This repo contains a proof-of-concept puzzle on integral octonions, and some scripts to explore octonions. This is a joint work with Roice Nelson. This puzzle is a proof of concept of non-associative puzzles. It was created so that we can have a taste of the difficulty level and non-associative solving experience. We are working on improving it and making it more intuitive. We are also working on other non-associative puzzles. An introduction can be found here.
How to run the integral octonions puzzle?
- Install Python, 2.* or 3.*.
- Install numpy.
- Run puzzle_python2.py or puzzle_python3.py depending on the Python version you installed.
- Follow the instructions to solve it. You may want to refer to the explanations below.
- Octonions are a number system extended from complex numbers and quaternions.
- Each octonion can be thought of as a vector of 8 real numbers, or a sum of 8 basis vectors.
- Addition, subtraction, multiplication, inverse, norm, conjugate can be defined similar to complex numbers, with a more complicated multiplication table.
- Octonion multiplication is not commutative (a * b != b * a) and not associative ((a * b) * c != a * (b * c)) in general.
They are octonions whose coordinates are all integers or all half-integers satisfying certain properties.
The minimum nonzero norm of integral octonions is 1. There are 240 integral octonions with norm = 1.
The identity octonion, (1, 0, 0, 0, 0, 0, 0, 0), is a unit norm integral octonion.
The set of 240 unit norm integral octonions is closed under octonion multiplication. That is, the product of two numbers in this set stays in the set.
If we start from three generators
- i = (0, 1, 0, 0, 0, 0, 0, 0),
- j = (0, 0, 1, 0, 0, 0, 0, 0),
- h = (0, 1, 1, 1, 1, 0, 0, 0)/2,
and use the octonion multiplication *, we can generate all 240 integral octonions with unit norm.
Summary of the Rubik's Cube, as an analogy:
- All the states of the Rubik's Cube are generated from generators U, D, L, R, F, B, and the group operation of concatenating moves.
- We draw a random one from all the states as the initial state or scramble.
- In each step
- We input any generator individually, or in general, any sequence of generators as an expression.
- (The new state) = (the old state) * (input).
- We win when the state is the identity or "solved" state.
About this integral octonions puzzle:
- All the 240 unit norm integral octonions are generated by i, j, and h.
- We pick a random unit norm integral octonion as the initial state or scramble.
- In each step
- We input an expression of i, j, h, and the octonion multiplication *, with parenthesis (). For example, h * (i * j).
- The input expression is evaluated. By construction, it is also a unit norm integral octonion.
- (The new state) = (the old state) * (input expression).
- We win when the state is the identity octonion (1, 0, 0, 0, 0, 0, 0, 0).
Why input expression rather than individual generators? Non-associativity:
- Since the multiplication is not associative: state * (a * b) != (state * a) * b, applying the expression (a * b) as input is different from applying a and then applying b.
- If you only input i or j or h individually and do not use multiplication in the input expression, it is unlikely (with 20% chance) you will be able to solve the puzzle.
- Therefore, we highly recommend trying at least (i * h) if you are stuck.