TY - JOUR

T1 - Skewers

AU - Tabachnikov, Serge

N1 - Funding Information:
Acknowledgments The ‘godfather’ of this paper is Richard Schwartz whose question was the motivation for this project, and who discovered Theorem 2 and helped with its proof. I am grateful to Rich for numerous stimulating discussions on this and other topics. I am also grateful to I. Dolgachev, M. Skopenkov, V. Timorin, and O. Viro for their insights and contributions. Many thanks to A. Barvinok who introduced me to the chains of circles theorems. I was supported by NSF grants DMS-1105442 and DMS-1510055. Part of this work was done during my stay at ICERM; it is a pleasure to thank the Institute for the inspiring, creative, and friendly atmosphere.

PY - 2016/6/1

Y1 - 2016/6/1

N2 - The skewer of a pair of skew lines in space is their common perpendicular. To configuration theorems of plane projective geometry involving points and lines (such as Pappus or Desargues) there correspond configuration theorems in space: points and lines in the plane are replaced by lines is space, the incidence between a line and a point translates as the intersection of two lines at right angle, and the operations of connecting two points by a line or by intersecting two lines at a point translate as taking the skewer of two lines. These configuration theorems hold in elliptic, Euclidean, and hyperbolic geometries. This correspondence principle extends to plane configuration theorems involving polarity. For example, the theorem that the three altitudes of a triangle are concurrent corresponds to the Petersen–Morley theorem that the common normals of the opposite sides of a space right-angled hexagon have a common normal. We define analogs of plane circles (they are 2-parameter families of lines in space) and extend the correspondence principle to plane theorems involving circles. We also discuss the skewer versions of the Sylvester problem: given a finite collection of pairwise skew lines such that the skewer of any pair intersects at least one other line at right angle, do all lines have to share a skewer? The answer is positive in the elliptic and Euclidean geometries, but negative in the hyperbolic one.

AB - The skewer of a pair of skew lines in space is their common perpendicular. To configuration theorems of plane projective geometry involving points and lines (such as Pappus or Desargues) there correspond configuration theorems in space: points and lines in the plane are replaced by lines is space, the incidence between a line and a point translates as the intersection of two lines at right angle, and the operations of connecting two points by a line or by intersecting two lines at a point translate as taking the skewer of two lines. These configuration theorems hold in elliptic, Euclidean, and hyperbolic geometries. This correspondence principle extends to plane configuration theorems involving polarity. For example, the theorem that the three altitudes of a triangle are concurrent corresponds to the Petersen–Morley theorem that the common normals of the opposite sides of a space right-angled hexagon have a common normal. We define analogs of plane circles (they are 2-parameter families of lines in space) and extend the correspondence principle to plane theorems involving circles. We also discuss the skewer versions of the Sylvester problem: given a finite collection of pairwise skew lines such that the skewer of any pair intersects at least one other line at right angle, do all lines have to share a skewer? The answer is positive in the elliptic and Euclidean geometries, but negative in the hyperbolic one.

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U2 - 10.1007/s40598-016-0037-7

DO - 10.1007/s40598-016-0037-7

M3 - Article

AN - SCOPUS:85034647630

SN - 2199-6792

VL - 2

SP - 171

EP - 193

JO - Arnold Mathematical Journal

JF - Arnold Mathematical Journal

IS - 2

ER -