Definition:Dipper Relation/Illustration

Illustration of Dipper Relation

When the stars of the Big Dipper are numbered as shown, the sequence:

$1, 1 +_{3, 4} 1, 1 +_{3, 4} 1 +_{3, 4} 1, \ldots$

traces out those stars in the order:

first the handle: $\text{Alkaid}, \text{Mizar}, \text{Alioth}$

then:

round the pan indefinitely: $\text{Megrez}, \text{Dubhe}, \text{Merak}, \text{Phecda}, \text{Megrez}, \ldots$

Hence $x \mathrel {\RR_{m, n} } y$ can be interpreted as:

Start at $\text{Alkaid}$ and count $x$ stars along the handle and then clockwise round the pan.

Then start at $\text{Alkaid}$ again and count $y$ stars along the handle and then clockwise round the pan.

$x \mathrel {\RR_{m, n} } y$ if and only if you end up at the same star.