01 - Grettings

My name is Joshua Chao.

This is an essay to detailedly elaborate the working principles of “Dot”, “Globe Chess” and “Chess on the Dot”, which are the chess variants with traditional chess rules but on the spherical chessboard.

A wooden model of "Chess on the Dot", designed and crafted by Joshua Chao.

This is an essay to detailedly elaborate the working principles of “Dot”, “Globe Chess” and “Chess on the Dot”, which are the chess variants with traditional chess rules but on the spherical chessboard.

How is it possible to play chess on a spherical chessboard with the traditional rules? Well, I found it’s not only possible but astonishingly suitable.

To be continued.

02 - Why

Continue with the previous article.

As you can see in Wikipedia “Chess variant” entry, there are plenty of chess variants all over the world. People use different boards, pieces, rules or different number of players to create lots of funny new games.

When I saw these chess variants, I asked myself: “If I’m going to make my own chess variant now, why do I do that after all? Is there any good reason to change such a perfect, classic board game?”

Soon I realized that when Hindus created “Chaturanga”—the origin of chess-like board games—in the sixth century, maybe their intention was to reconstruct an epic battle of armies (“Chaturanga” means “army” in ancient Hindu epic poetry) by a game on a flat board. And the most likely reason of why they used a flat board is because they believe the world is flat at that time (see here), and surely also for game-play convenience.

And here I found a pretty good reason: Nowadays we know the world is in fact an orb, not flat anymore. So how about play chess on a globe?

When I say “globe”, I mean an absolutely spherical form, like a ball. And then I made this wooden model:

An absolutely spherical chessboard. It can roll like a ball.

It’s not a cylinder or something similar to a ball. It’s a perfect sphere with no borders on it, just like our planet Earth.

That’s a good reason for me to make a chess variant now.

To be continued.

03 - How

Continue with the previous article.

In order to call it briefly, I gave this spherical chessboard a short name, “Dot”.

It takes three steps to transmogrify a traditional, flat chessboard to a Dot:
1) Connect east and west borders, to create a tube.
2) Cover north and south ends, to create a cylinder.
3) Squeeze and bulk it up to a sphere.

This process is similar to transform a Mercator-projection world map into a terrestrial globe, just like the animation below:

The chessboard-Dot transmogrification animation

So, there are still 64 squares, with two extra “ends” on the surface of Dot.

The two ends are labeled “N” and “S”, they are two poles on the Dot. They are not squares, neither are they white nor black (so I use middle gray to color them). They exist to intact the surface of Dot, and keep all squares four-sided.

Since the two poles are not squares, an important principle rises: Pieces must not stay on the poles.

And as long as the poles are meant to intact the surface of Dot, another important principle rises: Pieces may pass the poles when they move.

By the way, the origin of the name “Dot” is from a photograph of planet Earth taken in 1990 by the Voyager 1 space-probe from a record distance of about 3.7 billion miles away from Earth, which is called “Pale Blue Dot”.

To be continued.

04 - Diagram

Continue with the previous article.

Back to the previously mentioned question: “How is it possible to play chess on a spherical chessboard with the traditional rules?” To answer that, first I need to exhaustively display all 64 squares on the surface of Dot in a 2-dimensional diagram, so I can draw pictures to explain it well.

It takes two steps to make a diagram of Dot:
1) Divide the Dot into two hemispheres by splitting it from the equator.
2) Use “Azimuthal equidistant projection” to project them as two circles.

Make a diagram of Dot

In the diagram, the upper circle is the northern hemisphere and the lower circle is the southern hemisphere. The circumferences of two circles are connected with a single point on the diagram, but they are actually full-connected as a Dot (the equator of Dot). The animation below shows the relation of squares on two hemispheres more accurately.

The relation of squares on two hemispheres

In the future, consider the circles of all diagrams are rotatable, just like the gear-wheel animation above.

To be continued.

05 - Notation & Position

Continue with the previous article.

Remember how the Dot is transmogrified from the flat chessboard? If the notation of every single square remains during transmogrification, it will look like the diagram below:

Notation of squares on the chessboard and the Dot

The principles of notation on the chessboard is: “Ranks” are noted with numbers 1 to 8. “Files” are noted with letters a to h.

Likewise, the principles of notation on the Dot is: “Latitudes” are noted with numbers 1 to 8. “Longitudes” are noted with letters a to h.

