravity divides the route in two is always   = 1: 3 and the corner point is always the corner point of the longer of the two sections. ${\displaystyle \lambda }$ ${\displaystyle S({\mathcal {T}})}$ ${\displaystyle {\overline {{{S'}_{X}}X}}}$ ${\displaystyle \lambda }$ ${\displaystyle X}$

Generalizations

The state of affairs corresponding to Commandino's theorem applies to simplexes of any dimension :

If a -Simplex of any dimension is in and are its corner points, then the center lines , i.e. the straight lines connecting the -Simple points with the centers of gravity of the opposite -dimensional side surfaces , meet   exactly in the focus of the -Simplex. ${\displaystyle \Delta }$ ${\displaystyle p}$ ${\displaystyle p>1}$ ${\displaystyle \mathbb {R} ^{n}\;(p,n\in \mathbb {N} ,n\geq p)}$ ${\displaystyle X(0),X(1),\dots ,X(p)}$ ${\displaystyle m_{X(0)},m_{X(1)},\dots ,m_{X(p)}}$ ${\displaystyle \Delta }$ ${\displaystyle X(i)(i=0,1,\dots ,p)}$ ${\displaystyle S'_{X(i)}}$ ${\displaystyle (p-1)}$ ${\displaystyle \Delta '_{X(i)}}$ ${\displaystyle S(\Delta )}$ ${\displaystyle p}$

The ratio in which the center of gravity divides the route in two is the same    .   is the corner point of the longer of the two sections and the distance between and is always -fold the distance between and . ${\displaystyle S(\Delta )}$ ${\displaystyle {\overline {{S'_{X(i)}}{X(i)}}}}$ ${\displaystyle 1:p}$ ${\displaystyle X(i)}$ ${\displaystyle X(i)}$ ${\displaystyle S(\Delta )}$ ${\displaystyle {\tfrac {p}{p+1}}}$ ${\displaystyle X(i)}$ ${\displaystyle S'_{X(i)}}$

General proposition

In full generality, the following theorem applies, which shows a fundamental relationship that corresponds to the law of leverage in physics :

Given natural numbers and as well as that in a - vector space pairwise different points . ${\displaystyle m}$ ${\displaystyle k}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle {\mathcal {V}}}$   ${\displaystyle m+k}$ ${\displaystyle X_{1},\dots ,X_{m},Y_{1},\dots ,Y_{k}\in {\mathcal {V}}}$

The focus of these points is , while the focus of and the one who may be. ${\displaystyle m+k}$ ${\displaystyle S}$ ${\displaystyle S_{X}}$ ${\displaystyle X_{i}\;(i=1,\dots ,m)}$ ${\displaystyle S_{Y}}$ ${\displaystyle Y_{j}\;(j=1,\dots ,k)}$

Then:

${\displaystyle {\begin{aligned}S&=S_{X}+{\frac {k}{m+k}}(S_{Y}-S_{X})\\&={\frac {m}{m+k}}S_{X}+{\frac {k}{m+k}}S_{Y}\\\end{aligned}}}$

The focus is therefore on the route and divides it proportionally . ${\displaystyle S}$ ${\displaystyle {\overline {{S_{X}}{S_{Y}}}}}$ ${\displaystyle k:m}$

Reusch's theorem

The above general theorem includes not only the above generalization of Commandino's theorem (and thus this itself), but apparently also another interesting theorem about the centers of gravity of the tetrahedron, which according to the mathematical conversations of Friedrich Joseph Pythagoras Riecke on the Tübinger Professor of Physics Friedrich Eduard Reusch goes back and can be represented as follows:

The center of gravity of a tetrahedron is found by determining the center points of two pairs of opposite edges and connecting the two opposite edge centers with the associated center lines . The intersection of the two center lines obtained in this way is the center of gravity of the tetrahedron.

In connection with the fact that a tetrahedron has exactly three pairs of opposite edges, the following result can be derived from Reusch's theorem:

In a tetrahedron, the three center lines belonging to opposite edge centers intersect at one point, namely in the center of gravity of the tetrahedron.

