Skip to main content
SpringerLink
Log in

The Goat in the City

  • Article
  • Published:
The Mathematical Intelligencer Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6
Figure 7
Figure 8
Figure 9
Figure 10
Figure 11
Figure 12
Figure 13

Notes

  1. See the article on taxicab geometry in Wikipedia. Regarding the name, I prefer Chicago metric, because its streets are also a grid, it is outside my window, and goats and Chicago go together. A Chicago goat was the cause of the curse on the baseball team the Chicago Cubs that made them losers for a hundred years. Tavern owner Billy Siannis hexed the team after being denied admission to the ballpark (accompanied by his pet goat). Years later, the Billy Goat Tavern was the inspiration for the famous John Belushi/Dan Ackroyd Saturday Night Live sketch “Cheeseburgers, Cheeseburgers, Cheeseburgers,” not to be confused with the Girl and the Goat restaurant and its famous Top Chef Champion, Stefani Izard, a candidate for GOAT, like MJ.

  2. The text reads as follows: All Persons who are pleased to be CONTRIBUTORS, by answering the ENIGMAS, QUESTIONS, ETC., in this Diary, or by sending new Enigmas, Questions, Paradoxes, or other Subjects fitting for this WORK, are desired to send their Solutions with them before the End of April, 1748, directed for the Author, at Mr. Simpson’s, at Stationers Hall, LONDON. [Post Paid.]

  3. For details see [2, 6, 7, 18, 19]. At their meeting in London one imagines Simpson asking Boscovich, “Roger, enough about absolute values: got any good math problems for my Ladies Diary?”

  4. Simpson’s known aliases: Patrick O'Cavanaugh, Kubernetes, Anthony Shallow, Hurlothrumbo, Timothy Doodle Esq., Marmaduke, Hodgson [16, p. 49].

  5. Note that the city’s parallel grid, if exactly valid, needs to be embedded in a flat Earth. This contrasts with Earth cities, whose north–south streets in the northern hemisphere would, if extended, meet at the North Pole. Since such streets get closer when one is traveling north, the paths in Figure 6 would be Red > Green > Blue. Note that similar issues arise on the farm, where shortest Earth distances follow meridians. Alternatively, instead of traveling the meridians on top of the sphere, shortest-distance paths could follow straight-line paths by boring Elon Musk–like under the surface.

  6. \({{\varvec{X}}}^{C}\)=\({{\varvec{X}}}^{A}\)+a[1, 1] and \({{\varvec{X}}}^{C}\)+b[1,–1]=\({{\varvec{X}}}^{B}\), so \({{\varvec{X}}}^{A}{-{\varvec{X}}}^{B}=a[\mathrm{1,1}]-b[1,-1],\) or, \(2a=({x}_{1}^{A}-{x}_{1}^{B})+{(x}_{2}^{A}-{x}_{2}^{B})\) and \(2b={-(x}_{1}^{A}-{x}_{1}^{B})+{(x}_{2}^{A}-{x}_{2}^{B})\).

  7. The rotation of \({L}_{1}\) into \({L}_{\infty }\) is special to \({R}^{2}\) and does not extend to higher dimensions. In \({R}^{3}\), for example, the \({L}_{1}\) ball is polyhedral with 2-dimensional “faces” that are triangles. The \({L}_{\infty }\) unit ball is a cube with square faces. You cannot get from \({L}_{1}\) to \({L}_{\infty }\), because you cannot rotate a triangle into a square. Rotation works in \({R}^{2}\) because the one-dimensional faces of both unit balls collapse to intervals, which can be rotated into one another.

  8. See the Wikipedia article on Chebyshev distance.

References

  1. Joe Albree and Scott H. Brown. A valuable monument of mathematical genius: The Ladies’ Diary (1704–1840). Historia Mathematica 36 (2009), 10–47.

    Article  MathSciNet  Google Scholar 

  2. G. Bassett and R. Koenker. Asymptotic theory of least absolute error regression. Journal of the American Statistical Association 73:363 (1978), 618–622, https://doi.org/10.2307/2286611.

