Game

A chiralkine game can be played on game board divided into 8 x 8 token spaces.

Each token space can be owned by neither player, owned exclusively by one player (not the other) or owned jointly by both players. The object of the game is to be the first to secure ownership of a chain of token spaces linking opposed sides of the game board. Players compete by deploying tokens to change the ownership states in the token spaces.

Each token in the game corresponds with one of the six quaternions coding a face in a cube. Accordingly there are six tokens. A white token with a green centre indicates that a token space is jointly owned (+ + – -), i.e. a + + or L state. A black token with a green centre indicates that a token space is owned by neither player (- – + +), i.e. a – – or D state. A black token with a yellow or blue centre indicates a token space that is owned by the player of that colour, not the player of the other colour. It is a + – or C state.  A white token with a yellow or blue centre indicates a token space that is owned by the player of the other colour, not the player of that colour. It is a + – or A state.  The six quaternions are thus of four kinds of relationship state.

Each player starts with a set of C and A tokens of the same colour (yellow or blue). Thus, the tokens played by yellow player are in states C (+ – – +) and A (- + – +), while those for blue player are in states C (+ – + -) and A (- + + -). In a move, a player deploys one of each kind of token. A move is analogous to posting a debit entry and an opposite credit entry in accounting. Deployment of a token in a token space changes the state in that token space, depending on the starting state. An A token played on a C token changes the state into L, but changes a D or L state into A (itself). A C token played on an A token changes the state into D, but changes a D or L state into C (itself) [C turns A into D and A turns C into L]. Thus one part of the player’s move is selfish (for the player’s benefit) and one part is altruistic (for the opponent’s benefit). However, when a player uses an A token to change her opponent’s C token to an L token, the effect is to change the ownership state from exclusively the opponent’s to joint ownership. It is analogous to forcing an opponent to share ownership of something, like when a government collects taxes to invest in public services or nationalizes an industry. Players quickly learn to use the “altruistic” part of their move (playing an A token) to convert their opponent’s C spaces into shared, L spaces. Figuring out a way to counter this led to the discovery of another feature of the six quaternions that may have applications in the development of artificial intelligence.

When players look at the game board, they see the six different kinds of relationship states, for example as shown below. It is possible for two other people to play another separate, but connected game by interpreting the functions of the tokens differently. I call these other people hunters, because their objective is to hunt down L tokens and convert them directly into D tokens (a state change that the players cannot effect in one move – it corresponds with jumping directly from one local side of a Mobius strip to the other). The hunters cannot see the yellow and blue tokens.

In an electronic version, the players and hunters would see different displays on different screens. As the players convert C states into L states, the hunters simply see L states appearing. When the hunters capture and convert L states into D states, the players simply see L states turning into D states. The effect of this conversion is symmetric on the players in that the state change is from both own to neither own a token space. The game for the hunters can be made more interesting by providing that the A and C states present an obstacle to their movement. To them, the world is then in three states, like +1 (unblocked), 0 (prey) and -1 (blocked), which is how we teach our children to see it. There appears to be a parallel here between the workings of the conscious and unconscious mind.

In the game, the A and C tokens can be deployed using musculature on the left and right sides of the body, for example the left and right hands. In an electronic version of the game, players could affect state changes using a game controller having buttons adapted to receive inputs from the two hands. It could also be played using a neural headset positioned to pick up when a player visualises contracting either or both of the left and right hands, or not. (It is already known from LaFleur et. al., that a quadcopter can be controlled in flight in two dimensions simply through mental imagery of clenching the left and/or right hands).

The skeletal musculature, which controls rotation about joints, is organised into antagonist pairs, the members of which are known as the flexor and extensor. There are four basic states: both contracting (+ +), neither contracting (- -), flexor, not extensor contracting (+ -) and extensor, not flexor contracting (- +). These states are analogous to the ownership states denoted by tokens in token spaces of the game. Each of us can imagine contracting our own muscles (which can be detected using a neural headset) or those of another person (which presumably could also be detected using a neural headset). It would thus seem plausible that a person could code all six relationship states (equivalent to motion in 3D) using mental imagery of contracting their own or another person’s left and/or right muscles. For example, the states could be right, not left, me, not you (+ – + -); right, not left, not me, you (+ – – +); not right, left, me, you (- + + -); not right, left, not me, you (- + – +); right, left (+ + – -) and not right, not left (- – + +). Presumably the limbs of a robot could be controlled in this way as well. Thus, a dynamic link can be made between the co-ordination of movement of the body by the left and right skeletal musculature and states of ownership (property rights).

Playing the game forces the two sides of the brain to co-operate when planning strategy. I imagine that this could be exploited to assist patients who have suffered from brain injury or stroke to recover, by using the healthy side of the brain to support restructuring of the damaged side. The brain is plastic, so lost functionality can eventually be assigned to healthy tissue. I would very much like to see the game coded, so that this idea can be tested, for example using a brain scanner. There is good precedent for using virtual reality games in the treatment of stroke.

The connectivity that the game brings between the two sides of the brain can be appreciated by performing the state changes using the left and right hands to rotate a cube as shown below. I call it a chiralkine clock.