Drapeable unit arcs fit in the unit 30° sector
-
Yevgenya Movshovich
and
John E. Wetzel
Abstract
We show that a 30° circular sector of unit radius contains an isometric copy of every drapeable unit arc, and we describe the family of drapeable exit arcs of unit length in the sector. The conjecture that this sector is a cover for the family of all unit arcs remains unresolved.
Acknowledgements
We want to express our gratitude to Peter Moses (Redditch, Worcestershire, UK) and to Frederick Norwood (East Tennessee State University) for some helpful comments. We want also to acknowledge others who have claimed some progress on the question of whether the 30° unit sector is a cover for all unit arcs but have not published their arguments. They are Professors Patrick Coulton (Eastern Illinois University), George Poole (East Tennessee University), and Wacharin Wichirimala (Chulalongkorn University). In addition we are grateful for the supportive comments at significant times by our colleague and friend J. Ralph Alexander. Finally, we would like to express our gratitude for the exceptional efforts made by the referee, which helped us greatly to improve the article.
References
[1] P. Coulton, Y. Movshovich, Besicovitch triangles cover unit arcs. Geom. Dedicata123 (2006), 79–88. MR2299727 Zbl 1119.5201010.1007/s10711-006-9107-7Search in Google Scholar
[2] S. R. Finch, J. E. Wetzel, Lost in a forest. Amer. Math. Monthly111 (2004), 645–654. MR2091541 Zbl 1187.5101710.1080/00029890.2004.11920126Search in Google Scholar
[3] J. M. Maki, J. E. Wetzel, W. Wichiramala, Drapeability. Discrete Comput. Geom. 34 (2005), 637–657. MR2173931 Zbl 1091.5200210.1007/s00454-005-1189-8Search in Google Scholar
[4] Y. Movshovich, J. E. Wetzel, Escape paths of Besicovitch triangles. J. Comb. 2 (2011), 413–433. MR2913201 Zbl 1254.5200110.4310/JOC.2011.v2.n3.a4Search in Google Scholar
[5] J. E. Wetzel, Fits and covers. Math. Mag. 76 (2003), 349–363. MR2085394 Zbl 1059.5202210.1080/0025570X.2003.11953209Search in Google Scholar
[6] W. Wichiramala, Small convex covers for convex unit arcs. Chiang Mai J. Sci. 37 (2010), 185–194. MR2658224 Zbl 1203.52002Search in Google Scholar
© 2017 by Walter de Gruyter Berlin/Boston
Articles in the same Issue
- Frontmatter
- Ricci solitons on four-dimensional Lorentzian Walker manifolds
- The exterior splash in PG(6, q): carrier conics
- Central figure-8 cross-cuts make surfaces cylindrical
- Vieta’s formulae for regular polynomials of a quaternionic variable
- On characterizations of sausages via inequalities and roots of Steiner polynomials
- On maximal symplectic partial spreads
- Helly numbers of algebraic subsets of ℝd and an extension of Doignon’s Theorem
- A maximum problem of S.-T. Yau for variational p-capacity
- Drapeable unit arcs fit in the unit 30° sector
- Uniform Lie algebras and uniformly colored graphs
- A Krasnosel’skii-type theorem for an enlarged class of orthogonal polytopes
Articles in the same Issue
- Frontmatter
- Ricci solitons on four-dimensional Lorentzian Walker manifolds
- The exterior splash in PG(6, q): carrier conics
- Central figure-8 cross-cuts make surfaces cylindrical
- Vieta’s formulae for regular polynomials of a quaternionic variable
- On characterizations of sausages via inequalities and roots of Steiner polynomials
- On maximal symplectic partial spreads
- Helly numbers of algebraic subsets of ℝd and an extension of Doignon’s Theorem
- A maximum problem of S.-T. Yau for variational p-capacity
- Drapeable unit arcs fit in the unit 30° sector
- Uniform Lie algebras and uniformly colored graphs
- A Krasnosel’skii-type theorem for an enlarged class of orthogonal polytopes