    Article  MathSciNet  MATH  Google Scholar 

  3. Mónica Blanco. Thomas Simpson: weaving fluxions in 18th-century London. Historia Mathematica 41 (2014), 38–81.

    Article  MathSciNet  Google Scholar 

  4. Shelly Costa. The Ladies Diary: gender, mathematics and civil society in early eighteenth century England. Osiris 17 (2002), 49–73.

    Article  Google Scholar 

  5. B. Efron. Regression percentiles using asymmetric squared error loss. Statistica Sinica 1:1 (1991), 93–125.

    MathSciNet  MATH  Google Scholar 

  6. R. W. Farebrother. Studies in the history of probability and statistics XLII. Further details of contacts between Boscovich and Simpson in June 1760. Biometrika 77:2 (1990), 397–400.

  7. R. Koenker and G. Bassett. On Boscovich’s estimator. Annals of Statistics 13: 4 (1985), 1625–1628.

    Article  MathSciNet  Google Scholar 

  8. R. Koenker and G. Bassett. Regression quantiles. Econometrica, 46:1 (1978), 33–50, https://doi.org/10.2307/1913643.

    Article  MathSciNet  MATH  Google Scholar 

  9. The Ladies Diary: Or, The Womens Almanack, for the Year of Our Lord … . J. Wilde, 1745.

  10. Fraser Marshall. A tale of two goats. Mathematics Magazine 55:4 (1982), 221–227.

    Article  MathSciNet  Google Scholar 

  11. Fraser Marshall. The grazing goat in n dimensions. College Mathematics Journal 15:2 (1984), 126–134.

    Article  MathSciNet  Google Scholar 

  12. Charles E. Myers, J. W. Watson, W. L. Harvey, J. T. Fairchild, P. S. Berg, and John Dolman, Jr. [Solution to problem 30.] American Mathematical Monthly 1:11 (1894), 395–396.

  13. Mark D. Myerson. Return of the grazing goat in n dimensions. College Mathematics Journal 15:5 (1984) , 430–432.

    Article  Google Scholar 

  14. Steve Nadis. After centuries, a seemingly simple math problem gets an exact solution. Quanta Magazine, December 9, 2020.

  15. W. K. Newey and J.L. Powell. Asymmetric least squares estimation and testing. Econometrica 55:4 (1987), 819–847.

    Article  MathSciNet  Google Scholar 

  16. Teri Perl. The Ladies’ diary or woman’s almanac, 1704–1841. Historia Mathematica 6 (1979), 36–53.

    Article  MathSciNet  Google Scholar 

  17. Ritchie Robertson. The Enlightenment: The Pursuit of Happiness, 1680–1790. Harper, 2021.

  18. S. M. Stigler. The History of Statistics: The Measurement of Uncertainty before 1900. Harvard University Press, 1986.

  19. S. Stigler. Studies in the history of probability and statistics XL. Boscovich, Simpson and a 1760 manuscript note on fitting a linear relation. Biometrika, 71:3 (1984), 615–620, https://doi.org/10.2307/2336572.

  20. Frank J. Swetz. “The Ladies Diary”: a true mathematical treasure. Convergence, August, 2018. Mathematical Association of America, https://doi.org/10.4169/convergence20180827.

  21. Frank J. Swetz. The Impact and Legacy of The Ladies’ Diary (1704–1840): A Women’s Declaration. MAA press, 2021.

  22. I. Ullisch. A closed-form solution to the geometric goat problem. Mathematical Intelligencer 42:3 (2020), 12–16.

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

Thanks to Bill Farebrother, the anonymous reviewer, and the copyeditor, David Kramer, for helpful comments and suggestions.

Author information

Authors and Affiliations

  1. Department of Finance, University of Illinois at Chicago, Chicago, IL, USA

    Gilbert Bassett

Authors
  1. Gilbert Bassett
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gilbert Bassett.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bassett, G. The Goat in the City. Math Intelligencer 44, 1–6 (2022). https://doi.org/10.1007/s00283-021-10120-7

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00283-021-10120-7

Search

Navigation