In a room in Pembroke College, Cambridge, hangs a round picture in a square frame.
On another wall of the room I found a similarly curious picture:
(We were able to stay for a couple of nights in this room because I was once a student at this college.)
The first picture seems to be of a green lawn surrounded by buildings and then by sky; the other, its converse, sky surrounded by buildings and then by grass.
It took a little studying to be sure that the buildings are those surrounding the Old Court of the college. I could discern the great hall, the chapel, and other features. The lawn, closely mown in the fashion of ancient colleges, is in fact square; its corners are obtusified by the perspective. The pictures are of the same space, looking down and up, vertically. In one, the center is the nadir, the point below you; in the other, the view is straight up, to the zenith, or point overhead. In both, the field of view is more than 90 degrees in radius (more than 180 degrees in diameter). Here’s my crude cross-section of the upward view:
For the downward view, flip the blue outline.
The views are like those of a super-fisheye camera, taking in a bit more than a hemisphere. If the field of view were only 180 degrees, the upward-looking eye would see the upper parts of the buildings but not the lowest parts or any of the grass. (With the peripheral vision of both eyes included, I have a slightly more than 180° field.)
The pictures bore only a small indecipherable monogram. I was later able to find out that they, together with similar paired pictures of the Ivy Court and the Library Lawn, had been offered to the college in 2007 by Edward Hill, and I was amazed to find that he is only yards from where I at present live, and I have yet to visit his gallery. Also to ask him whether he based his drawings of the buildings on photography or sketching.
I was interested in the pictures because they reminded me of the projections I use for plotting astronomical charts. Here is one way of showing the sky,
centered on the zenith for a certain place and time (latitude 45 north, sidereal hour 6 to put Orion on the meridian, though it doesn’t really matter).
The chart’s radius is 90 degrees from the zenith to the horizon, with another 10 degrees to show “grass” on the planet. It’s simplified in that I’ve shown only few stars and no Milky Way or ecliptic or other details, but have included the constellation boundaries because they serve to give you a feel of how shapes get transformed by the mapping projection one chooses to use. I want to remark on mapping projections, but they need an irreducible minimum of explanation and I will do so in a further post.
The chart and the “up” picture of the college courtyard correspond: the one could have old buildings around its circumference, the other could have constellations instead of clouds across the middle. And with the same projection, centered on the nadir instead of the zenith, one could map the starry sky as a band around a shrunken Earth.
Indeed you’ve seen pictures like this. Some (I’ve seen charming elaborate examples though I can’t remember where) are of gardens and other scenery on which the viewer is looking down from a modest height, and the landscape spreads out with foreshortening, perhaps to an encircling sea and then the encircling horizon. But others are photographs from nearby space, of our whole planet. All are like charts, from increasing height, centered on the nadir.
And then there are Escher’s woodcuts of “planetoids.”