A Rölot tetrahedron is a body that is the intersection of four identical balls whose centers are located at the vertices of a regular tetrahedron , and the radii are equal to the side of this tetrahedron. This body is a spatial analogue of the Roelot triangle as the intersection of three circles on a plane.
However, unlike the Röhlo triangle, the Röhlo tetrahedron is not a body of constant width : the distance between the midpoints of the opposite boundary curved edges connecting its vertices is
times larger than the edge of the original regular tetrahedron [1] [2] .
Meissner bodies
The Ryelo tetrahedron can be modified so that the resulting body turns out to be a body of constant width. For this, in each of the three pairs of opposite curved edges, one edge is “smoothed” in a certain way [2] [3] . The resulting two different bodies (the three edges on which the substitutions occur, can be taken either from one vertex or forming a triangle [3] ) are called Meissner bodies , or Meissner tetrahedra [1] [4] . The hypothesis formulated by Tommy Bonnesen and Werner Fennel in 1934 [5] states that it is these bodies that minimize volume among all bodies of a given constant width, however (as of 2019) this hypothesis has not been proved [2] .
Notes
- ↑ 1 2 Weisstein EW Reuleaux Tetrahedron . MathWorld .
- ↑ 1 2 3 Kawohl B., Weber C. Meissner's Mysterious Bodies (English) // Mathematical Intelligencer. - 2011. - Vol. 33, no. 3 . - P. 94-101. - DOI : 10.1007 / s00283-011-9239-y .
- ↑ 1 2 Gardner. The Unexpected Hanging and Other Mathematical Diversions, 1991 , p. 218.
- ↑ Weber C. Meissner Bodies - interactive . SwissEduc . Archived March 22, 2013.
- ↑ Bonnesen T., Fenchel W. Theorie der konvexen Körper. - Berlin : Springer-Verlag , 1934 .-- P. 127-139. (German)
Literature
- Gardner M. Chapter 18: Curves of Constant Width // The Unexpected Hanging and Other Mathematical Diversions. - Chicago ; London : University of Chicago Press, 1991 .-- P. 212-221. - 264 p. - ISBN 978-0-2262-8256-5 . (eng.)
- Kawohl B. Convex Sets of Constant Width // Oberwolfach Reports. - 2009. - Vol. 6. - P. 390-393.
- Weber C. Bodies of Constant Width . SwissEduc . Archived March 22, 2013.