Can you find a continuous one-to-one function from the circle to the real line?

Put it another way: is it possible, for every longitude on the earth, to assign a time so that each longitude has a *different* time, but the times at nearby longitudes are close?

In fact, no. Any attempt to construct such a function will inevitably fail. This explains why the world has an “international date line”: assigning time to geographical location is a function from a circle (longitude) to an interval (time).

By convention, we normally assign times in discrete chunks (time zones), but the idea is the same. Our current method says that if it is 10 AM on Wednesday in Claremont, then it is 11 AM in Denver, and 12 noon in Minneapolis. Continuing around the globe in this manner, one finds that to keep nearby points having nearby times, one would have to assign Claremont a different time as well— 10AM on THURSDAY! The only other alternative is to give up the continuity in favor of one-to-oneness, and put the discontinuity along a longitude of the earth that would affect the fewest people, i.e., somewhere in the Pacific, and call the discontinuity a “date line”.

**Presentation Suggestions:**

Do the presentation above (adjusted to your locale). Then read from the diary of Magellan’s circumnavigation of the globe about returning to Europe (See Winfree, p.11):

- The 18 survivors of Magellan’s expedition around the world were the first to present this dilemma for the bewilderment of all Europe. After three years westward sailing, they first made contact with European civilization again on Wednesday 9 July, 1522 by ship’s log. But in Europe it was already Thursday! Pigafetta writes (translated in The First Voyage Around the World, Hakluyt Society, vol. 52, 1874, p. 161):

“In order to see whether we had kept an exact account of the days, we charged those who went ashore to ask what day of the week it was, and they were told by the Portuguese inhabitants of the island that it was Thursday, which was a great cause of wondering to us, since with us it was only Wednesday. We could not persuade ourselves that we were mistaken; and I was more surprised than the others, since having always been in good health, I had every day, without intermission, written down the day that was current.”

**The Math Behind the Fact:**

The fact that there does not exist any continuous one-to-one function from the circle onto the interval follows from the Borsuk-Ulam Theorem in dimension 1. Topologists often study 1-1 and onto functions which are continuous in both directions; such functions are called homeomorphisms and yield an equivalence relation for objects in topology.

**How to Cite this Page:**

Su, Francis E., et al. “Why an International Date Line?.” *Math Fun Facts*. <https://www.math.hmc.edu/funfacts>.

**References:**

A.T. Winfree, The Geometry of Biological Time, Springer-Verlag, NY, 1990.

**Fun Fact suggested by: **

Thomas LoFaro