Volume formula of the general tetrahedron

The volume formula of the general tetrahedron is a mathematical formula of stereometry . It was given by Leonhard Euler (1707–1793) in his famous treatise E 231 ( Demonstratio nonnullarum insignium proprietatum quibus solida hederis planis inclusa sunt praedita .). Euler treats and solves in point 20 of this paper the problem of giving a formula for the volume of the general tetrahedron with reference to the lengths of the six tetrahedron edges alone . The volume formula of the general tetrahedron is based on the same task as that of Heron's formula in triangular geometry .

The Euler formula

A tetrahedron is given , i.e. a pyramid with a triangular base . The belonging to the triangular base - edges were with referred and the opposite in the space of three -edges with . ${\displaystyle {\mathcal {T}}\subset \mathbb {R} ^{3}}$ ${\displaystyle {\mathcal {T}}}$ ${\displaystyle a,\,b,\,c}$ ${\displaystyle {\mathcal {T}}}$ ${\displaystyle a^{'},\,b^{'},\,c^{'}}$

Next is for each -edges the length of this edge referred. ${\displaystyle {\mathcal {T}}}$ ${\displaystyle x}$ ${\displaystyle |x|}$

Then the following applies to the tetrahedron volume : ${\displaystyle V=V({\mathcal {T}})}$

${\displaystyle V={\tfrac {1}{12}}\cdot {\sqrt {|a|^{2}\cdot |a^{'}|^{2}\cdot f_{a}+|b|^{2}\cdot |b^{'}|^{2}\cdot f_{b}+|c|^{2}\cdot |c^{'}|^{2}\cdot f_{c}-\delta }}}$

With

${\displaystyle f_{a}=|b|^{2}+|b^{'}|^{2}+|c|^{2}+|c^{'}|^{2}-|a|^{2}-|a^{'}|^{2}}$

${\displaystyle f_{b}=|a|^{2}+|a^{'}|^{2}+|c|^{2}+|c^{'}|^{2}-|b|^{2}-|b^{'}|^{2}}$

${\displaystyle f_{c}=|a|^{2}+|a^{'}|^{2}+|b|^{2}+|b^{'}|^{2}-|c|^{2}-|c^{'}|^{2}}$

${\displaystyle \delta =|a|^{2}\cdot |b|^{2}\cdot |c|^{2}+|a|^{2}\cdot |b^{'}|^{2}\cdot |c^{'}|^{2}+|a^{'}|^{2}\cdot |b|^{2}\cdot |c^{'}|^{2}+|a^{'}|^{2}\cdot |b^{'}|^{2}\cdot |c|^{2}}$

Simplified Euler formula for isosceles and regularity

For isosceles tetrahedron applies to each of the six edges . Here the Euler formula is simplified as follows: ${\displaystyle |x|=|x^{'}|}$ ${\displaystyle {\mathcal {T}}}$ ${\displaystyle x}$

${\displaystyle V={\tfrac {\sqrt {2}}{12}}\cdot {\sqrt {(|b|^{2}+|c|^{2}-|a|^{2})\cdot (|a|^{2}+|c|^{2}-|b|^{2})\cdot (|a|^{2}+|b|^{2}-|c|^{2})}}}$

The well-known volume formula for the regular tetrahedron results directly from this :

${\displaystyle V={\tfrac {\sqrt {2}}{12}}\cdot |a|^{3}}$

Simplified Euler formula for right-angled tetrahedra

For right-angled tetrahedra, the Pythagorean theorem etc. applies to each of the three edges of the base. Here the Euler formula with the spatial product of the opposite edges is simplified to: ${\displaystyle |a|^{2}=|b^{'}|^{2}+|c^{'}|^{2}}$ ${\displaystyle {\mathcal {T}}}$ ${\displaystyle a,\,b,\,c}$ ${\displaystyle (a^{'},b^{'},c^{'})}$

${\displaystyle V={\tfrac {1}{6}}\cdot |(a^{'},b^{'},c^{'})|={\tfrac {1}{6}}\cdot |a^{'}||b^{'}||c^{'}|}$.

Determinant representation

The following identities , which are based on determinants of symmetrical matrices , can also be used in an elegant way to represent the tetrahedron volume: ${\displaystyle V=V({\mathcal {T}})}$

${\displaystyle {\begin{aligned}288\cdot V^{2}&=\det {\begin{pmatrix}0&|c|^{2}&|b|^{2}&|a^{'}|^{2}&1\\|c|^{2}&0&|a|^{2}&|b^{'}|^{2}&1\\|b|^{2}&|a|^{2}&0&|c^{'}|^{2}&1\\|a^{'}|^{2}&|b^{'}|^{2}&|c^{'}|^{2}&0&1\\1&1&1&1&0\\\end{pmatrix}}\\&=\det {\begin{pmatrix}2|c|^{2}&|c|^{2}+|b|^{2}-|a|^{2}&|c|^{2}+|a^{'}|^{2}-|b^{'}|^{2}\\|c|^{2}+|b|^{2}-|a|^{2}&2|b|^{2}&|b|^{2}+|a^{'}|^{2}-|c^{'}|^{2}\\|c|^{2}+|a^{'}|^{2}-|b^{'}|^{2}&|b|^{2}+|a^{'}|^{2}-|c^{'}|^{2}&2|a^{'}|^{2}\\\end{pmatrix}}\\\end{aligned}}}$

The first determinant to appear is called a Cayley – Menger determinant (after the two mathematicians Arthur Cayley and Karl Menger ) .

