His polyhedral Butterfly World Map, like Buckminster Fuller's later Dymaxion map of 1943 and 1954, enabled all continents to be uninterrupted, and with reasonable fidelity to a globe. Cahill demonstrated this principle by also inventing a rubber-ball globe which could be flattened under a pane of glass in the "butterfly" form, then return to its ball shape.
"An Account of a New Land Map of the World" (The Scottish Geographical Magazine, 1909–09) pp. 449–469 [reproduced in 21 jpegs] The first publication and exposition of the Butterfly Map.
"Map of the World" (U.S. patent 1,054,276, 1913) Washington, DC: United States Patent Office, 1913-02-25; filed 1912-03-05
"Geographic Globe" (US Patent 1081207, 1913: rubber-ball globe which can flatten to a Butterfly Map, or return to ball shape.)