Summary

Chiralkine is a cyclic system for performing computation in 1s and 0s that is constructed logically on a new principle. It is internally consistent and solves a hitherto unrecognised problem inherent in computation performed on the principle of balance – the principle upon which all computation including artificial intelligence (AI) is presently being performed. It exploits bilateral symmetry in rings having three planes of symmetry said rings being of formula aBcAbC in which a, b and c are 0 and A, B and C are 1. It treats 1 and 0 mirror symmetrically as if they are two mutually exclusive sides of a unitary bilaterally symmetric relationship between opponents (1|0), like the mutually exclusive left and right sides of a human body (right|left) and our relationship (you|me).

It works relationally by drawing and clearing distinctions between opponents in ordered pairs of 1s and 0s, treating as mirror-opposites 01 and 10 and also 00 and 11 (01|10 ↔ 00|11) through rotation of a duplicated ring that acts as a unifying boundary separating the two mirror-opposite sides. The lower and upper case letters in the rings do not each map to one opponent, but instead the distinction between them maps to a distinction between opponents. Being duplicated the boundary possesses symmetry opposite to that of the two sides it is separating, but is itself mirror symmetric across another plane of symmetry.

Chiralkine is useful for performing computation where conservation of mirror symmetry between the two sides of a relationship between opponents is required throughout the drawing and clearing of distinctions in ordered pairs of 1s and 0s (01|10 ↔ 00|11).

Clearing a distinction between the opponents of one pair also draws a distinction between another pair, so for example the clearing of the distinction between left and right can draw a distinction between above and below or front and back, corresponding with a rotation in 3D space.

This relativism can be disturbing if you first meet it thinking only in absolutes. The logical construction applies across all opponent pairs. Rotations connect the two sides of an opponent pair such that they can be cleared in favour of another pair, even distinct and same or unitary and divided. However, it is important to keep in mind that the objective of chiralkine is to offer a system that ensures that the “you” and “me” in a system are treated mirror symmetrically by computation, not to privilege relativism over absolutism.

A conscious observer can observe only one side of a distinction at a time. The unobserved side nevertheless remains an inseparable mirror-opposite within the same relationship. Every distinction exists only as part of a temporally ordered cycle of drawn and cleared distinctions, so its meaning depends on what preceded it and what follows it. For example, if we see blue, yellow is the unseen opponent. Blue and yellow derive their meanings from their relative positions within a colour wheel of distinctions. Because consciousness presents only one side of a distinction at a time, we are inherently biased towards one side of a mirror-opposite pair. For example, most people are right-handed and there is a long-standing cultural prejudice against left (sinister) in relation to right (dexter). This may well explain why privileging one side over another in counting operations has become accepted, but the right and left sides of the body are one unitary body: one side does not keep growing at the expense of the other. You and me are mirror symmetric in our relationship, so counting operations must treat us accordingly. A valid counting operation should preserve the symmetry of the relationship it represents.

Chiralkine solves a hitherto unrecognised problem with counting operations computed on the principle of balance: such counting operations are incomplete, which allows freedom for one of the two sides of a distinction defining a relationship (you|me) to be privileged over the other by counting one side independently of the other. Chiralkine removes this freedom. Once the solution to the problem has been understood, it is not difficult to see privileging taking place in plain sight across all fields of human endeavour and the impact it is having on the integrity of relationships.

Discussion

A plane of symmetry divides a unitary, bilaterally symmetric object into two mutually exclusive sides.

Chiralkine is constructed from coupled rings possessing three planes of symmetry that can rotate about a central axis. Each ring is a regular skew hexagon of formula:

in which a, b and c represent 0 and A, B and C represent 1. The ring corresponds to six corners of a cube linked aBcAbC such that the remaining two opposed corners D and d define a rotation axis.

The ring possesses three planes of symmetry that in turn duplicate a different pair of letters (aA then Bb then cC) lying in a mirror plane when a ring is rotated stepwise from one mirror plane to the next:

Note that although the planes of symmetry in the regular skew hexagon are 60° apart, the links in a regular skew hexagon are orthogonal. 3D Cartesian number lines (+x, -x, +y, -y, +z and –z) also possess orthogonal links, but they connect with one another once, through one absolute zero (0) in accordance with working on the principle of balance.

