Pentagon set

The pentagon theorem , English Pentagon theorem , is a theorem from the mathematical branch of Euclidean geometry . It deals with a property of certain pentagons in - dimensional Euclidean space . ${\displaystyle 3}$

Formulation of the sentence

The sentence says the following:

Forming five equal length vectors in a closed pentagon such that the vectors of these included angles are also equal, so they are coplanar . ${\displaystyle \mathbb {R} ^{3}}$

In short:

A three-dimensional pentagon with all angles and sides of the same size is necessarily a flat geometrical structure .

Remarks

1. The facts presented in the pentagonal theorem can neither be transferred to quadrilaterals nor to polygons with six or more corner points : Here you can find such polygons with all angles and sides of equal size, the corner points of which are nevertheless not in one plane.
2. Ostermann and Wanner also call the theorem as the pentagonal theorem from van der Waerden ( English van der Waerden's pentagon theorem ). The first strict proof of the theorem, however, is said not to have been given in the work submitted by Bartel Leendert van der Waerden in 1970 (see below), but in 1961 in a Russian specialist journal, after the theorem was replaced by a in 1957 by the Russian Mathematician Vladimir Igorevich Arnold had been suggested the problem posed. Van der Waerden himself became aware of the sentence through conversations with the British chemist Jack David Dunitz . Chemists had apparently long suspected the validity of the pentagonal theorem after investigations into the arsenic compounds (AsCH ₃ ) _n .
3. The mathematician Stanislav Šmakal presented a brief elementary proof of the theorem by means of volume calculations using the Gram's determinant in 1972 .
4. Independent of each other, the two mathematicians found Gerrit Bol and Harold Scott MacDonald Coxeter an elegant proof of Fünfecksatzes which on the of in 1970. Leonhard Euler based given set that the orientation-preserving orthogonal images of dimensional Euclidean space exact on to rotations of space correspond ${\displaystyle 3}$ .

Sources and literature

• 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 , pp. 69 , doi : 10.1007 / 978-3-642-29163-0 . MR2918594
• S. Šmakal : A comment on a theorem about spatial pentagons . In: Elements of Mathematics . tape 27 , 1972, p. 62-63 . MR0333916
• BL van der Waerden : A sentence about spatial pentagons . In: Elements of Mathematics . tape 25 , 1970, pp. 73-78 . MR0266026

References and footnotes

1. ↑ Alexander Ostermann, Gerhard Wanner: Geometry by Its History. 2012, pp. 280, 299
2. ^ Ostermann, Wanner, op.cit., P. 280
3. ↑ Ostermann, Wanner, op. Cit., Pp. 280-281
4. ^ Ostermann, Wanner, op.cit., P. 299
5. ↑ Ostermann, Wanner, op.cit., Pp. 304, 306