The Varignon Theorem

In connection with the above general theorem, in addition to Reusch's theorem, a related theorem of Pierre de Varignon about the centers of gravity of quadrangles in Euclidean space should be mentioned. This theorem, also known as the Varignon Theorem , says the following:

In is given a square with four different corner points, which do not necessarily have to be in one plane . ${\displaystyle \mathbb {R} ^{n}\;(n\geq 2)}$

Then:

The two center lines, i.e. the two connecting lines of opposite side centers, intersect in the corner centroid of the four corner points and are halved by this.

See also

• Varignon set
• Diagonal set

literature

• Nathan Altshiller-Court: Modern Pure Solid Geometry . 2nd Edition. Chelsea Publishing Company, Bronx NY 1964, OCLC 1597161 .
• HSM Coxeter: Immortal Geometry . Translated into German by JJ Burckhardt (=  Science and Culture . Volume 17 ). Birkhäuser Verlag, Basel / Stuttgart 1963 ( MR0692941 ).
• Howard Eves: An Introduction to the History of Mathematics . 5th edition. Saunders College Publishing, Philadelphia [et al. a.] 1983, ISBN 0-03-062064-3 .
• Egbert Harzheim : Introduction to combinatorial topology (=  mathematics. Introductions to the subject matter and results of its sub-areas and related sciences ). Scientific Book Society, Darmstadt 1978, ISBN 3-534-07016-X ( MR0533264 ).
• Friedrich Joseph Pythagoras Riecke (Hrsg.): Mathematische Unterhaltungen . Second issue. Dr. Martin Sendet, Walluf near Wiesbaden 1973, ISBN 3-500-26010-1 (unchanged reprint of the Stuttgart edition 1867–1873).
• Harald Scheid (Ed.): DUDEN: Rechnen und Mathematik . 4th, completely revised edition. Bibliographical Institute, Mannheim / Vienna / Zurich 1985, ISBN 3-411-02423-2 .

References and comments

1. ^ Nathan Altshiller-Court: Modern Pure Solid Geometry . 2nd Edition. Chelsea Publishing Company, Bronx NY 1964, OCLC 1597161 , p. 57, 339 .
2. ^ Howard Eves: An Introduction to the History of Mathematics . 5th edition. Saunders College Publishing, Philadelphia [et al. a.] 1983, ISBN 0-03-062064-3 , pp. 438 .
3. ^ Nathan Altshiller-Court: Modern Pure Solid Geometry . 2nd Edition. Chelsea Publishing Company, Bronx NY 1964, pp. 57 .
4. ↑ Here the focus is always to be understood as the corner focus .
5. ^ Nathan Altshiller-Court: Modern Pure Solid Geometry . 2nd Edition. Chelsea Publishing Company, Bronx NY 1964, pp. 57-58 .
6. ↑ Egbert Harzheim : Introduction to Combinatorial Topology (=  Mathematics. Introductions to the subject matter and results of its sub-areas and related sciences ). Scientific Book Society, Darmstadt 1978, ISBN 3-534-07016-X , p. 33 ( MR0533264 ).
7. ↑ Egbert Harzheim : Introduction to Combinatorial Topology (=  Mathematics. Introductions to the subject matter and results of its sub-areas and related sciences ). Scientific Book Society, Darmstadt 1978, ISBN 3-534-07016-X , p. 31 .
8. ↑ Egbert Harzheim : Introduction to Combinatorial Topology (=  Mathematics. Introductions to the subject matter and results of its sub-areas and related sciences ). Scientific Book Society, Darmstadt 1978, ISBN 3-534-07016-X , p. 31 ff .
9. ↑ See article about Riecke on Wikisource
10. ↑ ^{a b} Friedrich Joseph Pythagoras Riecke (Ed.): Mathematische Unterhaltungen. Second issue. 1973, pp. 100, 128
11. ↑ In the Mathematische Unterhaltungen (Second Booklet, p. 128) reference is made to p. 36 of Reusch's treatise The Pointed Arch.
12. ↑ Coxeter, op.cit., P. 242
13. ↑ DUDEN: arithmetic and mathematics. 1985, p. 652