References
- Arkani-Hamed, N., Bourjaily, J., Cachazo, F., Goncharov, A., Postnikov, A., Trnka, J. (2016). Grassmannian Geometry of Scattering Amplitudes. Cambridge, UK: Cambridge Univ. Press.
- Avishalom, D. (1963). The perimetric bisection of triangles. Math. Mag. 36(1): 60–62. doi.org/10.2307/2688140 DOI: 10.2307/2688140.
- Blaschke, W. (1917). Über affine Geometrie XI: Lösung des “Vierpunktproblems” von Sylvester aus der Theorie der geometrischen Wahrscheinlichkeiten. Leipz. Ber. 69: 436–453.
- Bonk, M., Schramm, O. (2000). Embeddings of Gromov hyperbolic spaces. Geom. Funct. Anal. 10(2): 266–306. doi.org/10.1007/s000390050009 DOI: 10.1007/s000390050009.
- Cantarella, J., Deguchi, T., Shonkwiler, C. (2014). Probability theory of random polygons from the quaternionic viewpoint. Commun. Pure Appl. Math. 67(10): 1658–1699. doi.org/10.1002/cpa.21480 DOI: 10.1002/cpa.21480.
- Cantarella, J., Duplantier, B., Shonkwiler, C., Uehara, E. (2016). A fast direct sampling algorithm for equilateral closed polygons. J. Phys. A: Math. Theory. 49(27): 275202. doi.org/10.1088/1751-8113/49/27/275202 DOI: 10.1088/1751-8113/49/27/275202.
- Cantarella, J., Grosberg, A. Y., Kusner, R., Shonkwiler, C. (2015). The expected total curvature of random polygons. Amer. J. Math. 137(2): 411–438. doi.org/10.1353/ajm.2015.0015 DOI: 10.1353/ajm.2015.0015.
- Cantarella, J., Shonkwiler, C. (2016). The symplectic geometry of closed equilateral random walks in 3-space. Ann. Appl. Probab. 26(1): 549–596. doi.org/10.1214/15-AAP1100 DOI: 10.1214/15-AAP1100.
- De Morgan, A. (1871). On infinity; and on the sign of equality. Trans. Camb. Philos. Soc. 11: 145–189.
- Draper, B., Kirby, M., Marks, J., Marrinan, T., Peterson, C. (2014). A flag representation for finite collections of subspaces of mixed dimensions. Linear Algebra Appl. 451: 15–32. doi.org/10.1016/j.laa.2014.03.022 DOI: 10.1016/j.laa.2014.03.022.
- Edelman, A., Strang, G. (2015). Random triangle theory with geometry and applications. Found. Comput. Math. 15(3): 681–713. doi.org/10.1007/s10208-015-9250-3 DOI: 10.1007/s10208-015-9250-3.
- Ghys, E., de la Harpe, P., eds. (1990). Sur les Groupes Hyperboliques d’après Mikhael Gromov, Progress in Mathematics, Vol. 83. Boston, MA: Birkhäuser. doi.org/10.1007/978-1-4684-9167-8
- Gromov, M. (1987). Hyperbolic groups. In: Gersten, S.M., ed. Essays in Group Theory. New York: Springer, pp. 75–263. doi.org/10.1007/978-1-4613-9586-7_3
- Guy, R. K. (1993). There are three times as many obtuse-angled triangles as there are acute-angled ones. Math. Mag. 66(3): 175–179. doi.org/10.2307/2690963 DOI: 10.2307/2690963.
- Hausmann, J.-C., Knutson, A. (1997). Polygon spaces and Grassmannians. L’Enseign. Math. (2) 43(1–2): 173–198. doi.org/10.5169/seals-63276
- Howard, B., Manon, C., Millson, J. J. (2011). The toric geometry of triangulated polygons in Euclidean space. Canad. J. Math. 63(4): 878–937. doi.org/10.4153/CJM-2011-021-0 DOI: 10.4153/CJM-2011-021-0.
