Contributed by
Arturo Magidin
This starts as an alternate history short story, in which Lord Halifax became Prime Minister of England in 1940 and reaches an accommodation with Germany; Germany holds sway over Europe and Russia, Japan over Asia, and the U.S. never entered the war. As the story progresses, however, it becomes apparent that things are very different: nobody has ever crossed the Pacific. The story takes place in 1950, as an experimental aircraft that can travel for several weeks without stopping starts a journey to try the crossing. Though they travel many times the (estimated) distance from the Earth to the Moon, they do not seem to be able to cross the Pacific.
Towards the end, the narrator character suggests that the Earth's geometry is hyperbolic, so that Earth's radius is finite, but has an infinite surface area, and thus the Pacific cannot be crossed; there is a "Fold" in the middle of it, and as the characters get closer to the fold, they grow "smaller", so that the finite radius takes longer and longer to traverse.

First published in "The Mammoth Book of Extreme SF" edited by Mike Ashley.
Also appears in "The Year's Best Science Fiction: Twenty Fourth Annual Collection" edited by Gardner Dozois. Contributed by
David Lepus
I think that math was not a main concept of the story until the end of the story. The mystery and its possible solutions was the driving point of the story. Mathematics expressed as geography or geology is disturbing to this reader. Searching for discussions about the math involved in this story brought this reader to your wonderful website.
There was a brief concept introduced in the story about how the alternate Earth still casts a circular shadow on the Moon. I have searched for a geometric object which would represent this Earth. Would the Torricelli's Trumpet be a good example?

David, I think the object you should view as a "good example" here is Henri PoincarĂ©'s Disc Model of hyperbolic space. Here's how you can think of it: In usual Euclidean plane geometry, if you draw a point next to a line, then there is only one line through that point which is parallel to the first (i.e. lies in the same infinite plane but never meets it). However, it was realized in the 19th century that there were spaces for which this was not true...so called "nonEuclidean geometries". In the case of hyperbolic space, for example, there are MANY lines through that point which never meet the given line. Of the various models of hyperbolic space, the disc model is perhaps the easiest to imagine. Beginning with a disc (just a filled circle), consider it to be a space unto itself. Suppose, for instance, that there are beings inhabiting it. However, to them it appears to be infinitely large. This is because as they approach the boundary of the disc, they get smaller. Consequently, no matter how many steps they take towards the boundary, they will never reach it...just getting smaller and smaller and taking smaller and smaller steps instead. If these beings tried to draw what they considered to be a straight line, it would look like the arc of a circle to us looking down on the disc from our world. This is because their sense of measurement is affected by the sense that they shrink as they near the boundary.
As far as I can tell, the intent in Pacific Mystery is to imagine a similar kind of shrinking. However, instead of occurring at the circular boundary of a disk, it occurs along a longitude line in the middle of the pacific. No matter how long you travel towards it, you can never reach or cross it because the metric of space (the way lengths are measured at each point) changes so that the distance appears infinite. I believe the remark about the circular shadow cast on the moon is supposed to illustrate that  like the disc model  the Earth still appears to be a sphere when viewed from outside using a different sort of measurement, the metric of space in the universe which is what the light travelling past the Earth follows.
This may all sound ridiculously abstract, but as far as we can tell, the "metric" of space does really change from one point to the next. This is the basis of the general theory of relativity, and our usual name for the effect of this changing metric is "gravity".
Frequent contributor Vijay Fafat has written with a note about a similar work that is not quite mathematical enough to garner its own entry on this website:
Contributed by
Vijay Fafat
Rene Daumal  Mount Analogue, A Tale of NonEuclidean and Symbolically Authentic Mountaineering Adventures
This incomplete book by Daumal is considered to be a minor classic, symbolic of a man's desire to reach for the unattainable (I have to admit it didn't carry any such force for me, though the writing is lyrical at times). It describes the journey of a group of people who have postulated the existence of Mount Analogue, an enormously tall mountain on the surface of the earth, located at the antipode of the CG of earth's surface land mass...somewhere near New Zealand....It has not been visible till now because a shell of curved space around it shields it from outside view. This makes it take on a hyperbolic geometry, making its peak inaccessible. They find a way to reach the base of the mountain and begin a steep climb.
This is where the novel ends, with a hanging coma. Evidently, Daumal was interrupted by a visitor while writing the novel in 1944 and he left the book midsentence. After that, he was never well enough to pick up a pen and continue writing (Shades of the story, "Person from Porlock" in real life!)
A couple of striking lines in the novel:
"Mount Analogue  Its highest summit touches the sphere of eternity, and its base branches out in manifold foothills into the world of mortals. Its summit must be inaccessible but its base must be accessible to human beings, as nature had made them. The gateway to the invisible must be visible..."
"In the beginnning, the sphere and the tetrahedron were united in a single, unthinkable, unimaginable form."
ecountrib

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