10 - Diagonal Loop

Continue with the previous article.

Now, we have already seen latitudinal loop and longitudinal loop of rook on the Dot. How about diagonal loop of bishop on the Dot?

Let’s figure it out. One step a time.

In this part, I have to show the route on 2DFARCSOD first. And in order to show it as clear as possible, I put magenta lines on the charts to visualize the routes.

Step 1 of 3: If a piece, e.g., a bishop, on square d2 on the Dot and moves along diagonal toward North-East, the route on 2DFARCSOD will look like this:

Bishop starts its journey by moving toward North-East.

Step 2 of 3: According to the chess rule, “A bishop always moves on squares with the same color”. Therefore, after the bishop passes the North pole, it should still be on a black square and moves toward South, which means it can only move toward South-West. Now the route looks like this:

After passes the North pole, the bishop moves toward South-West.

Step 3 of 3: According to the previously mentioned rule, after the bishop passes the South pole, it should still be on a black square and moves toward North, which means it can only move toward North-East, and that allows the bishop back to its starting square d2. Now the route looks like this:

Bishop moves along diagonal on the Dot

So, the bishop starts from d2 and then passes e3, f4, g5, h6, a7, b8, N, f8, e7, d6, c5, b4, a3, h2, g1, S, c1 and eventually moves back to d2.

And here is the single-chessboard view:

Bishop moves along diagonal on the Dot

And here is the diagram of diagonal loop on the Dot:

Bishop moves along diagonal on the Dot

The diagonal loop is some kind of spiral on the diagram. And this route keeps bishop always on the squares with the same color.

To be continued.