- Ingleby, C. M. (1866). Correction of an inaccuracy in Dr. Ingleby’s note on the four-point problem. Math. Quest. Solut. Educ. Times. 5: 108–109.
- Kapovich, M., Millson, J. J. (1996). The symplectic geometry of polygons in Euclidean space. J. Differential Geom. 44(3): 479–513. doi.org/10.4310/jdg/1214459218 DOI: 10.4310/jdg/1214459218.
- Kendall, D. G. (1985). Exact distributions for shapes of random triangles in convex sets. Adv. Appl. Probab. 17(2): 308–329. doi.org/10.2307/1427143 DOI: 10.2307/1427143.
- Kleiman, S. L., Laksov, D. (1972). Schubert calculus. Amer. Math. Monthly. 79(10): 1061–1082. doi.org/10.2307/2317421 DOI: 10.2307/2317421.
- Mackay, J. S. (1893). Formulae connected with the radii of the incircle and the excircles of a triangle. Proc. Edinb. Math. Soc. 12: 86–105. doi.org/10.1017/S0013091500001711 DOI: 10.1017/S0013091500001711.
- Marrinan, T., Beveridge, J. R., Draper, B., Kirby, M., Peterson, C. (2014). Finding the subspace mean or median to fit your need. 2014 IEEE Conference on Computer Vision and Pattern Recognition, Columbus, OH, pp. 1082–1089. doi.org/10.1109/CVPR.2014.142
- Needham, T. (2016). Grassmannian geometry of framed curve spaces. Ph.D. dissertation. University of Georgia, Athens, GA.
- Needham, T. (2018). Kähler structures on spaces of framed curves. Ann. Glob. Anal. Geom. 54(1): 123–153. doi.org/10.1007/s10455-018-9595-3 DOI: 10.1007/s10455-018-9595-3.
- Pfiefer, R. E. (1989). The historical development of J. J. Sylvester’s four point problem. Math. Mag. 62(5): 309–317. doi.org/10.2307/2689482 DOI: 10.2307/2689482.
- Portnoy, S. (1994). A Lewis Carroll pillow problem: probability of an obtuse triangle. Statist. Sci. 9(2): 279–284. doi.org/10.1214/ss/1177010497 DOI: 10.1214/ss/1177010497.
- Suzuki, T., Yamamoto, T., Tezuka, Y. (2014). Constructing a macromolecular K3,3 graph through electrostatic self-assembly and covalent fixation with a dendritic polymer precursor. J. Am. Chem. Soc. 136(28): 10148–10155. doi.org/10.1021/ja504891x DOI: 10.1021/ja504891x.
- Sylvester, J. J. (1864). Algebraical researches, containing a disquisition on Newton’s rule for the discovery of imaginary roots, and an allied rule applicable to a particular class of equations, together with a complete invariantive determination of the character of the roots of the general equation of the fifth degree, &c. Philos. Trans. R. Soc. London. 154: 579–666. doi.org/10.1098/rstl.1864.0017
- Sylvester, J. J. (1864). Mathematical question 1491. Educ. Times J. Coll. Precept. 17: 20.
- Sylvester, J. J. (1866). On a special class of questions on the theory of probabilities. Rep. Br. Assoc. Advmt. Sci. 35: 8–9.
- Watson, S. (1862). Answer to mathematical question 1987. The Lady’s and Gentleman’s Diary 159: 66–68.
- Wilson, J. M. (1866). On the four-point problem and similar geometrical chance problems. Math. Quest. Solut. Educ. Times 5: 81.
- Woolhouse, W. S. B. (1861). Mathematical question 1987. The Lady’s and Gentleman’s Diary 158: 76.
- Woolhouse, W. S. B. (1865). Mathematical question 1835. The Educational Times and Journal of the College of Preceptors 18: 189.
- Younes, L., Michor, P. W., Shah, J., Mumford, D. (2008). A metric on shape space with explicit geodesics. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 19(1): 25–57. doi.org/10.4171/RLM/506