Non-Euclidean Chess: Part 1

Most traditional grid-based board games are played in Euclidean space. This is straightforward and easy to understand but topologically boring.

Why not spruce up the classic grid-based game of chess by modifying it to be played on a non-Euclidean board?

The simplest way to transform a standard board into a non-Euclidean board is to connect the sides of the board together. Connecting the squares on adjacent sides to each other creates a board with an inherently spherical topology. In contrast, connecting the squares on opposite sides can lead to a board with a toroidal structure. A twist in the way the squares are connected can lead to a board with the topology of a klein bottle.

chess   torus1


Connecting opposite sides of a chess board (in the diagram above, red sides connect to blue sides) leads to a board with a toroidal structure.

While these macro structures are interesting, they are not clearly well suited for chess. In particular, none of these boards have edges. Thus, it is impossible for a Pawn to reach the edge of the board and become a Queen. Actually, for all we know chess has always been played on boards with these macrostructures, and we just “walled off” a section that was locally equivalent to a subsection of a Euclidean space.

How can we create a non-Euclidean board that does not lose the edge constraints of the original Chess board? One way is to connect random squares to each other by the edges. We will refer to these kinds of connections as “chutes” in homage to a classic board game with these kinds of connections – Chutes and Ladders. Chutes will be shown as blue lines in the diagrams below.

The side of the tile going into the “top” of a chute connects to the side of the tile coming out of the “bottom” of that chute.


For example, the green “pawn”, above, moves forward one square to the red position. Notice that the pawn is now moving sideways, and can be Queen-ed on its next move. This variant of non-Euclidean chess is best played with pieces with some obvious indication of directionality on them, so that you don’t lose track of their traveling directions.

In order for the board to be connected, we map the tiles that come out of the tops and bottoms of each chute and ladder to each other.


The green piece above ends up where the red piece is when it moves forward one space.

At this point, it is pretty clear how to travel across the sides of the tiles, but what it means to travel diagonally, like a bishop, is less well defined. We will use a “slide-left” metric* to define diagonal moves.

Consider any diagonal move. It can be represented as moving forward one space in one of the four cardinal directions and then sliding to the left one space repeatedly until the destination is reached.


By describing each diagonal move as a series of horizontal and vertical moves, the diagonal movement of pieces through chutes becomes clear.


In the figure above, the green piece begins a diagonal move by moving 1 forward through a chute, then sliding left. The diagonal move can continue until it hits the edge of the board and can no longer complete a “forward one, left one” series.

Note that knights still get to choose whether they move over two and up one or vice versa, and so may easily have more valid destinations than normal.

It is notable that this sort of non-Euclidean chess board is rather unlikely to be fair if you use randomly chosen chutes (perhaps an advantage if you want to handicap players of uneven ability). Can you design a chess board with a single chute such that White as a mate in one?

There are many other fascinating games that you can try playing along similar lines. In one variant of this game, players take turns moving, then adding a new chute to the board. The second post in this series will explore playing chess on a hyperbolic checkerboard (an idea first mentioned to me by Roice Nelson). Are there any other interesting boards that you can imagine playing chess on?

* As David Dalrymple and other people with a better ability to not confuse terminology than me might note: “slide-left” is not a “metric”. It is a homomorphism from the free group on {(1,0),(1,1),(0,1),(-1,1)} to the free group on {(1,0),(0,1)} which preserves actions on the square lattice.

3Doodling with my 3Doodler

Like tens of thousands of other people imagining epic 3-dimensional doodling abilities, I purchased a 3Doodler on Kickstarter. Last weekend, I decided to sit down and actually try doodling something cool.


The first problem that I discovered when 3Doodling is that it really isn’t very easy to doodle accurately in the air. The plastic takes just enough time to dry that you still have to deal with sag. On the other hand, the plastic does detach easily from paper. People seem to mostly suggest tracing 2-D templates and then putting the pieces together.

My 3Doodled piece was made out of 20 interwoven 3-pronged pieces that were doodled onto a flat 2D template that I sketched out quickly beforehand. It is based off of the Medial Triambic Icosahedron. The red outline in my template below is representative of the full face of the medial triambic icosahedron. The blue line is the template I traced. This is a portion of the face selected such that the faces would spiral around each other rather than intersect in my final model.


The second problem that is that (at least on my device) the feed mechanism is flaky and plastic comes out in spurts followed by nothing, which makes drawing accurately even on paper rather difficult. This problem makes drawing in air seem pretty impossible. My device also has a relatively minor issue of the top “fast” button having a tendency to catch and get stuck in the “on” position.

You can see the resultant unevenness in this close-up of one vertex where some of the lines are notably thinner or blobbier than others.


The last issue is that the 3Doodler is an ergonomic nightmare. Holding the 3Doddler for several hours while pressing the “extrude” button with my thumb gave me sufficiently severe thumb and wrist pain that I am unlikely to do much 3Doodling in the future.

All of this is unfortunate because despite it’s issues, the 3Doodler is really quite nifty and fun to play with. If I was able to use it without wrist pain, I would be seriously considering how to purchase more plastic (as you can see, I actually ran out of red towards the end of my model and started connecting corners with black). There is definitely something rewarding about drawing something very solid and physical, and as far as end results go, I was actually reasonably pleased with my doodle.

As I’ve mentioned before, a lot of models like this one cast impressive shadows. Here are some pictures of this model’s shadow.


Shortbraid and other geometric cookies

What happens when you combine three recreational math artists with a fantastic pastry chef?

Last weekend, Vi Hart, Gwen FisherRuth Fisher and I got together to try this experiment. Ruth provided several colors of shortbread dough and white chocolate “glue” and a few assorted cookie cutters. Gwen took photos (shown with permission below) and Vi took a bunch of video.

Our first idea was to make polyhedra cookies. The hexagon cookie cutter seemed to suggest truncated tetrahedra, octahedra, and icosahedra. We tried making one of each.


Assembly was a bit tricky and required the use of an assortment of cardboard jigs to keep pieces in place while the chocolate dried.

cookies3  cookies1

While our hexagon cookies were baking, Vi got started making a rhombic dodecahedron. We used a straight-edge to cut rhombi with the correct dimensions.


cookies9  cookies4

She wanted to have little cut-out holes in her polyhedron, and used a rhombic cookie cutter that happened to have just the correct dimensions to tile the plane nicely.

cookies8  cookies6

Which, of course, got us thinking about more interesting geometric tilings. For example, non-periodic Penrose tilings. Everyone likes the kite dart tiling and we made ourselves some cookie cutters out of card stock to generate kites and darts quickly.


A real math cookie party isn’t complete with out some fractals (here we have some Sierpinski Tetrahedra)…

cookies10and some shortbread “shortbraids”. The three rings here form an 18-crossing Brunnian link.