Mobius configuration

Configuration example. The face planes of the red tetrahedron are shown in the upper figure. The face planes of the blue tetrahedron are shown in the bottom. The coordinates of the vertices of the red tetrahedron:

(0,0,0),

{\ displaystyle (0,0,0),}

{\ displaystyle (0,0,0),}

(0,0,1),

{\ displaystyle (0,0,1),}

{\ displaystyle (0,0,1),}

(0,1,0)

{\ displaystyle (0,1,0)}

(0,1,0)

and

(1,0,0)

{\ displaystyle (1,0,0)}

(1,0,0)

. The coordinates of the vertices of the blue tetrahedron are

(0,-\gamma ,\gamma ),

{\ displaystyle (0, - \ gamma, \ gamma),}

{\ displaystyle (0, - \ gamma, \ gamma),}

(\gamma ,0,-\gamma ),

{\ displaystyle (\ gamma, 0, - \ gamma),}

{\ displaystyle (\ gamma, 0, - \ gamma),}

(-\gamma ,\gamma ,0)

{\ displaystyle (- \ gamma, \ gamma, 0)}

{\ displaystyle (- \ gamma, \ gamma, 0)}

and

(\lambda ,\lambda ,\lambda ),

{\ displaystyle (\ lambda, \ lambda, \ lambda),}

{\ displaystyle (\ lambda, \ lambda, \ lambda),}

where

\gamma ={\frac {1}{\sqrt {2}}}

{\ displaystyle \ gamma = {\ frac {1} {\ sqrt {2}}}}

{\ displaystyle \ gamma = {\ frac {1} {\ sqrt {2}}}}

and

\lambda ={\frac {1}{3}}

{\ displaystyle \ lambda = {\ frac {1} {3}}}

{\ displaystyle \ lambda = {\ frac {1} {3}}}

.

In geometry, a Möbius configuration or Möbius tetrahedron is a configuration in Euclidean space or projective space consisting of two mutually inscribed tetrahedra - each vertex of one tetrahedron lies on a plane passing through the face of the other tetrahedron and vice versa. Thus, in the resulting system of eight points and eight planes, each point lies on four planes (three planes define the vertex of the tetrahedron, and the fourth plane is the plane passing through the face of the second tetrahedron on which the vertex lies), and each plane contains four points ( three vertices of the tetrahedron face and the vertex of another tetrahedron lying on the same plane).

Mobius theorem

The configuration is named after Augustus Ferdinand Mobius , who proved in 1828 that if two tetrahedrons have the property that seven of their vertices lie on the corresponding planes of the faces of another tetrahedron, then the eighth vertex also lies on the plane of the corresponding face, forming the Mobius configuration. This also true in a more general three-dimensional projective space if and only if Papp's theorem holds in this space ( Reidemeister , ), and it holds in a three-dimensional space built on the body in volume and only if the commutative law is satisfied, and therefore the group must be a field (Al-Dhahir). Due to projective duality, the Mobius result is equivalent to the statement that if seven out of eight planes of two tetrahedra passing through the faces contain the corresponding vertices of another tetrahedron, then the plane of the eighth face also contains another vertex.

Build

Coxeter ( Coxeter 1950 ) described the simple construction of a configuration. We start from an arbitrary point p in Euclidean space. Let A , B , C, and D be four planes passing through p , no three of which intersect in one straight line. We place six points q , r , s , t , u, and v on six lines formed by the pairwise intersection of these planes in such a way that no four points lie on the same plane. For any plane A , B , C, and D, four of the seven points p , q , r , s , t , u, and v lie on this plane and three lie outside it. We construct the planes A ' , B' , C ' and D' through triples of points lying outside the planes A , B , C and D, respectively. Then, in the dual form of the Mobius theorem, these four new planes intersect at one point w . Eight points p , q , r , s , t , u , v and w and eight planes A , B , C , D , A ' , B' , C ' and D' form the Mobius configuration.

Similar Constructions

Gilbert and Cohn-Vossen 1952 (without reference) state that there are five configurations having eight points and eight planes with four points on each plane and with four planes passing through each point that can be realized in three-dimensional Euclidean space - such configurations are designated $8_{4}$ ${\ displaystyle 8_ {4}}$ . Information on these configurations can be obtained from Steinitz's article ( Steinitz 1910 ). Based on the findings of Muth ( Muth 1892 ), Bauer ( Bauer 1897 ) and Martinetti ( Martinetti 1897 ), the article actually claims that there are five $8_{4}$ ${\ displaystyle 8_ {4}}$ configurations with properties that a maximum of two planes have two common points and the dual property that a maximum of two points belong to two planes. (This condition means that any three points do not lie on one straight line and the dual three planes do not intersect on one straight line.) However, there are ten others $8_{4}$ ${\ displaystyle 8_ {4}}$ configurations for which this condition is not satisfied, and all fifteen configurations can be implemented in three-dimensional space. Of interest are the configurations in which two tetrahedra participate, each inscribed and described in each other, and these are precisely those configurations that satisfy the above property. Thus, there are five configurations with tetrahedra, and they correspond to the five conjugacy classes of the symmetric group $S_{4}$ ${\ displaystyle S_ {4}}$ . One can obtain the permutation of the four vertices of one tetrahedron S = ABCD into itself as follows: each vertex P of the tetrahedron S lies on a plane containing three vertices of the other tetrahedron T. The remaining point of the tetrahedron T lies on the plane containing three points of the tetrahedron S, and the point Q of the tetrahedron S lies outside this plane. We obtain a map P → Q. Five conjugacy classes of permutations $S_{4}$ ${\ displaystyle S_ {4}}$ Is e, (12) (34), (12), (123), (1234) and, of these five classes, the Mobius configuration corresponds to the conjugacy class e. It is designated Ke. Steinitz claims that if two Ke tetrahedra are $A_{0},B_{0},C_{0},D_{0}$ ${\ displaystyle A_ {0}, B_ {0}, C_ {0}, D_ {0}}$ and $A_{1},B_{1},C_{1},D_{1}$ ${\ displaystyle A_ {1}, B_ {1}, C_ {1}, D_ {1}}$ then eight planes $A_{i},B_{j},C_{k},D_{l}$ ${\ displaystyle A_ {i}, B_ {j}, C_ {k}, D_ {l}}$ these tetrahedra are given by odd sum indices $i+j+k+l$ ${\ displaystyle i + j + k + l}$ .