The animation below shows the relation of noted squares on two hemispheres more accurately.

The relation of noted squares on two hemispheres

Now if I put all pieces in initial position on the chessboard and the Dot, it will look like the diagram below:

All pieces in initial position on the chessboard and the Dot

It’s interesting that each side of pieces on the Dot surround their poles; white pieces surround the south pole, and black pieces surround the north pole.

The animation below shows the relation of pieces in initial position on two hemispheres more accurately.

The relation of pieces in initial position on two hemispheres

For future reference, the 8 squares alongside a pole are jointly called as “polar circle”. This noun will be mentioned later when it comes to “castling” and “pawn promotion” rules.

To be continued.

06 - Essence

Continue with the previous article.

Before we get any further, I have to clarify the essence of the Dot.

Although the Dot is a 3-dimensional object, which has height, width and depth, however the surface of the Dot is just a 2-dimensional membrane, without borders.

How to define the dimension of an object? It depends on how many sets of value is needed to pinpoint a location on that object. See this article if you want to know more.

On a spherical surface, such as the surface of planet Earth, it takes two sets of value to pinpoint a location. For example, the coordinates of Eiffel Tower in Paris, France is (48°51’29.6”N, 2°17’40.2”E), and the coordinates of Great Pyramid of Giza in Egypt is (29°58’45.03”N, 31°08’03.69”E).

Likewise, it takes two sets of value to note a square on the chessboard or the Dot. For example, the square the white king occupies in initial position is (e, 1), and the square the black queen occupies in initial position is (d, 8). We simply use “e1” and “d8” for chess notation.

Since the surface of the Dot is essentially a 2-dimensional membrane just like the flat chessboard, it must can be displayed as a 2-dimensional and rectangular chart just like the flat chessboard, can it not?

Yes, it can. And I call it “2DFARCSOD”, which represents “2-Dimensional, Flat And Rectangular Chart of Squares On the Dot”.

To be continued.

07 - 2DFARCSOD

Continue with the previous article.

“2DFARCSOD”, which represents “2-Dimensional, Flat And Rectangular Chart of Squares On the Dot”.

Before I start to make a 2DFARCSOD, I need to put some colors on the chessboard to show the process as clear as possible.

My sincere apology to people with color vision deficiency, but color is the best way I can ever imagine to make this chart easier to understand.

(Just for a quick memorize: North is red because it’s crowded. South is blue because it’s iced. West is green because it’s scientific. East is purple because it’s mysterious.)

A colored chessboard

It takes seven steps to make a 2DFARCSOD:
1) Replicate infinite times of chessboard and make them all West-East connected, to create a stream of chessboards. We call it “stream-alpha”.
2) Duplicate the “stream-alpha”.
3) Perform horizontal reflection to a “stream-alpha”, to make it only North-South reversed but not East-West reversed. We call it “stream-beta”.
4) Shift the “stream-beta” by 4 squares, to align rank a with e, b with f, c with g, and d with h.
5) Connect their South borders, to create a new stream. We call it “stream-gamma”.
6) Duplicate the “stream-gamma”, and connect their north borders.
7) Repeat step 6 infinite times, to eventually create a 2DFARCSOD, which is an infinite big, 2-dimensional chessboard continuum without border.

The animation of making a 2DFARCSOD

The relative positions between all squares on 2DFARCSOD are exactly the same as they are on the Dot.

The 2DFARCSOD is an infinite continuum because the surface of the Dot is border-less. The only way to make a border-less 2-dimensional chart is to make it infinite big.

By the way, you can use the same method to create a 2-dimensional, flat and rectangular chart of the surface of terrestrial globe:

2-dimensional, flat and rectangular chart of terrestrial globe

On the Dot, any piece moves along a single direction will find itself arriving at the very same square where it left from over and over again, because the piece is walking a loop. The same thing happens on the 2DFARCSOD.

To be continued.

08 - Latitudinal Loop

Continue with the previous article.

If an aircraft on the equator of planet Earth flies along latitude and keeps its route straight, it will round the equator and eventually back to its starting point. Likewise, if a piece, e.g., a rook, on square d5 on the Dot moves along latitude and keeps its route straight, it will pass e5, f5, g5, h5, a5, b5, c5 and eventually move back to d5.