The four letter chains containing duplicated letters that are in the process of being generated from triplets one after another in turn through symmetry breaking across the mirror planes are chiral (not superimposable on their mirror images). They are herein called chiralkines. “Chiral” comes from the Greek for hand, an object, and “kine”, as in kinetic comes from the Greek for movement, a process. Chiralkines possess object/process duality.

As the cube is rotated in order aBcAbC, the letters jump back and forth between the two planes, but, being ordered, it is as if they are all advancing or retreating in columns one step at a time towards or away, like a screw being turned clockwise or anticlockwise.

Put together they form a ring.

Each of the six faces of the cube corresponds with a different permutation of four letters, two in upper case and two in lower case. There are exactly six ways of ordering four different objects in a ring.

The letters oscillate upper case and lower case on each face, such that each face encodes 0101|1010. However, in the skew hexagon the faces are coded as if the letters are always in the same alphabetic order, so each of the cube faces is encoding a different permutation of two 1s and two 0s. Each face of the cube is a different cyclic permutation of the first four letters of the alphabet, but by following around the skew hexagon is transcribed so that the letters are ordered alphabetically (a or A, then b or B then c or C).

First recognise that the letters duplicated by symmetry breaking across successive planes of symmetry is the rotation axis d/D.

Next place the letters of each row in alphabetical order, and substitute the lower and upper case letters with the 0 and 1 that they encode.

Now look at the pattern of 1s and 0s in the table. It corresponds with interpenetrating truth tables for Boolean XOR (three 0s in a column) and its mirror opposite XNOR (three 1s in a column). Each row is in order a first input, a second input and an output in a truth table for XOR or XNOR. The order in which the adjacent inputs are taken matters: reversing the order inverts the output. It is noncommutative. These two truth tables provide mirror opposite logic for drawing and clearing distinctions: in XOR 0 means the same and 1 means different, while in XNOR 1 means the same and 0 means different. In the aA, bB and cC components of the three rows making up a truth table, two columns change sign and one does not.

Each row in the interpenetrating truth tables for XOR and XNOR is a chiralkine. The logic that is being used to draw and clear distinctions mirror symmetrically is a mirror symmetric combination of XOR, where 0 means same and 1 means distinct, and XNOR, where 1 means same and 0 means distinct.

The system works in three steps by drawing and clearing distinctions in two ordered pairs of 1s and 0s through coupled stepwise rotations of one ring that is duplicated through symmetry breaking relative to two other rings in mirror opposite configurations. The duplicated ring that rotates is a dynamic boundary (|) that separates two mirror-opposite sides (1 and 0). It provides a mechanism for tunnelling between mutually exclusive sides, coupling them together in a unitary system.

The system thus behaves as if three objects (left, right and boundary) are also four objects (two ordered pairs). The boundary that mediates this 3 x 4 behaviour is being generated by symmetry breaking. Note that although the ordered pairs consist of four of each upper and lower case letter, the symmetry breaking orders the upper and lower cases of letters in a three to one ratio.

Thus the boundary that is separating left from right possesses the opposite symmetry to left and right. It is not a line at which left and right are balanced. It is nevertheless mirror symmetric within the mirror plane.

The rings establish a first ordered pair, a second ordered pair and a third ordered pair, the first and third ordered pairs each sharing one of the duplicated rings of the second ordered pair.

As the pair of rings defining the boundary and second ordered pair is rotated through three steps, a distinction defined by the first ordered pair is cleared and that defined by the third ordered pair is drawn.

The system is therefore useful for coupling the clearing of distinctions defining one common bilaterally symmetric relationship (01|10 → 00|11) with the drawing of distinctions defining another (00|11 → 01|10), thereby enabling synchronised changes of state without privileging one side of a relationship over the other. Three steps are required instead of one, because each step only changes one of the three letters in a triplet.

It is instructive to look at counting first, because this provides a template for clearing all kinds of distinction quantitatively.