Application of the Cayley-Menger determinant

The Cayley-Menger determinant representation of the tetrahedron volume can be used to formulate a classical theorem of Leonhard Euler, namely Euler's so-called four - point theorem :

Four spatial points (not necessarily different from one another ) lie in one plane if and only if the relationship ${\displaystyle P_{1},P_{2},P_{3},P_{4}}$

${\displaystyle \det {\begin{pmatrix}0&a_{12}^{2}&a_{13}^{2}&a_{14}^{2}&1\\a_{12}^{2}&0&a_{23}^{2}&a_{24}^{2}&1\\a_{13}^{2}&a_{23}^{2}&0&a_{34}^{2}&1\\a_{14}^{2}&a_{24}^{2}&a_{34}^{2}&0&1\\1&1&1&1&0\\\end{pmatrix}}=0}$

applies, where each denotes the Euclidean distance between the points and . ${\displaystyle a_{ij}=a_{ji}\;(i,j=1,2,3,4)}$ ${\displaystyle P_{i}}$ ${\displaystyle P_{j}}$

The statement of Euler's four-point theorem is therefore the following:

Four spatial points lie in one plane if and only if the tetrahedron formed by them has degenerated and thus has the volume . ${\displaystyle {\mathcal {T}}={\mathcal {T}}(P_{1},P_{2},P_{3},P_{4})\subset \mathbb {R} ^{3}}$ ${\displaystyle V({\mathcal {T}})=0}$

literature

Original work

• Leonhard Euler: Demonstration nonnullarum insignium proprietatum quibus solida hederis planis inclusa sunt praedita . In: Novi commentarii academiae scientiarum Petropolitanae (1752/53) . tape 4 , 1753, pp. 140-160 .

Monographs

• Nathan Altshiller-Court: Modern Pure Solid Geometry . 2nd Edition. Chelsea Publishing, Bronx NY 1964, OCLC 1597161 .
• György Hajós : Introduction to Geometry . BG Teubner Verlag, Leipzig (Hungarian: Bevezetés A Geometriába . Translated by G. Eisenreich [Leipzig, also editing]).
• Maximilian Miller: Stereometry (Crantz Collection) . BG Teubner Verlag, Leipzig 1957.
• Alexander Ostermann, Gerhard Wanner: Geometry by Its History (=  Undergraduate Texts in Mathematics. Readings in Mathematics ). Springer Verlag, Heidelberg / New York / Dordrecht / London 2012, ISBN 978-3-642-29162-3 , doi : 10.1007 / 978-3-642-29163-0 ( MR2918594 ).
• Andreas Speiser et al. (Editor): Leonhardi Euleri Opera omnia. Series great. Opera mathematica. Volume XXVI: Commentationes geometricae. Volume I . Orell Füssli, Zurich 1953.

References and comments

1. ^ Maximilian Miller: Stereometry. 1957, p. 41
2. ↑ The title of the treatise E 231 reads roughly as follows in German: Presentation of some characteristic properties with which bodies enclosed by flat surfaces are equipped . In this treatise Euler gives the first proof of the polyhedron formula , which he already gave in an earlier treatise ( E 230 , printed under Elementa doctrinae solidorum , Novi commentarii academiae scientiarum Petropolitanae 4, pp. 109-140; cf. Introduction to the Commentationes geometricae ) mentioned but not yet proven.
3. ^ Andreas Speiser et al .: Leonhardi Euleri Opera omnia. Series great. Opera mathematica. Volume XXVI: Commentationes geometricae. Volume I. 1953, pp. 106-107
4. ↑ This results from considering the formula ${\displaystyle (-\alpha +\beta +\gamma )\cdot (\alpha -\beta +\gamma )\cdot (\alpha +\beta -\gamma )=\alpha ^{2}\cdot (-\alpha +\beta +\gamma )+\beta ^{2}\cdot (\alpha -\beta +\gamma )+\gamma ^{2}\cdot (\alpha +\beta -\gamma )-2\cdot \alpha \cdot \beta \cdot \gamma .}$
5. ^ Nathan Altshiller-Court: Modern Pure Solid Geometry. 1964, p. 112
6. ↑ Miller, op.cit., P. 46
7. ^ IN Bronstein, KA Semendjajev et al .: Taschenbuch der Mathematik. 2008, p. 157
8. ^ György Hajós: Introduction to Geometry. 1970, p. 383
9. ↑ Alexander Ostermann, Gerhard Wanner: Geometry by Its History. 2012, p. 297
10. ^ György Hajós : Introduction to Geometry . BG Teubner Verlag, Leipzig, p. 384 (Hungarian: Bevezetés A Geometriába . Translated by G. Eisenreich [Leipzig, also editorial]).