Steinitz also claims that only one $8_{4}$ ${\ displaystyle 8_ {4}}$ the Mobius configuration corresponds to the geometric theorem. However, Glynn disputes this fact ( Glynn 2010 ) - he showed, using computer search, that there are exactly two $8_{4}$ ${\ displaystyle 8_ {4}}$ , one corresponds to the Möbius configuration, for the second configuration (corresponding to the conjugacy class (12) (34) above), the theorem also holds for all three-dimensional projective spaces over the field , but not over common bodies . There are other similarities between the two configurations, including the fact that they are self-dual in the sense of the duality of matroids . In abstract terms, the second configuration has “points” 0, ..., 7 and “planes” 0125 + i, (i = 0, ..., 7), where integers are taken modulo eight. This configuration, like the Mobius band, can be represented as two tetrahedra, mutually inscribed and described - in the representation as whole tetrahedra there can be 0347 and 1256. However, these two $8_{4}$ ${\ displaystyle 8_ {4}}$ the configurations are not isomorphic, since the Mobius configuration has four pairs of planes that do not contain common configuration points, while the second configuration does not have such planes.

The Levi graph of the Mobius configuration has 16 vertices, one for each point and plane, and the edges correspond to the incidence of vertices and planes (a pair is a plane and a vertex lying on it). The graph is isometric to the hypercube graph with 16 vertices Q ₄ . A close Mobius – Cantor configuration , formed by two mutually inscribed quadrangles, has a Mobius – Cantor graph , a subgraph of Q ₄ , as the Levy graph.

Notes

Literature

MW Al-Dhahir. A class of configurations and the commutativity of multiplication. - The Mathematical Gazette. - The Mathematical Association, 1956. - T. 40. - S. 241–245. - DOI : 10.2307 / 3609605 . .
G. Bauer. {{{title}}} // München Ber .. - 1897. - T. 27 . - S. 359 . .
HSM Coxeter. Self-dual configurations and regular graphs // Bulletin of the American Mathematical Society. - 1950 .-- T. 56 , no. 5 . - S. 413–455 . - DOI : 10.1090 / S0002-9904-1950-09407-5 . .
DG Glynn. Theorems of points and planes in three-dimensional projective space // Journal of the Australian Mathematical Society. - 2010 .-- T. 88 . - S. 75–92 . - DOI : 10.1017 / S1446788708080981 . .
David Hilbert, Stephan Cohn-Vossen. Geometry and the Imagination. - 2nd. - Chelsea, 1952. - S. 184. - ISBN 0-8284-1087-9 . .
V. Martinetti. Le configurazioni (8 ₄ , 8 ₄ ) di punti e piani (Italian) // Giornale di Matematiche di Battaglini. - 1897. - T. 35 . - S. 81–100 . .
AF Möbius. Kann von zwei dreiseitigen Pyramiden einejede in Bezug auf die andere um- und eingeschriehen zugleich heissen? // Journal für die reine und angewandte Mathematik . - 1828. - T. 3 . - S. 273–278 . . In collected works (1886), vol. 1, pp. 439–446.
P. Muth. // Zeitschrift Math. Phys .. - 1892. - T. 37 . - S. 117 . .
K. Reidemeister. Zur Axiomatik der 3-dimensionalen projektive Geometrie // Jahresbericht der Deutschen Mathematiker-Vereinigung. - 1929 .-- T. 38 . - S. 71 . .
K. Reidemeister. Aufgabe 63 (gestellt in Jahresbericht DMV 38 (1929), 71 kursiv). Lösung von E. Schönhardt // Jahresbericht der Deutschen Mathematiker-Vereinigung. - 1931.- T. 40 . - S. 48-50 . .
Ernst Steinitz. Konfigurationen der projektiven Geometrie. 6. Konfigurationen von Punkten und Ebenen // Enzyklopädie der mathematischen Wissenschaften. - 1910. - T. 3-1-1 AB 5a . - S. 492–494 . .