Chiralkine can be used to count a quantity (#) of objects (1, 2, 3, 4 . . . #). Each object is a relationship created by drawing distinction between what it is and is not. The quantities of “is” and “not” are the same, so there are (#) “is” and (#) “not”. Each object can be counted only once. So every object can be associated with two different numbers selected from (1, 2, 3, 4 . . . #), one for “is” and one for “not”. Counting then proceeds by linking objects pairwise through like numbers, 1 with 1, 2 with 2, 3 with 3, up to # with #, at which point a ring is formed. All of the objects are now associated with the same number for “is” as “not” and the distinction between counted and uncounted has been cleared.

Before counting starts and after it has been completed, the objects are indistinguishable. They are in a ring. During counting symmetry is broken, so objects are distinguishable in the sense that they can be connected once or twice. Chiralkine tracks this. Objects in a ring are all in mirrored states aA and AA. Objects with one link are in mirrored states ab and bA. Objects with two links are in mirrored states aC and CA. Counting is complete when mirrored states aA has switched to state aa, clearing the distinction between counted and uncounted, and mirrored state AA has switched to aA.

A quantity (#) consists of # objects, each defined by a distinction being drawn between what it is and is not: # is and # not, but the quantity (#) is also an object defined by a distinction being drawn between what it is and is not. Hence 4 at the end of a count 1, 2, 3, 4 is different from 4 during a count 1, 2, 3, 4, 5. The two 4s differ in symmetry. Quantity (#) behaves as frequency.

Counting quantities of objects performed on the principle of an equation is incomplete, because it does not conserve the relational symmetry inherent in the distinctions being drawn to define those objects. Therein lies a fundamental cause of problems now manifesting in the relationships between people and in their collective relationship with the planet – the natural world.

Nature is mirror symmetric, not balanced as in equations, and it operates in cycles.

When a quantity of objects is counted in accordance with the principle of mirror symmetry, each object is visited three times – singly linked, doubly linked then ring. This is not in accordance with what you observe when you move potatoes from one bucket to another counting one each time, the act of counting 1, 2, 3, 4, 5. It is not at all intuitive that each potato must be visited three times to complete the count instead of just once!

Over millennia mathematicians have developed a system for counting based upon the principle of balance – the equation. We have integrated this into every aspect of our lives: science, engineering, finance, voting and so on. It accords with what we observe when counting. However, to make this system work, we have to work in ordered pairs. This requires each object to be visited twice, corresponding with making a debit and credit entry in double entry bookkeeping. A potato in a bucket is coded 01 and the absence of a potato is coded 00. When the potato is subtracted from a bucket we post a 1 on the left side of 01 to give 11, which being balanced cancels down to 00. When the potato is added to the second bucket we post a 1 on the right side of 00 to afford 01. Thus the counting of one potato exchanges 01 and 00 (01 ↔ 00) and this requires two visits. The process is independent of the quantity (#) of potatoes being counted. It forms the logical basis for equations and arithmetic, which is a counting operation: 01 (+1) + 10 (-1) = 11 = 00 (0) and 00 (0) + 01 (+1) = 01 (+1) – a count of 1.

If you think deeply about it, a potato is a mental construct. A complete description of a potato (an object) is a relationship between what it is and what it is not that emerges from a distinction being drawn as another is being cleared by a process produced at a boundary.

But if you continue with that deep thinking, the quantity of potatoes to be counted is also an object defined by a relationship between what it is and is not.

In chiralkine the potato is treated as a trinity: a mirrored relationship between what it is and is not being separated by a boundary. The boundary must rotate three steps in order to exchange 01|10 and 00|11 (01|10 → 00|11). This requires three visits. The process is dependent of the quantity (#) of potatoes being counted, because symmetry is broken at the start of the count and is restored at the end. Thus, for example, 4 on a count up to the quantity 5 (#) is different from 4 at the end of a count up to 4 (#).

In this way the distinction between offers and wants can be cleared in an economy without requiring any form of currency (credit and debt created from nothing by double entry accounting posts).

People wanting to exchange goods or services they own for those that they want (01|10) can be coupled together into a circuit and thereby effect unitary transfer of ownerships such that what they own is what they want (00|11) without using any imaginary store of value (currency). During the process they link once to a person having what they want, once to a person wanting what they have to offer, and finally also to all at once when the offer and want ends of the chain link to form a ring. The people at the ends of the chain can find one another if the original link is given an identifier, such as a number, bar code or frequency. Online search engines with AI could be used to help guide people to identify chain ends and so speed up ring formation. Note that no currency is used in this system, and that all participants are treated mirror symmetrically.

