Shadows
The
Troll Calendar is designed to predict the spring equinox (for
simplicity, the following description assumes the Troll Calendar is
located north of the equator). The spring equinox occurs when there is
a full stack of nine tiles and the shadows of all the tiles align with
each other.
There are four other times in the year where there is a tower of nine tiles, but at these times, the shadows don't align:
In January, the sun's position is in the southern sky, so the shadows extend to the north of the tower.
In
June and August, the sun's position is north of where it is at the
equinox, so the northern edges of the tiles are illuminated.
In October, the sun as descended again to the south, putting the northern face in shadow again.
On
the spring equinox, the shadows align on the north edge of the tiles
all day long. Here is a sequence showing the shadows at 8AM, 10AM,
noon, 2PM, and 4PM.
This
is accomplished by the shape of the tiles used in the Towers of Hanoi
puzzles. Each tile is wider by some number, deltaW, than the next
smaller sized tile. Additionally, all tiles have the same thickness, T.
The tangent of the angle of Latitude is the ratio deltaW/(2*T). If you
would like to construct your own, just cut the tiles so that the width
of each tile is 2*T*tan(Latitude) wider than the next smallest tile.