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The Goat in the City

The Mathematical Intelligencer volume 44, pages 1–6 (2022)Cite this article

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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.

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Acknowledgments

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

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

    Gilbert Bassett

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

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