Viewed from this perspective, double entry bookkeeping, used to solve the double coincidence of wants problem inherent in barter through the creation, exchange and redemption of money, can be seen to be incomplete counting, because it leaves free the ability to privilege positive (+1, credit, money) over negative (-1, debt). These opponents are properly two sides of a mirror symmetric distinction. Privileging one side over the other causes asymmetry to grow between traders, at the levels of individuals and nation States.

The way votes are processed can also be seen to be incomplete counting. Voting is intended to be a counting operation that enables voters to take a decision collectively, pooling the contributions of all their perspectives. However, the way we take collective decisions democratically is not the way nature (a real person as opposed to an artificially constructed legal entity) takes binary decisions through crossing a threshold, because it is not mirror symmetric and it is also not complete.

Properly voters share a common mirror symmetric distinction between what they are for and what they are against. The objective should be to gather the views of all voters for and against and clear them. However, unlike in simple counting where letters start out sorted in ordered pairs of like kinds, in voting they start out mixed up. This is because people need not simply to be able to vote for a candidate, but also against. This is important, because it safeguards against electing a candidate that, while the most popular, is disliked by a majority of voters. Furthermore, many voters may need to communicate their likes and dislikes through active and passive abstention, because the options presented to them do not match up with their perspectives. Chiralkine can accommodate these different perspectives and through an adjustment of candidates after a first vote, can provide a mechanism whereby voters can reach a consensus.

One way to understand chiralkine is to think of potential energy stored in a battery. The potential energy (potential difference) depends upon a distinction (01|10): it is a relationship. The distinction is cleared (to 11|00) as the battery does work, for example lighting a bulb. Note that the lower and upper case letters in a chiralkine cycle do not each map to one side of a potential difference, but instead the distinction between upper and lower case maps to a distinction inherent in potential difference.

A simple circuit can be constructed from a battery, bulb and switch. First a link is made between say one terminal of the battery and the bulb, such that each has one link. Then a link is made between the other terminal of the battery and an open switch, so now the battery has two links and the switch has one. Next the bulb is linked to the open switch, so now both the bulb and battery have two links, but there are two chains terminating at the open switch. When the switch is closed, a third delocalised link is created as a circuit is formed. Each increase in the number of links of a component corresponds with one rotation for that component. When the circuit is formed, the rotations are synchronized and all of the distinctions (01|10) are cleared (00|11).

Another way to develop intuition of the system is to imagine the ring as a Mobius strip. A Mobius strip has two local sides and one global side. Imagine further that the Mobius strip consists of an outer ring and an inner ring. Both rings are of formula aBcAbC. The two outer local sides have their letters opposed: aA, Bb, cC, Aa, bB and Cc. The two local sides of the inner ring have their letters aligned aa, BB, cc, AA, bb, CC. So the local sides of the inner and outer rings form two ordered pairs of letters in rings, one set aA, Bb, cC, Aa, bB and Cc and the other set aa, BB, cc, AA, bb, CC. Now imagine the inner ring rotating with respect to the outer ring, mixing the letters in the pairs. This effects the local clearing and drawing of distinctions while conserving the global quantity of distinctions.

Conclusion

The principle of balance is inherently incomplete because equating quantities does not conserve the relational symmetry through which the distinctions defining those quantities are drawn.

Chiralkine is a cyclic system constructed logically in ordered 1s and 0s for performing computation on the principle of bilateral symmetry instead of balance. Counting performed on the principle of balance is incomplete and so can be used to privilege one side of a relationship over the other. This is taking place across the systems of computation in use today – from the asymmetric way in which we treat credit and debt in finance and for and against in voting to the asymmetric way in which we treat the two sides of our relationship with the planet. Chiralkine treats the two sides of a relationship mirror symmetrically, so neither side can be privileged over the other.

The Chiralkine Project

The chiralkine project was conceived – and has always been intended – as a social enterprise: a tool to make people’s lives better by restoring relational meaning to the systems we rely on.

Contact: Martin A. Hay
📧 martin@chiralkine.com 👉 Explore the rest of the site to see how chiralkine logic applies to physics, economics, computing, and more.

A New Principle for Performing Computation