Rook moves along latitude on the Dot

Now if I change the diagram above into the 2DFARCSOD below, it will look like this:

Rook moves along latitude on the Dot

And here is the single-chessboard view:

Rook moves along latitude on the Dot

In case of it’s not easy to notice, let me point it out that all the rooks on the 2DFARCSOD above are actually the same one, because all chessboards are in fact replicas from the single, original one.

To be continued.

09 - Longitudinal Loop

Continue with the previous article.

If an aircraft on planet Earth flies along longitude and keeps its route straight, it will reach a pole, and then another pole, and then eventually back to its starting point.

Likewise, if a piece, e.g., a rook, on square d7 on the Dot moves along longitude and keeps its route straight, it will pass d8, N, h8, h7, h6, h5, h4, h3, h2, h1, S, d1, d2, d3, d4, d5, d6 and eventually move back to d7.

Rook moves along longitude on the Dot

Now if I change the diagram above into the 2DFARCSOD below, it will look like this:

Rook moves along longitude on the Dot

Like previously mentioned, all rooks on the 2DFARCSOD above are actually the same one, because all chessboards are in fact replicas from the single, original one.

And here is the single-chessboard view:

Rook moves along longitude on the Dot

Due to the “North-South reversed” of some chessboards, the moving direction of piece is also North-South reversed when it pass the pole.

To be continued.

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.

11 - Answer

Continue with the previous article.

Finally, it’s time to answer the previously mentioned question: How is it possible to play chess on a spherical chessboard with the traditional rules?

And here is the answer: According to explanation above, the essence of the surface of the spherical chessboard (a.k.a. the Dot) can be considered as 2-dimensional, flat and rectangular. So all rules of chess can perfectly apply to it without change.

The total 20 loops on the Dot, in single-chessboard view.

To be continued.

12 - Rules

Continue with the previous article.

The following rules in texts are exactly the same to traditional chess rules, and the diagrams are the implementations of traditional chess rules on the Dot.

On the diagrams, little black dots on the squares mark the piece’s destinations. Lines between piece and dots mark all routes from piece to its destinations.

Rook: The rook can move any number of squares along any rank (latitude) or file (longitude), but may not leap over other pieces. It moves the same way when capture an opponent’s piece.

Rook's move and capture on the Dot

Bishop: The bishop can move any number of squares along any diagonal, but may not leap over other pieces. It moves the same way when capture an opponent’s piece.

Bishop's move and capture on the Dot

Queen: The queen can move any number of squares along any rank (latitude), file (longitude) or diagonal, but may not leap over other pieces. It moves the same way when capture an opponent’s piece.

Queen's move and capture on the Dot

King: The king can move only one square along any rank (latitude), file (longitude) or diagonal. It moves the same way when capture an opponent’s piece.

King's move and capture on the Dot

Knight: The knight can move to any of the closest squares that are not on the same rank, file, or diagonal, thus the move forms an “L-shape”: two squares along rank (latitude) and one square along file (longitude), or two squares along rank (latitude) and one square along file (longitude). The knight is the only piece that can leap over other pieces. It moves the same way when capture an opponent’s piece.

Knight's move and capture on the Dot

Pawn: The pawn may move forward (approach to opponent’s pole) to the unoccupied square immediately in front of it on the same file (longitude), or on its first move it may advance two squares along the same file (longitude) provided both squares are unoccupied. The pawn may not move backwards (away from opponent’s pole).

Pawn's move on the Dot

The pawn may capture an opponent’s piece on a square diagonally in front (approach to opponent’s pole) of it on an adjacent file (longitude), by moving to that square. The pawn may not move to these squares if they are vacant.

Pawn's capture on the Dot

Castling: This consists of moving the king two squares along the “polar circle” toward a rook and then placing the rook on the last square the king has just crossed.

Castling on the Dot

En passant: A special pawn capture which can occur immediately after a player makes a double-step move from its starting position, and an opponent’s pawn could have captured it had the pawn moved only one square forward (approach to opponent’s pole).

En passant on the Dot

Promotion: The transformation of a pawn that reaches its opponent’s “polar circle” into the player’s choice of a queen, knight, rook, or bishop of the same color. Every pawn that reaches its eighth rank (opponent’s polar circle) must be promoted.

Promotion on the Dot

To sun up, the only difference between “Chess” and “Chess on the Dot ” is the topological shape of the chessboard. A chess player can play “Chess on the Dot” without learning any new rules.

The End