Area Difference Bounds for Dissections of a Square into an Odd Number of Triangles

References

• [Abrams and Pommersheim 14] A. Abrams and J. Pommersheim. “Spaces of Polygonal Triangulations and Monsky Polynomials,” Discrete Comput. Geom. 51:1 (2014), 132–160.
   Web of Science ®Google Scholar
• [Aigner and Ziegler 14] M. Aigner and G. M. Ziegler. Proofs from THE BOOK. 5th ed., Berlin: Springer, 2014.
   Google Scholar
• [Bates et al. 13] D. Bates, J. Hauenstein, A. Sommese, and C. Wampler. Bertini: Software for numerical algebraic geometry, 2013, bertini.nd.edu/BertiniUsersManual.pdf.
   Google Scholar
• [Brinkmann and McKay 11] G. Brinkmann and B. McKay. plantri: program for generation of certain types of planar graph, 2011, cs.anu.edu.au/bdm/plantri.
   Google Scholar
• [Chapman 09] R. Chapman. “Solution of Problem 11266: Partitioning Values of a Polynomial into sets of Equal Sum,” Amer. Math. Monthly 116:2 (2009), 181–183.
   Google Scholar
• [de Loera et al. 96] J. A. de Loera, S. Hoşten, F. Santos, and B. Sturmfels. “The Polytope of all Triangulations of a Point Configuration,” Documenta Math.—J. DMV 1 (1996), 103–119.
   Google Scholar
• [Emiris et al. 10] I. Z. Emiris, B. Mourrain, and E. P. Tsigaridas. The DMM bound: multivariate (aggregate) separation bounds, ISSAC 2010—Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation, ACM Press, 2010, pp. 243–250.
   Google Scholar
• [Firla and Ziegler 99] R. T. Firla and G. M. Ziegler. “Hilbert Bases, Unimodular Triangulations, and Binary Covers of Rational Polyhedral Cones,” Discrete & Computational Geometry 21:2 (1999), 205–216.
   Web of Science ®Google Scholar
• [Kleist 16] L. Kleist. Drawing planar graphs with prescribed face areas, Graph-Theoretic Concepts in Computer Science (Berlin, Heidelberg) (Pinar Heggernes, ed.), Lecture Notes in Computer Science, vol. 9941, Berlin: Springer-Verlag, 2016, pp. 158–170.
   Google Scholar
• [Lothaire 97] M. Lothaire. Combinatorics on words, Cambridge Mathematical Library, Cambridge University Press, Cambridge,1997, reprint with minor revisions; original edition, Reading, MA: Addison-Wesley, 1983.
   Google Scholar
• [Mansow 03] K. Mansow. Ungerade Triangulierungen eines Quadrats von kleiner Diskrepanz, Diploma thesis, Technische Universität Berlin, December 2003, i+48 pp.
   Google Scholar
• [Monsky 70] P. Monsky. “On Dividing a Square into Triangles,” Amer. Math. Monthly 77:2 (1970), 161–164.
   Web of Science ®Google Scholar
• [Praton 02] I. Praton. “Cutting Polyominos into Equal-Area Triangles,” Amer. Math. Monthly 109:9 (2002), 818–826.
   Web of Science ®Google Scholar
• [Prouhet 51] E. Prouhet. “Mémoire sur quelques relations entre les puissances des nombres,” Comptes Rendus Acad. Sci. Sér. I 33 (1851), 225.
   Google Scholar
• [Ringel 90] G. Ringel. Equiareal graphs, Contemporary Methods in Graph Theory — In honour of Prof. Dr. Klaus Wagner (Rainer Bodendiek, ed.), BI Wissenschaftsverlag, 1990, pp. 503–505.
   Google Scholar
• [Richman and Thomas 67] F. Richman and J. Thomas. “Problem 5479,” Amer. Math. Monthly 74:9 (1967), 329.
   Google Scholar
• [Rudenko 13] D. Rudenko. “On Equidissection of Balanced Polygons,” J. Math. Sci., New York 190:3 (2013), 486–495.
   Google Scholar
• [Rudenko 14] D. Rudenko. Arithmetic of 3-valent graphs and equidissections of flat surfaces, preprint, arXiv:1411.0285 (2014), 19 pp.
   Google Scholar
• [Schulze 11] B. Schulze. “On the Area Discrepancy of Triangu- 1465 lations of Squares and Trapezoids,” Electron. J. Combin. 18:1 (2011), Paper 137, 16 pp.
   Google Scholar
• [Stein et al. 14] W. A. Stein et al., Sage mathematics software (version 6.1.1), The Sage Development Team, 2014, www.sagemath.org.
   Google Scholar
• [Stein and Szabó 94] S. K. Stein and S. Szabó. Algebra and Tiling: Homomorphisms in the Service of Geometry, Carus Mathematical Monographs, vol. 25, Mathematical Association of America, 1994.
   Google Scholar
• [Stein 99] S. K. Stein. “Cutting a Polyomino into Triangles of Equal Areas,” Amer. Math. Monthly 106:3 (1999), 255–257.
   Web of Science ®Google Scholar
• [Stein 00] S. K. Stein. “A Generalized Conjecture about Cutting a Polygon into Triangles of Equal Areas,” Discrete Comput. Geom. 24:1 (2000), 141–145.
   Web of Science ®Google Scholar
• [Stein 04] S. K. Stein. “Cutting a Polygon into Triangles of Equal Areas,” Math. Intelligencer 26:1 (2004), 17–21.
   Web of Science ®Google Scholar
• [Thomas 68] J. Thomas. “A Dissection Problem,” Math. Mag. 41:4 (1968), 187–191.
   Google Scholar
• [Thomassen 80] C. Thomassen. “Planarity and Duality of Finite and Infinite Graphs,” J. Combinat. Theory, Ser. B 29 (1980), 244–271.
   Web of Science ®Google Scholar
• [Tutte 60] W. T. Tutte. “Convex Representations of Graphs,” Proc. London Math. Soc. 10 (1960), 304–320.
   Google Scholar
• [Wright 59] E. M. Wright. “Prouhet’s 1851 Solution of the Tarry–Escott Problem of 1910,” Amer. Math. Monthly 66:3 (1959), 199–201.
   Google Scholar
• [Ziegler 06] G. M. Ziegler. “Problems in Discrete Differential Geometry, Open Problem 10,” Oberwolfach Reports no. 1 (2006), 693–694.
   Google Scholar